Science, Faculty of
Mathematics, Department of
DSpace
UBCV
Chernobai, Mikhail
Doctor of Philosophy - PhD
University of British Columbia
This thesis contains two separate parts. The first part is for the three dimensional incompressible Navier-Stokes equations where we obtain a global existence result and eventual regularity for local energy weak solutions with a large class of initial data that allows growth at spatial infinity. Work is motivated by results of Bradshaw, Kukavica and Tsai, where was proven global existence of solutions for initial data in the Herz-type spaces which involve weighted integrals centered at the origin. Our results bridge theexistence theorems of Lemarié-Rieusset in the uniformly local L² space and of Bradshaw, Kukavica and Tsai. The second part of the thesis is about weak solvability and properties of weak solutions to the Dirichlet problem for a scalar elliptic equation Δu +b(𝝰) ·∇u = f in a bounded domain Ω ⊂ ℝⁿ, n = 2, 3, containing the origin. We focus on the case of critical drift b(𝝰) := b − 𝝰x/|x|² in case n = 2 and b(𝝰) := b − 𝝰x′/x′|² , where x′ = (x1, x2, 0) for the three dimensional case, with div b = 0. The parameter α can be positive or negative.
On problems of regularity andexistence for critical drift ellipticequations and Navier-Stokes equationsbyMikhail ChernobaiSpecialist, Saint Petersburg State University, 2019A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)July 2024© Mikhail Chernobai 2024The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:On problems of regularity and existence for critical drift ellipticequations and Navier-Stokes equationssubmitted byMikhail Chernobai in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Mathematics.Examining Committee:Tai-Peng Tsai, Professor, Mathematics, UBCSupervisorSven Bachmann, Associate Professor, Mathematics, UBCSupervisory Committee MemberStephen Gustafson, Professor, Mathematics, UBCSupervisory Committee MemberMatthew W Choptuik, Professor, Physics and Astronomy, UBCUniversity ExaminerJingyi Chen, Professor, Mathematics, UBCUniversity ExaminerDoyoon Kim, Professor, Mathematics, Korea UniversityExternal ExamineriiAbstractThis thesis contains two separate parts. The first part is for the three di-mensional incompressible Navier-Stokes equations where we obtain a globalexistence result and eventual regularity for local energy weak solutions witha large class of initial data that allows growth at spatial infinity. Work ismotivated by results of Bradshaw, Kukavica and Tsai, where was provenglobal existence of solutions for initial data in the Herz-type spaces whichinvolve weighted integrals centered at the origin. Our results bridge theexistence theorems of Lemarie´-Rieusset in the uniformly local L2 space andof Bradshaw, Kukavica and Tsai.The second part of the thesis is about weak solvability and properties ofweak solutions to the Dirichlet problem for a scalar elliptic equation ∆u +b(α) · ∇u = f in a bounded domain Ω ⊂ Rn, n = 2, 3, containing the origin.We focus on the case of critical drift b(α) := b − αx|x|2 in case n = 2 andb(α) := b− αx′|x′|2 , where x′ = (x1, x2, 0) for the three dimensional case, withdiv b = 0. The parameter α can be positive or negative.iiiLay SummaryIn this thesis we study global existence of local energy solutions for NavierStokes equations, which describe a motion of incompressible fluid. Thenovelty of our result is the new class of initial data for which we construct oursolutions. We also research existence and regularity of solutions to ellipticequations with critical drift, in this thesis we proved existence, uniquenessand higher regularity of such solutions.ivPrefaceThis thesis contains several pre-prints and a published work. chapter 2 is a joint work with Prof. Tai-Peng Tsai and Prof. ZacharyBradshaw. My contribution was proving and writing down local energyand pressure estimates, as well as global existence of solutions,someproperties of intermediate spaces and proofreading other sections. Pro-fessors Tsai and Bradshaw worked on local regularity result as wellas approximation lemma and properties of intermediate spaces, lo-cal pressure estimates, introduction section and proofreading the finalversion. chapter 3 is my solo work on a problem suggested by Prof. Tai-PengTsai. chapter 4 and chapter 5 are joint works with Prof. Timofei Shilkin,my contribution was proving and writing down a-prioriW 1,p estimatesand existence of p-weak solutions in both cases of positive and negativedivergence of the drift function, while Prof. Shilkin worked on Ho¨ldercontinuity and boundness of the solutions, introduction section andproofreading. chapter 4 is published online on 29 Sep 2020, Scalarelliptic equations with a singular drift, Complex Variables and EllipticEquations,https://doi.org/10.1080/17476933.2020.1816980.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . ixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Navier-Stokes flows . . . . . . . . . . . . . . . . . . . . . . . 11.2 Elliptic equations with critical drift . . . . . . . . . . . . . . 92 Global Navier-Stokes flows in intermediate spaces . . . . . 172.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Eventual regularity . . . . . . . . . . . . . . . . . . . . . . . 232.3 The intermediate spaces . . . . . . . . . . . . . . . . . . . . . 252.4 Pressure estimate . . . . . . . . . . . . . . . . . . . . . . . . 402.5 A priori bounds . . . . . . . . . . . . . . . . . . . . . . . . . 442.6 Local and global existence of weak solutions . . . . . . . . . 472.7 Eventual regularity . . . . . . . . . . . . . . . . . . . . . . . 513 Global Navier-Stokes flows in weighted spaces . . . . . . . 563.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2 Pressure estimate . . . . . . . . . . . . . . . . . . . . . . . . 613.3 Local Energy Inequality . . . . . . . . . . . . . . . . . . . . . 633.4 Global Existence . . . . . . . . . . . . . . . . . . . . . . . . . 683.5 Eventual regularity . . . . . . . . . . . . . . . . . . . . . . . 73viTable of Contents4 Scalar elliptic equations with a singular drift . . . . . . . . 754.1 Introduction and Main Results . . . . . . . . . . . . . . . . . 754.2 Higher Integrability Results . . . . . . . . . . . . . . . . . . . 794.3 Ho¨lder Continuity of p–Weak Solutions . . . . . . . . . . . . 854.4 Proof of Main Results . . . . . . . . . . . . . . . . . . . . . . 965 Elliptic equations with a singular drift from a weak Morreyspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.1 Introduction and Main Results . . . . . . . . . . . . . . . . . 1035.2 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . 1115.3 Higher integrability of weak solutions . . . . . . . . . . . . . 1135.4 Modified De Giorgi classes . . . . . . . . . . . . . . . . . . . 1195.5 Ho¨lder continuity of weak solutions . . . . . . . . . . . . . . 1296 Concluding Chapter . . . . . . . . . . . . . . . . . . . . . . . . 131Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133AppendicesA Imbedding Theorem . . . . . . . . . . . . . . . . . . . . . . . . 141B Approximation Proposition . . . . . . . . . . . . . . . . . . . 142viiList of Figures2.1 The (α, λ) parameter region for global existence: For (α, λ)in the left part we need to assume (2.1.10). For (α, λ) in theright part we don’t. . . . . . . . . . . . . . . . . . . . . . . . . 212.2 The spaces F˚2α,λ when αλ = 1/3, which is fixed for illustra-tive purposes. Note that M˚2,αλ = F˚21,αλ by Lemma 2.3.9.These spaces imply eventual regularity in regions with thesame algebraic bounds, up to constants. Spaces are strictlyincreasing in λ. The conclusion of Theorem 2.2.1 is new inall but the points (α = 1, λ = 1/3). . . . . . . . . . . . . . . . 24viiiAcknowledgementsI would like to acknowledge and give my warmest thanks to my supervisorTai-Peng Tsai, as well as Timofei Shilkin who made this work possible, andwere guiding me throughout these years.ixDedicationThank you to everyone who has supported me throughout this journey.xChapter 1IntroductionThis thesis contains of two parts. Part one is about global existence andeventual regularity of local energy and is a subject of chapter 2, chapter 3.Part two contains results regarding existence and higher integrability forsolution of elliptic equations with critical drifts and is a subject of chapter 4,chapter 5.1.1 Navier-Stokes flowsNavier-Stokes equations describe the evolution of velocity and pressure (u, p)of a fluid, solving the following system in the sense of distributions:{∂tu−∆u+ (u · ∇)u+∇p = 0,div u = 0in R3 × (0, T ) (1.1.1)for T > 0. Navier–Stokes existence and smoothness problem is one of wellknown problems in mathematics and is one of the Millennium Prize Prob-lems selected by the Clay Mathematics Institute. There are several typesof solutions for Navier-Stokes equation and the main difficulty is to eitherprove that a smooth solution exist globally or to prove regularity of weaksolution which is known to exist. An alternative approach to the problemis constructing a blow up, or unbounded solution using computer modelingand approximations. Historically first approach to Navier-Stokes equationswas to construct a weak solution, which is not smooth but exists globally,has minimal regularity and includes sufficiently large class of initial data.Initially J.Leray constructed weak solutions in [21] for any divergence freeinitial data u0 ∈ L2(R3). Later in [18] Lemarie´-Rieusset proved the similarresult for the local initial data u0 ∈ L2uloc. Where the uniform local space isa larger space than standard L2 space and is defined as following:∥u0∥L2uloc = supx∈R3∥u0∥L2(B(x,1)) <∞. (1.1.2)These results are concerned with the problem of finding a broad space ofinitial data, for which there exist global solution. The goal of my thesis11.1. Navier-Stokes flowsis to prove local and global existence of local energy solutions with a largeclass of initial data that might not decay on spatial infinity. First, we willgive a definition of local energy solutions, this is a class of solutions toNavier-Stokes equations that have sufficient regularity and most commonproperties that we want from solutions to satisfy. For any point x0 ∈ R3and radius r > 0 we denote cube Q = Q(x0, r) = {x ∈ R3| |xi − xi,0| ≤ r}as well as Q∗ = Q(x0, 5r/4) a slightly bigger cube with the same center andQ∗∗ = Q(x0, 4r/3) is similarly a small extension of Q∗.Definition 1.1.1. A vector field u ∈ L2loc(R3 × [0, T )), where T > 0 is alocal energy solution to (1.1.1) with initial data u0 ∈ L2loc(R3) if the followingholds: u ∈ ∩R>0L∞(0, T ;L2(BR(0))), ∇u ∈ L2loc(R3 × [0, T )), there is p ∈ L32loc(R3 × [0, T )) such that u, p is NSE solution in a senseof distributions, For all compacts K ⊂ R3 we have u(t)→t→0 u0 in L2(K), u satisfies local energy inequality: for all ϕ ∈ C∞0 (R3× (0, T )), ϕ ≥ 0,2∫ ∫|∇u|2ϕ dx dt ≤∫ ∫|u|2(∂tϕ+∆ϕ) dx dt+∫ ∫(|u|2 + 2p)(u · ∇ϕ) dx dt,(1.1.3) the function t 7→ ∫ u(x, t)w(x) dx is continuous on [0, T ) for any w ∈L2(R3) with a compact support, for every cube Q ⊂ R3, there exists pQ(t) ∈ L 32 (0, T ) such that forx ∈ Q∗ and 0 < t < T ,p(x, t)− pQ(t) = −13|u(x)|2+p.v.∫y∈Q∗∗Kij(x− y)(ui(y, s)uj(y, s)) dy+∫y/∈Q∗∗(Kij(x− y)−Kij(xQ − y))(ui(y, s)uj(y, s)) dy,(1.1.4)in L3/2(Q∗×(0, T )), here xQ is the center of Q and Kij = ∂i∂j(4π|y|)−1.21.1. Navier-Stokes flowsHere the class u ∈ L∞(0, T ;L2(R3)), ∇u ∈ L2(R3× [0, T )) is a standardchoice for the weak solution, since it allows the equation to be properlydefined, other properties are very similar to Leray solutions and are com-monly used in theory of Navier-Stokes equations, with the exception of lastproperty which is the pressure decomposition formula. It is not always clearhow pressure is related to the velocity field and this representation providesexact local formula for the pressure depending on velocity field v up to timedependent function pQ(t). This definition was used in [4] and differs fromprevious definitions in [7, 14, 15, 18, 26] since we don’t require u0 ∈ L2uloc.This is important since we need a class of solutions that is defined for ini-tial data outside of uniform local space, otherwise the theory is covered byresults of Lemarie´-Rieusset [18]. Global-in-time local energy solutions of(1.1.1) for initial data which do not decay fast at spatial infinity was firstconstructed in local uniform L2 spaces [15, 16, 18, 24]. The condition ofdecay is crucial and without it we can only prove local in time existenceof solutions. It was then extended to data whose norms at a sequence ofscales grow under certain rate (Herz spaces) [4, 5, 7, 11, 25], or is boundedin certain weighted spaces [9, 13, 19]. To summarize, up to this point theglobal existence of local energy solutions was proven either in certain Herzspaces or in uniform local Lq spaces and the goal of this thesis is to closethe gap between these two results and find intermediate spaces that stillallow to prove global existence of local energy solutions. For this purpose,let us reformulate the definitions of the local uniform Lq space and the Herzspaces and later introduce the intermediate spaces that are the main focusof our work. For q ≥ 1 introduce the definition of local uniform Lq space asLquloc ={f ∈ Lqloc(R3); supx∈R3∫B1(x)|f |q <∞}. (1.1.5)All global existence results required some decay on infinity, so we denote thelocal uniform space with decay as following:Eq ={f ∈ Lqloc(R3); lim|x|→∞∫B1(x)|f |q = 0}. (1.1.6)Next we define Herz spaces considered in [4] for λ ≥ 0,M q,λ ={f ∈ Lqloc(R3),∫B1(0)|f |q + sup|x|>11|x|λ∫B(x,|x|/2)|f |q <∞}.(1.1.7)31.1. Navier-Stokes flowsSimilarly to uniform local spaces, we define a subset of Herz spaces thatvanishes on infinityM˚ q,λ ={f ∈ Lqloc(R3), lim|x|→∞1|x|λ∫B(x,|x|/2)|f |q = 0}. (1.1.8)Previously global existence was shown in M˚2,λ for 0 ≤ λ ≤ 2 in [4], theresult contains only case λ = 2 but the same argument can be applied to0 < λ < 2. Next, we introduce the intermediate spaces between E2 andM2,λDefinition 1.1.1. Let 1 ≤ p <∞, 0 ≤ α ≤ 1 and 0 ≤ λ <∞. Define F pα,λto be the space of f ∈ Lploc(R3) with finite norm ∥f∥F pα,λ(R) where for anyR ≥ 1,∥f∥F pα,λ(R) = supx∈R3(1rλx∫B(x,rx)|f |p)1/p, rx =18max{R, |x|}α. (1.1.9)We denote by F pα,λ,σ the space of divergence-free vector fields in R3 with allcomponents in F pα,λ. Furthermore, we denote by F˚pα,λ and F˚pα,λ,σ the closureof C∞c (R3) and C∞c,σ(R3) = {ζ ∈ C∞c (R3;R3),div ζ = 0} in F pα,λ-norm,respectively.This is a logical choice of intermediate spaces, since for α = 0, rx = 1/8and for any λ ≥ 0,F 20,λ = L2uloc, F˚20,λ = E2,and when α = 1, rx =18 max(R, |x|),F 21,λ =M2,λ, F˚ 21,λ = M˚2,λ.Moreover for 0 < α < 1 there are examples of functions in F 2α,λ that arenot part of either E2 and M2,λ, therefore spaces F 2α,λ serve as connectingbridge between E2 and M2,λ and a new class of solutions. In our definitionclass F˚ pα,λ represents condition of spatial decay, since we can approximatethe function with a sequence of compactly supported functions. In cases ofE2 and M2,λ it is equivalent condition to the decay properties we definedearlier, but in intermediate spaces it is not automatic without additionalassumptions, therefore we used such definition of F˚ pα,λ as our initial data forglobal existence.41.1. Navier-Stokes flowsBesides spaces F 2α,λ, we will also consider another weighted spaces, forthat let us define the following set of cubes. For n ∈ N, let S0 = {x : |xi| ≤2; i = 1, 2, 3} and Rn = {x : |xi| ≤ 2n; i = 1, 2, 3}. Denote Sn = R¯n+1 \Rnfor n ∈ N \ {1} and S1 = R¯2 \ S0. Then |Sn| = 56 · 23n. We separate S0 =∪641 Qk00 into 64 cubes with side-length 1 and Sn = ∪56kn=1Qknn into 56 cubeswith side-length 2n, the set of all these cubes we denote as C = ∪∞n=0∪knQknn .This set will satisfy the following properties Side-length of each cube is proportional to the distance to the origin. Adjacent cubes have proportional volume. The distance between cubes Q,Q′ is proportional to max{|Q| 13 , |Q′| 13 }. The amount of cubes Q′, such that |Q′| < |Q| is bounded above by|Q| 13Using the set of cubes C let us introduce the following spaces:Definition 1.1.2. Let q ≤ 2 and r > 2, we define spaces F qr :F qr ={f ∈ L2loc(R3)∣∣∣ ∑n,kn( 1|Qknn | q3∫Qknn|f |2 dx) r2<∞}(1.1.10)Where Qknn are cubes in set C, the norm is defined as following∥f∥rF qr =∞∑n,kn( 1|Qknn | q3∫Qknn|f |2 dx) r2Notices that unlike spaces M2,λ and F 2α,λ where we take supremum overthe balls in R3 in spaces F qr we take a sum of integrals not the supremum.The following are our main theorems on the local and global existence oflocal energy weak solutions in F qr and F 2α,λ.Theorem 1.1.1. Let q ≤ 2, 2 < r <∞ and assume u0 ∈ F qr is divergencefree. Then there exists u : R3×(0,∞)→ R3 and p : R3×(0,∞)→ R so that(u, p) is a local energy solution to the Navier-Stokes equations on R3×(0,∞)with initial data u0.Theorem 1.1.2. Assume0 ≤ α ≤ 1, 0 ≤ λ ≤ 2. (1.1.11)51.1. Navier-Stokes flowsFor any divergence-free initial data u0 ∈ F˚ 2α,λ, there exist T+ > 0 (withan explicit lower bound) and a local energy solution (u, p) to the Navier-Stokes equation (1.1.1) with initial data u0 for t ∈ (0, T+) such that u(t) ∈C(0, T ;F 2α,λ(R)) for any 0 < T < T+ and 1 < R <∞.If we further assumelimρ→∞ ρα(λ−2)∥u0∥2F 2α,λ(ρ) = 0 (1.1.12)when (3 − λ)α < 1, (no condition when (3 − λ)α ≥ 1), then we can takeT+ =∞, the solution is globally defined and satisfies the a priori boundess sup0<s<Tsupx∈R31|B(x, rx)|λ/3∫B(x,rx)|u(s)|2+supx∈R31|B(x, rx)|λ/3∫ T0∫B(x,rx)|∇u|2 ≤ C0∥u0∥2F 2α,λ(R).(1.1.13)For R = Rn → ∞, T = Tn → ∞, here rx is defined the same way as in(1.1.9)Comments on Theorem 1.1.2:1. Conditions (1.1.11) are imposed since, for α, λ outside of that rangethe spaces F 2α,λ are trivial or do not present any scientific interest.Cases α = 0, α = 1 present uniform local space E2 and M2,λ, whilein between we have intermediate spaces F 2α,λ.2. Notice that property (1.1.12) is automatically satisfied for (3−λ)α > 1but otherwise we needed it as additional assumption on the initial data.3. Inequality (1.1.13) provides control of the solution by initial data inspaces F 2α,λ4. The proof of global existence works uniformly for 0 < α ≤ 1, 0 ≤ λ ≤2, which does not cover case α = 0, when F 20,λ = L2uloc. Note thatglobal existence when α = 0 was shown by [18] for initial data in E2by iterative extension, which can not be adapted to our method. Onthe other hand in case of λ = 0 space F 2α,0 is a strict subset of L2uloc,so solutions in F 2α,0 have more control.The proof relies on use of Gronwall lemma and constructing approximationsequence that satisfies a priori bound similar to (1.1.13). With this result weconnected the previously known results of global existence for initial data61.1. Navier-Stokes flowsin local unform spaces and Herz spaces. We had to impose additional decaycondition (1.1.12), since it is not clear how to prove that our space F˚ 2α,λautomatically satisfy it for all α, λ.Another topic that is covered in this thesis is eventual regularity of so-lutions from F 2α,λ. It is classical that solutions in the Leray class eventuallyregularize in the sense that there exists a time T so that the solution in viewis smooth on R3 × (T,∞). The weighted local spaces considered in [4, 5, 7]also have results about eventual regularity of local energy solutions. Usuallyeventual regularity is a consequence of uniform in time a priori bound, forexample with Leray solutions it is the following estimate for all T > 0,∫ T0∫|∇u|2 dx dt ≤ ∥u0∥22. (1.1.14)With help of interpolation and Sobolev inequalities we can conclude that thesolution has small L3 norm at some large time, and the small-data globalwell-posedness theory in L3 and weak-strong uniqueness together guaranteethe solution becomes smooth for all large times. On the other hand, eventualregularity is not known for local energy solutions. This is due to the lackof an estimate that extends to arbitrarily large times. Indeed, Lemarie´-Rieusset’s solutions, which have data in L2uloc have only a short time estimate[18, 20]. The smaller M2,λ and F˚ 2α,λ spaces, on the other hand, allow foran estimate in Theorem 1.1.2 which extends up to arbitrarily large timesfor certain choices of parameters. These a priori bounds can be used as asubstitute for (1.1.14) in proving eventual regularity, as has been exploredin [4, 5, 7]. In [4] eventual regularity is proven above a paraboloid for datain M˚2,1(R3). In [5] corresponding estimates are carried out in M˚2,λ(R3+)where λ ≤ 1. In this case, the paraboloid region can be replaced by alarger algebraic region with exponents depending on λ. As was known tothe authors of [5], the theorem in [5] can be improved in M˚2,λ(R3) yieldingan algebraic region of the form{(x, t) : |x|2λ ≲ t and t > M},for a non-universal constant M = M(u0). Note that the classical eventualregularity result for Leray weak solutions would correspond to q = 0.It is natural to ask what happens if we seek eventual regularity withinthe finer F 2α,λ scale of spaces, noting thatM2,λ corresponds to F 21,λ. For thiswe need to introduce new sub-classes of our F 2α,λ spaces. LetF2α,λ := {f ∈ F 2α,λ : supR≥1∥f∥F 2α,λ(R) <∞}, (1.1.15)71.1. Navier-Stokes flowsandF˚2α,λ := {f ∈ F2α,λ : limR→∞∥f∥F 2α,λ(R) = 0}. (1.1.16)The latter is necessarily a subspace of M˚2,λ, which can be easily seen usingan alternative definition of the M2,λ norm given in [4].Theorem 1.1.3 (Eventual regularity). Assume that u0 ∈ F˚2α,λ is divergencefree and α ∈ (0, 1], λ ∈ (0, 1]. Let (u, p) be a local energy solution with initialdata u0 on R3 × (0,∞), satisfying the following inequality for some R > 0and for all t <∞ess sup0<s<tsupx∈R31|B(x, rx)|λ/3∫B(x,rx)|u(s)|2+supx∈R31|B(x, rx)|λ/3∫ t0∫B(x,rx)|∇u|2 <∞.(1.1.17)Here rx =18 max{R, |x|}α. Then, there exists M = M(u0) so that u isregular within a region of the form{(x, t) : |x|2αλ ≤ t, t ≥M},where the suppressed constants depend on λ and α, whileM is determined bythe length scale ρ at which ∥u0∥F 2α,λ(ρ) becomes small compared to a universalsmall constant. Additionally, for (x, t) in the preceding region, |u(x, t)| ≲t−1/2.The condition (1.1.17) is needed to ensure that the a priori bound(1.1.13) holds. This is satisfied by the solutions constructed in Theorem1.1.2.The region of regularity here is already established for M˚2,αλ-solutions.Our result identifies larger classes which have the same regularity region,insofar as the degree of the algebraic boundary is concerned, as the M˚2,αλcase. To support this we show that if αλ = q ∈ (0, 1], thenF˚21,q = M˚2,q ⊊ F˚2α ̸=1,λ ̸=1 ⊊ F˚2q,1.The largest space in Theorem 1.1.3 which admits a regularity region with|x|2αλ ≲ t for large t is, therefore, F˚2q,1. Similarly we have a result for spacesF qr81.2. Elliptic equations with critical driftTheorem 1.1.4. (Eventual Regularity) Assume q < 1 and u0 ∈ F qr isdivergence-free, and let (u, p) be a local energy solution on R3 × (0,∞) withinitial data u0. Assume additionally that:∥u(·, t)∥rF qr +∞∑m=1( 1|Qm|q3∫ t0∫Qm|∇u|2 dx dt) r2<∞, (1.1.18)for all t <∞. Then for any δ > 0, there exists a time τ depending on u0, δso that u is smooth on{(x, t), t > max{δ|x|2, τ}} (1.1.19)To summarise our result connects previously known global existence re-sults for solutions with initial data from local uniform Lquloc spaces and Herzspaces, as well as provides eventual regularity of local energy solutions. Theproofs of Theorems 1.1.2, 1.1.3 will be given in chapter 2, while Theorems1.1.1,1.1.4 are in chapter 3.1.2 Elliptic equations with critical driftMy second research topic in this thesis is elliptic equations with critical drift.It is widely researched area in partial differential equations, since equationswith drift occur in a lot of areas of mathematics. Assume Ω ⊂ Rn is abounded Lipschitz domain, n ≥ 2. We consider the following boundaryvalue problem: {−∆u+ b · ∇u = −div f in Ω,u|∂Ω = 0.(1.2.1)Here u : Ω→ R is unknown, b : Ω→ Rn and f : Ω→ Rn are given functions.The motivation for this project came from Navier-Stokes equations, sinceunder assumption of axially symmetry some components of the solutionsatisfy an equation with critical drift. Our final goal is to prove results forparabolic case, but due to technical difficulties we started with elliptic, timeindependent case. With this motivation we are particularly interested in thedrift of typen = 2, b(x) = −α x|x|2 , α ∈ R. (1.2.2)n = 3, b(x) = −α x′|x′|2 , x′ = (x1, x2, 0), α ∈ R. (1.2.3)91.2. Elliptic equations with critical driftWe will first focus on three dimensional case, to include these drifts weassume that the drift b satisfies the conditionb ∈ L2,n−2w (Ω), (1.2.4)where Lp,λw (Ω) is the weak Morrey space equipped with the quasi-norm∥b∥Lp,λw (Ω):= supx0∈ΩsupR<diamΩR−λp ∥b∥Lp,w(BR(x0)∩Ω),and Lp,w(Ω) is the weak Lebesgue space equipped with the quasi-norm∥b∥Lp,w(Ω) := sups>0s |{ x ∈ Ω : |b(x)| > s }| 1p , (1.2.5)where |Ω| denotes the n-dimensional Lebesgue measure of Ω. This choicespace for drift is logical, since it includes certain drifts that we encounter inNavier-Stokes theory but also have not been covered by other results so far.Note also that both scales of spaces Lp,n−p(Ω) and Lp,n−pw (Ω) are criticalones, i.e. these spaces are invariant under the scalingbλ(x) = λ b(λx), λ > 0 =⇒ ∥bλ∥Lp,n−pw (Ωλ) = ∥b∥Lp,n−pw (Ω), (1.2.6)where we assume Ω is star-shaped and denote Ωλ := {x ∈ Rn : x/λ ∈ Ω }.This scaling is important, since Navier Stokes equations are invariant underit, so we need our drift to satisfy the same property. For more properties ofweak Morrey spaces we refer to [69], [84]. Next we define the bilinear formB[u, η] corresponding to the drift term in equation (1.2.1) byB[u, η] :=∫Ωη b · ∇u dx. (1.2.7)Note that for b satisfying (1.2.4) the bilinear form (1.2.7) generally speakingis not well-defined for u ∈W 12 (Ω) and η ∈ C∞0 (Ω) whereW 1p (Ω) := {u ∈ Lp(Ω) : ∇u ∈ Lp(Ω) }, p ∈ [1,+∞],is the usual Sobolev space and C∞0 (Ω) is the linear space of C∞–smoothfunctions compactly supported in Ω. Therefore we can not take the usualW 12 (Ω) class for our solution. Nevertheless, the bilinear form B[u, η] is well-defined at least for u ∈W 1p (Ω) with p > 2 and η ∈ Lq(Ω) with q > 2pp−2 . So,instead of the standard notion of weak solutions from the energy classW 12 (Ω)we introduce the definition of p-weak solutions to the problem (1.2.1), seealso [38], [40], [81]:101.2. Elliptic equations with critical driftDefinition 1.2.1. Assume p > 2, b ∈ Lp′(Ω), p′ = pp−1 , and f ∈ L1(Ω).We say u is a p-weak solution to the problem (1.2.1) if u ∈◦W 1p(Ω) and usatisfies the identity∫Ω∇u · ∇η dx + B[u, η] =∫Ωf · ∇η dx, ∀ η ∈ C∞0 (Ω). (1.2.8)In three dimensional result we always assume that the vector field b hasa sign-defined divergence:div b ≤ 0 in D′(Ω), (1.2.9)where D′(Ω) is the space of distributions on Ω. In the case (1.2.3) thecondition (1.2.9) corresponds to α ≥ 0. The important impact of the con-dition (1.2.9) is due to the sign of the quadratic form B[u, u] provides. Itis well-known that in the case of a non-regular drift b the violation of thecondition (1.2.9) potentially can ruin the uniqueness for the problem (1.2.1)even in the class of smooth solutions. For example, in the case Ω = B whereB := {x ∈ Rn : |x| < 1 } the functionsu(x) = c(|x|2 − 1), b(x) = n x|x|2 , b ∈ Ln,w(B), div b ≥ 0 in D′(B)(1.2.10)satisfy to (1.2.1) with f ≡ 0. Note also that in the regular case b ∈ Ln(Ω)the uniqueness for the problem (1.2.1) holds regardless of the sign of div bdue to the maximum principle, see, for example, [33] and reference there.Together with (1.2.4) and (1.2.9) we will assume that b also possess thefollowing approximation property:∃ {bk}∞k=1 ⊂ C∞(Ω¯) : div bk ≤ 0 in Ω,∥bk∥L2,n−2w (Ω) ≤ c ∥b∥L2,n−2w (Ω), ∥bk − b∥L1(Ω) → 0.(1.2.11)This approximation property is quite important for existence of solutions,and is automatically satisfied as long the domain Ω satisfies certain prop-erties, which is an interesting research topic by itself. The main difficultyfor general domain is preserving condition (1.2.9) in (1.2.11). We includedone case where this construction is proven, if Ω is a star-shaped domain theapproximation can be obtained easily by the dilatation and mollification.The main results of this part of the thesis are the following two theoremsfor three dimensional case:111.2. Elliptic equations with critical driftTheorem 1.2.1. Assume b satisfies (1.2.4), (1.2.9), (1.2.11). Then thereexist p > 2 depending only on n, Ω and ∥b∥L2,n−2w (Ω)such that for anyf ∈ Lp(Ω) there exists a unique p-weak solution u ∈ W 1p (Ω) to the problem(1.2.1). Moreover, this solution satisfies the estimate∥u∥W 1p (Ω) ≤ c ∥f∥Lp(Ω), (1.2.12)with a constant c > 0 depending only on Ω, n and ∥b∥L2,n−2w (Ω).Theorem 1.2.2. Assume b satisfies (1.2.4), (1.2.9) and p > 2. Then for anyq > n there exists µ ∈ (0, 1) depending only on n, p, q, Ω and ∥b∥L2,n−2w (Ω)such that if u is a p–weak solution to the problem (1.2.1) corresponding tothe right-hand side f ∈ Lq(Ω) then u is Ho¨lder continuous on Ω¯ with theexponent µ and the estimate∥u∥Cµ(Ω¯) ≤ c ∥f∥Lq(Ω), (1.2.13)holds with the constant c > 0 depending only on q, p, n, Ω and ∥b∥L2,n−2w (Ω).As for two dimensional case, we use drift from weak L2 spaceb(α) = b− α x|x|2 , b ∈ L2,w(Ω), div b = 0. (1.2.14)Where L2,w is defined the same way as in (1.2.5) and α ∈ R. Similarly to(1.2.7) we define the bilinear form Bα[u, η] byBα[u, η] :=∫Ωη b(α) · ∇u dx, ∀η ∈ C∞0 (Ω). (1.2.15)Notice, that for two dimensional case we did not assume sign of α, this iscrucial because sign of α corresponds to the sign of div(b(α)), and as wementioned earlier it can cause difficulties with uniqueness of the solution.The main two results for two dimensional case are the following theorems:Theorem 1.2.3. Assume α ≥ 0, b satisfies (1.2.14) and q > 2. Then thereexists p > 2 depending only on q, Ω, α and ∥b∥L2,w(Ω) such that for anyf ∈ Lq(Ω) there exists a unique p-weak solution u to the problem (1.2.1).Moreover, this solution satisfies the estimate∥u∥W 1p (Ω) ≤ C ∥f∥Lq(Ω), (1.2.16)with the constant C depending only on q, p, α, Ω and ∥b∥L2,w(Ω).121.2. Elliptic equations with critical driftTheorem 1.2.4. Assume α < 0, b satisfies (1.2.14) and q > 2. Then thereexists p > 2 depending only on q, Ω, α and ∥b∥L2,w(Ω) such that for anyf ∈ Lq(Ω) there exists a unique p-weak solution u to the problem (1.2.1) sat-isfying the condition u(0) = 0. Moreover, this solution satisfies the estimate(1.2.16) with the constant C depending only on q, p, α, Ω and ∥b∥L2,w(Ω).Remark 1.2.5. In the case α ≥ 0 our method allows us to prove similarly tothree dimensional case that there exists p > 2 such that for any f ∈ Lp thereexist p-weak solution, but for case α < 0 we prove higher integrability resultbased on Ho¨lder continuity of the solutions and therefore we need right handside f ∈ Lq, q > 2. This approach does not give us precise control over thehigher integrability of the solution, therefore for readers convenience andsimplicity we formulated two dimensional results for f ∈ Lq, q > 2.The study of the weak solvability and properties of weak solutions todrift-diffusion equations has a long history. Below we focus on the briefoverview of results concerning the existence of weak solutions, their unique-ness, boundedness and Ho¨lder continuity.Very often the study of the problem (1.2.1) is coupled with the study ofthe dual problem {−∆v − div(bv) = div g in Ω,v|∂Ω = 0.(1.2.17)In the case of the divergence–free driftdiv b = 0 in D′(Ω) (1.2.18)problems (1.2.1) and (1.2.17) are equivalent, but for the general drifts inis not the case. The reason is that the dual problem (1.2.17) is presentedin the divergent form. Hence weak solutions, depending on the regularityof b, in principle are well-defined in a wider class than the energy spaceW 12 (Ω) (one can consider classes of very weak solutions, renormalized so-lutions etc.) For example, in a series of papers [61], [62], [63] (see also[64] and reference there) the existence of weak (distributional) solutions forthe problem (1.2.17) in various functional classes under various critical andsupercritical assumptions on b was studied.Another approach to the solvability of the problem (1.2.1) with the roughcoefficients was developed in a series of papers [76], [77], [78], [79] (see alsoreferences there). This approach is applicable to elliptic operators in thenon-divergent form (with the main part −ajk(x) ∂2u∂xj∂xk with a non-smooth131.2. Elliptic equations with critical driftuniformly elliptic matrix a(x) = (ajk(x)) instead of the Laplacian) and thedrift b belonging to the critical Morrey spaces. In this approach the existenceof solutions in Sobolev spaces W 2p (Ω) with some p > 1 is investigated andthe equation (1.2.1) is understood a.e. in Ω.For the problem (1.2.1) weak solutions from W 12 (Ω) formally are well-defined only for b ∈ L2(Ω) which does not cover the case (1.2.4). Neverthe-less, weak solutions to elliptic and parabolic equations with rough coefficientscommonly known to possess the higher integrability property. This meansthat the gradient of a weak solution is integrable with some exponent whichis slightly greater than 2. One can ask what are the optimal conditions onthe drift b which provide the higher integrability of weak solutions to theproblem (1.2.1)? The first results in this direction were obtained earlierin [39], [40], [81] where the critical drift b ∈ Ln,w(Ω) satisfying (1.2.9) wasstudied. So, our Theorem 1.2.1 can be viewed as a far away extension ofthese results. Note also that the 2D case was investigated separately in [65]and [82].All results concerning weak solvability and properties of weak solutionsto the problems (1.2.1) and (1.2.17) formally can be split onto three groups:the results related to the subcrtitical drifts, the results for critical driftsand supercritical results. We remind that we call some functional classX supercritical if for b ∈ X and bλ defined in (1.2.6) the norm ∥bλ∥X(B)computed over a unit ball B blows up as λ→ 0.The results concerning the drift b belonging to subcritical Lesbeguespaces are classical one, see [43]. The same is true for drifts belongingto the critical Lebesgue spaceb ∈ Ln(Ω). (1.2.19)We call drifts satisfying (1.2.19) regular. In the regular case the drift termprovides a compact perturbation of the Laplace operator and hence the prob-lem (1.2.1) possesses the Fredholm property (i.e. the existence for (1.2.1)follows from the uniqueness). Moreover, in the case (1.2.19) weak solutionsare Ho¨lder continuous and the maximum principle holds (which provides theuniqueness of weak solutions), see the survey of the results, for example, in[33]. So, we can consider the regular case as a starting point of our investiga-tion. Note that the necessary and sufficient conditions for the boundednessof the bilinear form (1.2.7) on the Sololev space W 12 (Rn) were investigatedin [45].The counterexample (1.2.10) shows that even the minimal relaxation ofthe condition (1.2.19) (still within the critical scale) can ruin the maximum141.2. Elliptic equations with critical driftprinciple and the uniqueness for the problem (1.2.1). So, to recover theuniqueness we need to impose some additional assumptions on b. For ex-ample, one can consider the condition (1.2.9) or its particular case (1.2.18).(Motivated by the properties of the quadratic form B[u, u] we call the con-dition (1.2.9) “non-spectral”.) In this thesis for three dimensional case wefocus only on the critical “non-spectral” case and for two dimensional casewe do both.For b ∈ L2(Ω) satisfying (1.2.18) the uniqueness of weak solutions inthe energy class W 12 (Ω) can be found in [58] (see also [57] for the paraboliccase). In [58] a counterexample to the uniqueness for the problem (1.2.1)with a divergence-free drift was also constructed if the condition b ∈ L2(Ω)is violated (see [67] and references there for the further developments inthis direction). For b ∈ L2(Ω) satisfying condition (1.2.9) the correspondinguniqueness result in the energy class W 12 (Ω) can be easily obtained by astraightforward adaptation of the methods of [57], [58]. For b ∈ L2,w(Ω)satisfying (1.2.9) our result provides uniqueness for the problem (1.2.1) in aclass of weak solutions which possess the higher integrability property.The next issue is the boundedness of weak solutions. It is well-knownthat even in the divergence–free case (1.2.18) the local boundedness of weaksolutions requires some extra regularity of the drift, see [47], [33], [71], [59],see also a recent survey [60] and reference there. Nevertheless, we showedthat for a drift b ∈ L2,w(Ω) satisfying (1.2.9) p–weak solutions to the bound-ary value problem (1.2.1) are always bounded without any extra assumptionson the regularity of b. We emphasize that our result is global (i.e. it requiressmooth boundary conditions) and can not be localized. The phenomena ofthe dependence of local boundedness of weak solutions on their behaviouron the boundary of the domain is discussed in [33].The last issue we are going to discuss is Ho¨lder continuity of weak so-lutions. Note that under the critical condition (1.2.19) Ho¨lder continuityis the optimal regularity. For the problems (1.2.1), (1.2.17) with a drift bbelonging to the subcritical Morrey space b ∈ L2,λ(Ω) with λ ∈ (n − 2, n)Ho¨lder continuity of weak solutions is known for a long time, see [68] and[83]. On the other hand, the counterexample in [32] shows that even in thedivergence–free case (1.2.18) the smallest violation of the critical scale canlead to the loss of continuity by weak solutions. So, in this thesis we focuson results concerning drifts belonging to critical spaces.For the divergence-free drifts from critical spaces the best known for to-day results were obtained in [31], [51] (see also [53]) where Ho¨lder continuityof weak solutions was proved for the heat equation with a drift term sat-isfying (1.2.18) (see also the related result in [44]). In its elliptic version,151.2. Elliptic equations with critical driftthese results assume that the drift b satisfying (1.2.18) belongs to the spaceBMO−1(Ω), i.e. there exists a skew-symmetric matrix A ∈ BMO(Ω) suchthat b = divA.For the non–divergence free critical drifts satisfying the condition (1.2.9)local a priori estimates of the Ho¨lder norms for Lipschitz continuous solutionsto the problem (1.2.1) were obtained earlier in [47] under the additionalrestriction b ∈ Lr,n−r(Ω) with r ∈ (n2 , n]. So, in our Theorem 1.2.2 we removethis restriction. To summarize we studied elliptic equations with criticaldrift, for both spectral case (div b ≥ 0) and non-spectral case (div b ≤ 0)in two and three dimensional cases and proved existence and uniquenessof solutions with higher integrability, as well as Ho¨lder regularity of suchsolutions.To summarize we have proven existence and higher regularity of solutionsto elliptic equations with drift from critical spaces both in two and threedimensional cases. Two dimensional Theorems 1.2.3,1.2.4 are contained inchapter 4, while three dimensional Theorems 1.2.1,1.2.2 are in chapter 5.16Chapter 2Global Navier-Stokes flows inintermediate spacesIn this chapter we construct global weak solutions of the three dimensionalincompressible Navier-Stokes equations in intermediate spaces between thespace of uniformly locally square integrable functions and Herz-type spaceswhich involve weighted integrals centered at the origin. Our results bridgethe existence theorems of Lemarie´-Rieusset and of Bradshaw, Kukavica andTsai. An application to eventual regularity is included which generalizes theprior work of Bradshaw, Kukavica and Tsai as well as Bradshaw, Kukavicaand Ozanski. Work is done with Professors Zachary Bradshaw and Tai-PengTsai2.1 IntroductionConsider the incompressible Navier-Stokes equations in R3,∂tu−∆u+ u · ∇u+∇p = 0,∇ · u = 0, (2.1.1)with divergence-free initial data u0. Global-in-time local energy solutionsof (2.1.1) for initial data which do not decay “rapidly” at spatial infinitywas first constructed in local uniform L2 spaces [15, 16, 18, 24]. It was thenextended to data whose norms at a sequence of scales grow under certainrate (Herz spaces) [4, 5, 7, 11, 25], or is bounded in certain weighted spaces[9, 13, 19]. They were also constructed in the framework of Wiener amalgamspaces [6, 8] which are intermediate spaces between L2, which is the initialdata space for Leray’s original solutions [21], and L2uloc. The main feature ofthese solutions is that they satisfy the local energy inequality, the pressuresatisfies the local pressure decomposition and they possess a priori boundsin spaces tailored to the initial data. The goal of this thesis is to identifya continuous family of spaces between uniform local and Herz spaces wherewe can construct global-in-time local energy solutions.172.1. IntroductionFor this purpose, we first reformulate the definitions of the local uniformLq space and the Herz spaces for comparison. We may reformulate thedefinition of local uniform Lq space asLquloc ={f ∈ Lqloc(R3); supx∈R3∫B1(x)|f |q <∞}. (2.1.2)Its subset of functions vanishing at spatial infinity isEq ={f ∈ Lqloc(R3); lim|x|→∞∫B1(x)|f |q = 0}. (2.1.3)Similarly, the Herz spaces considered in [4] (see also discussion in [12])are, for λ ≥ 0,M q,λ ={f ∈ Lqloc(R3),∫B1(0)|f |q + sup|x|>11|x|λ∫B(x,|x|/2)|f |q <∞}.(2.1.4)Its subset of functions vanishing at spatial infinity isM˚ q,λ ={f ∈ Lqloc(R3), lim|x|→∞1|x|λ∫B(x,|x|/2)|f |q = 0}. (2.1.5)Global existence was shown in M˚2,λ for 0 ≤ λ ≤ 2 in [4]. This builtupon a local existence result in [3]. Although details are only given forλ = 2, the other cases λ < 2 can be proved in the same way. Observe thatLr(R3) ⊂ M˚2,2 if and only if r ≤ 6. This problem was studied in 2D byBasson [1].We now consider intermediate spaces between E2 and M2,λ. They mustbe subsets of L2loc(R3) so that the local energy inequality may be valid upto time zero, and we may get a priori estimates in terms of the initial data.Definition 2.1.1. Let 1 ≤ p <∞, 0 ≤ α ≤ 1 and 0 ≤ λ <∞. Define F pα,λto be the space of f ∈ Lploc(R3) with finite norm ∥f∥F pα,λ(R) where for anyR ≥ 1,∥f∥F pα,λ(R) = supx∈R3(1rλx∫B(x,rx)|f |p)1/p, rx =18max{R, |x|}α. (2.1.6)We denote by F pα,λ,σ the space of divergence-free vector fields in R3 with allcomponents in F pα,λ. Furthermore, we denote by F˚pα,λ and F˚pα,λ,σ the closureof C∞c (R3) and C∞c,σ(R3) = {ζ ∈ C∞c (R3;R3),div ζ = 0} in F pα,λ-norm,respectively.182.1. IntroductionComments on Definition 2.1.1:1. We will show in Lemma 2.3.2 that finiteness of ∥f∥F pα,λ(R) implies finite-ness of ∥f∥F pα,λ(ρ) for any R, ρ ≥ 1. We will show in Lemma 2.3.5 thatthe space F˚ pα,λ,σ is the same as the space of diverence-free vector fieldswith all components in F˚ pα,λ.2. Note that when α = 0, rx = 1/8, for any λ ≥ 0,F 20,λ = L2uloc, F˚20,λ = E2,and when α = 1, rx =18 max(R, |x|),F 21,λ =M2,λ, F˚ 21,λ = M˚2,λ.If λ = 0 < α, the space F 2α,0 is strictly contained in L2uloc as its norminvolves bigger balls. We do not consider α > 1, in which case theball B(x, rx) will cover the origin for |x| large enough. Hence it is notlocalized around x.3. For fixed α, λ and R, the family of balls {B(x, rx) : x ∈ R3} has theimportant property that, if B(x, rx) ∩ B(y, ry) is nonempty, then rxand ry are comparable, no matter whether |x| ≲ R or |x| ≫ R.4. Simple examples show that F pα,λ is not a subset of either L2uloc or M2,λfor 0 < α < 1, see Examples 2.3.6 and 2.3.7. Hence F pα,λ is a newsolution class.We now introduce the notion of local energy solutions, similarly to [3, 4].Unlike [15, 16, 18], the following definition does not assume uniform in spacebounds of the local energy. For a given ball B of radius r, let B∗ and B∗∗denote concentric balls of radii 98r and54r, respectively.Definition 2.1.2. A vector field u ∈ L2loc(R3 × [0, T )), T > 0, is a localenergy solution to (2.1.1) with initial data u0 ∈ L2loc(R3) if the followingholds: u ∈ ∩R>0L∞(0, T ;L2(BR(0))), ∇u ∈ L2loc(R3 × [0, T )), there is p ∈ L32loc(R3×[0, T )) such that u, p is NSE solution in the senseof distributions, For all compacts K ⊂ R3 we have u(t)→t→0 u0 in L2(K),192.1. Introduction u is a Caffarelli-Kohn-Nirenberg solution, for all ϕ ∈ C∞0 (R3×(0, T )), ϕ ≥0,2∫ ∫|∇u|2ϕ dx dt ≤∫ ∫|u|2(∂tϕ+∆ϕ) dx dt+∫ ∫(|u|2 + 2p)(u · ∇ϕ) dx dt,(2.1.7) the function t 7→ ∫ u(x, t)w(x) dx is continuous on [0, T ) for any w ∈L2(R3) with a compact support, for every ball B ⊂ R3, there exists pB(t) ∈ L 32 (0, T ) such that forx ∈ B∗ and 0 < t, T ,p(x, t)− pB(t) = −13|u|2(x, t) + p.v.∫y∈B∗∗Kij(x− y)uiuj(y, t) dy+∫y/∈B∗∗[Kij(x− y)−Kij(xB − y)]uiuj(y, t) dy,(2.1.8)where xB is the center of B and Kij = ∂i∂j(4π|y|)−1.It is a local energy solution for 0 ≤ t <∞ if it is a local energy solutionfor 0 ≤ t < T for any T <∞.The following is our main theorem on the local and global existence oflocal energy weak solutions in F 2α,λ.Theorem 2.1.3. Assume0 ≤ α ≤ 1, 0 ≤ λ ≤ 2. (2.1.9)For any divergence-free initial data u0 ∈ F˚ 2α,λ, there exist T+ > 0 (withan explicit lower bound) and a local energy solution (u, p) to the Navier-Stokes equation (2.1.1) with initial data u0 for t ∈ (0, T+) such that u(t) ∈C(0, T ;F 2α,λ(R)) for any 0 < T < T+ and 1 < R <∞.If we further assumelimρ→∞ ρα(λ−2)∥u0∥2F 2α,λ(ρ) = 0 (2.1.10)when (3 − λ)α < 1, (no condition when (3 − λ)α ≥ 1), then we can takeT+ = ∞, the solution is globally defined and satisfies the a priori bound(2.6.4).202.1. IntroductionComments on Theorem 2.1.3:1. It will follow from Lemma 2.3.2 that for ∥u0∥2F 2α,λ(R=1) = A,ρα(λ−2)∥u0∥2F 2α,λ(ρ) ≲ ρα(λ−2)ρα(1−α)(3−λ)A = (ρα)1−(3−λ)αA.Hence (2.1.10) is automatic if (3 − λ)α > 1 (right region of Figure2.1). It can be proved when (3− λ)α = 1 using u0 ∈ F˚ 2α,λ, see Lemma2.3.8. When (3 − λ)α < 1 (left region of Figure 2.1), Example 2.3.7gives functions u0 ∈ F˚ 2α,λ that fail (2.1.10), and hence (2.1.10) has tobe assumed for global existence.λα0 0.25 0.5 0.75 100.511.52 (1, 2)(13 , 0)(3− λ)α ≥ 1(3− λ)α < 1Figure 2.1: The (α, λ) parameter region for global existence: For (α, λ) inthe left part we need to assume (2.1.10). For (α, λ) in the right part wedon’t.2. Suppose u0 satisfies (2.1.10). This impliesρα(λ−2)ραλ∫B(0,ρα)|u0|2 dx = 1ρ2α∫B(0,ρα)|u0|2 dx→ 0.Using [4, Lemma 2.1], it implies u0 ∈ M˚2,2(R3). Time-global existenceis already known in this space in [4]. The main merit of Theorem 2.1.3is to construct global weak solutions in the more restrictive class F 2α,λ.3. Our construction is based on a priori bounds, whose proof essentiallyuses the condition λ ≤ 2. See Lemma 2.5.2 and Remark 2.5.1.212.1. Introduction4. Our proof works uniformly for 0 < α ≤ 1 and 0 ≤ λ ≤ 2. When α = 0and F 20,λ = L2uloc, our proof only gives a local in time solution and notthe global existence, as the time upper limit Tn defined (2.5.8) cannotgo to infinity as n→∞ when α = 0.5. If λ = 0 < α, the space F 2α,0 is strictly contained in L2uloc as its norminvolves bigger balls. Hence solutions in F 2α,0 have more control.The proof is based on a priori estimates of the solution in F pα,λ(R) for asequence Rn = 2n, n ∈ N0. This type of estimate extends ideas in [14]. Wedenote the set of ballsBn ={B(z, rz) : rz =18max(2n, |x|)α}. (2.1.11)Definean(t) = ∥u(t)∥2F 2α,λ(R=2n) = supB∈Bn1|B|λ/3∫B|u(x, t)|2 dx (2.1.12)andbn(t) = supB∈Bn1|B|λ/3∫ t0∫B|∇u(x, s)|2 dx ds. (2.1.13)Our goal is to show a family of a priori estimatesess sup0<t<Tnan(t) + bn(Tn) ≤ Can(0), n ∈ N, (2.1.14)with C independent of n, for some Tn →∞.The strategy of the proof is following: we establish pressure estimatesin spaces F pα,λ first. To do that we use the pressure decomposition formula(2.1.8) for any ball B.Next we separate pressure in several parts p = pnear(x, t) + pfar(x, t) andapply Calderon-Zygmund inequality combined with estimates of the solutionover annulus B2n \ B2n−1 . We use local energy inequality combined withGronwall lemma and pressure estimate to get a priori bound for solutionsin F pα,λ. The first step would be to obtain the following inequality for norman(t),an(t) ≲ an(0) + C1(n)∫ t0an(s) ds+ C2(n)∫ t0a3n ds.After that we apply Gronwall lemma to get uniform bound up to some timeT which will depend on norm ∥u0∥F pα,λ(R) and R. This allows us to prove222.2. Eventual regularitylocal in time existence for all parameters α, λ and under conditions (2.1.9)we get the global existence. The construction of local energy solution isdone by approximation, where we first approximate the initial data with L2sequence {un0}, for every n we have global energy solutions un correspondingto initial data un0 and after applying a priori bounds in Fpα,λ we can constructa subsequence of {(un, pn)} that will converge to local energy solution (u, p).2.2 Eventual regularityThe weighted local spaces considered in [4, 5, 7] have a natural applicationto the problem of eventual regularity. It is classical that solutions in theLeray class eventually regularize in the sense that there exists a time T sothat the solution in view is smooth on R3 × (T,∞). This follows from thefact that, for all T > 0, ∫ T0∫|∇u|2 dx dt ≤ ∥u0∥22. (2.2.1)Since this holds as T →∞, by Sobolev embedding the L6 norm must becomesmall at some large times. Since u is also bounded in L2, interpolationimplies u is small in L3 at certain large times. The small-data global well-posedness theory in L3 and weak-strong uniqueness together guarantee thesolution becomes smooth for all large times. On the other hand, eventualregularity is not known for local energy solutions. This is due to the lackof an estimate that extends to arbitrarily large times. Indeed, Lemarie´-Rieusset’s solutions, which have data in L2uloc have only a short time estimate[18, 20]. The smaller M2,q and F˚ 2α,λ spaces, on the other hand, allow foran estimate in Theorem 2.5.4 which extends up to arbitrarily large timesfor certain choices of parameters. These a priori bounds can be used as asubstitute for (2.2.1) in proving eventual regularity, as has been explored in[4, 5, 7]. In [4] eventual regularity is proven above a paraboloid for datain M˚2,1(R3). In [5] corresponding estimates are carried out in M˚2,q(R3+)where q ≤ 1. In this case, the paraboloid region can be replaced by alarger algebraic region with exponents depending on q. As was known tothe authors of [5], the theorem in [5] can be improved in M˚2,q(R3) yieldingan algebraic region of the form{(x, t) : |x|2q ≲ t and t > M},232.2. Eventual regularityλα000.250.250.50.50.750.7511 F˚2αλ,1αλ = 1/3F˚21,αλF˚2α̸=1,λ ̸=1Figure 2.2: The spaces F˚2α,λ when αλ = 1/3, which is fixed for illustrativepurposes. Note that M˚2,αλ = F˚21,αλ by Lemma 2.3.9. These spaces implyeventual regularity in regions with the same algebraic bounds, up to con-stants. Spaces are strictly increasing in λ. The conclusion of Theorem 2.2.1is new in all but the points (α = 1, λ = 1/3).for a non-universal constant M =M(u0).1 Note that the classical eventualregularity result for Leray weak solutions would correspond to q = 0.It is natural to ask what happens if we seek eventual regularity withinthe finer F 2α,λ scale of spaces, noting that M2,q corresponds to F 21,q. For thiswe need to introduce new sub-classes of our F 2α,λ spaces. LetF2α,λ := {f ∈ F 2α,λ : supR≥1∥f∥F 2α,λ(R) <∞}, (2.2.2)andF˚2α,λ := {f ∈ F2α,λ : limR→∞∥f∥F 2α,λ(R) = 0}. (2.2.3)The latter is necessarily a subspace of M˚2,λ, which can be easily seen usingan alternative definition of the M2,q norm given in [4].Theorem 2.2.1 (Eventual regularity). Assume that u0 ∈ F˚2α,λ where α ∈(0, 1], λ ∈ (0, 1]. Let (u, p) be a local energy solution with initial data u0 onR3 × (0,∞), satisfyingess sup0<s<tan(s) + bn(t) <∞, for all t <∞, (2.2.4)1A proof is not written down but it follows directly from the argument in [5].242.3. The intermediate spacesfor one fixed n ∈ N0. Then, there exists M = M(u0) so that u is regularwithin a region of the form{(x, t) : |x|2αλ ≤ t, t ≥M},where the suppressed constants depend on λ and α, whileM is determined bythe length scale ρ at which ∥u0∥F 2α,λ(ρ) becomes small compared to a universalsmall constant. Additionally, for (x, t) in the preceding region, |u(x, t)| ≲t−1/2.The condition (2.2.4) is (2.5.6) and is needed to ensure that the a pri-ori bound (2.6.4) holds. This is satisfied by the solutions constructed inTheorem 2.1.3.The region of regularity here is already established for M˚2,αλ-solutions.Our result identifies larger classes which have the same regularity region,insofar as the degree of the algebraic boundary is concerned, as the M˚2,αλcase. To support this, in Lemmas 2.3.9 and 2.3.10 we show that if αλ = q ∈(0, 1], thenF˚21,q = M˚2,q ⊊ F˚2α ̸=1,λ ̸=1 ⊊ F˚2q,1.The largest space in Theorem 2.2.1 which admits a regularity region with|x|2αλ ≲ t for large t is, therefore, F˚2q,1. This is illustrated in Figure 2.2.The rest of the chapter is structured as follows: In Section 2.3 we studyproperties of the intermediate spaces F pα,λ. In Section 2.4 we show thepressure estimate. In Section 2.5 we show the a priori bounds. In Section2.6 we prove the main theorem on the local and global existence of weaksolutions. Finally in Section 2.7 we study eventual regularity.2.3 The intermediate spacesIn this section we explore properties of spaces F pα,λ defined in Definition2.1.1. For ρ > 0, we denote the ball and annulusBρ = B(0, ρ), Aρ = B2ρ \Bρ.Lemma 2.3.1 (annulas covering). Fix α ∈ [0, 1], R ≥ 1, and let rx =18 max(R, |x|)α for x ∈ R3. For any ρ ≥ R, the annulus Aρ can be coveredby N(ρ) balls B(xk, rxk), xk ∈ Aρ, k = 1, . . . , N(ρ),Aρ ⊂ ∪N(ρ)k=1 B(xk, rxk), with N(ρ) ≤ Cρ3−3α,where C is independent of R and ρ. Similarly, the ball BR can be covered byN(R) balls B(xk, rxk), xk ∈ BR, k = 1, . . . , N(R), with N(R) ≤ CR3−3α.252.3. The intermediate spacesProof. Denote the latticeΛ = rZ3, r := 116ρα ≤ 12 infx∈Aρ rx.Any point y ∈ Aρ belongs to at least one cube of side length r and verticesin Λ, and at least one of these vertices, denoted as x, is in Aρ. Thus |y−x| ≤√3r and hencey ∈ B(x,√3r) ⊂ B(x, rx).We choose xk, k = 1, . . . , N(ρ), to be all points in Λ∩Aρ. Let m = ⌈2ρ/r⌉ ≤Cρ1−α. We have Λ∩Aρ ⊂ Λρ, where Λρ = {r(z1, z2, z3) : zi ∈ Z, |zi| ≤ m, i =1, 2, 3}. HenceN(ρ) ≤ #Λρ = (2m+ 1)3 ≤ Cρ3−3α.The proof for the ball BR is similar.Lemma 2.3.2. Let 1 ≤ p < ∞, 0 ≤ α ≤ 1 and 0 ≤ λ < 3. Let 1 ≤ R <ρ <∞. Then ∥f∥F pα,λ(R) <∞ if and only if ∥f∥F pα,λ(ρ) <∞, andC1 ∥f∥pF pα,λ(R) ≤ ∥f∥pF pα,λ(ρ)≤ C2 ∥f∥pF pα,λ(R) (2.3.1)for C1 =( ρR)−αλand C2 = C(ρR+ρα)α(3−λ)≲ ρα(1−α)(3−λ).Remark. This lemma shows that the space F pα,λ defined by the Fpα,λ(R)-norm does not depend on the choice of R. Similar bound (2.3.1) hold forλ ≥ 3, but the formula for C2 will change.Proof. Denote Rx =18 max(R, |x|)α and ρx = 18 max(ρ, |x|)α. Note Rx ≤ρx ≤ (ρ/R)αRx.Suppose ∥f∥F pα,λ(ρ) <∞. For any x ∈ R3,1Rλx∫B(x,Rx)|f |p dy ≤ ρλxRλx1ρλx∫B(x,ρx)|f |p dy ≤( ρR)αλ ∥f∥pF pα,λ(ρ).This shows the first inequality of (2.3.1) with C1 =( ρR)−αλ.Suppose now ∥f∥F pα,λ(R) <∞. We may assume ∥f∥F pα,λ(R) = 1.If |x| > ρ, ρx = Rx, and 1ρλx∫B(x,ρx)|f |p dy ≤ ∥f∥pF pα,λ(R)= 1.It remains to consider |x| < ρ with ρx = 18ρα. If 2ρx < |x| < ρ, ally ∈ B(x, ρx) has |y| ∼ |x| and hence Ry ∼ Rx. Hence B(x, ρx) can be262.3. The intermediate spacescovered by K balls B(yk, Ryk) with K ≲ (ρx/Rx)3, and1ρλx∫B(x,ρx)|f |p dy ≤ 1ρλxK∑k=1∫B(yk,Ryk )|f |p dy ≤ 1ρλxK∑k=1Rλyk ≲ (ρx/Rx)3−λ≈(ρR+ |x|)α(3−λ)≲(ρR+ ρα)α(3−λ).(2.3.2)If |x| < 2ρx = 14ρα, then B(x, ρx) ⊂ B(0, ρα). In the case R > ρα,Ry =18Rα in B(0, ρα), and B(x, ρx) can be covered by K balls B(yk, Ryk)with K ≲ (ρx/Rx)3. We have the same bound (2.3.2).In the case R < ρα, we cover B(0, ρα) by BR(0) and K annulusAk = {y ∈ R3 : 2k−1R ≤ |y| < 2kR}, k = 1, . . . ,K,with 2K−1R ≤ ρα < 2KR. Each Ak can be covered by Nk balls Bk,j =B(yk,j , Ryk,j ) with yk,j ∈ Ak, Ryk,j ∼ (2kR)α, and Nk ≲ (2kR)3(1−α). Simi-larly, BR(0) can be covered by N0 balls B0,j with j ≤ N0 ∼ N1. Thus1ρλx∫B(x,ρx)|f |p dy ≤ 1ρλxK∑k=0Nk∑j=1∫Bk,j|f |p dy ≤ 1ρλxK∑k=0Nk∑j=1(2kR)αλ≲ 1ραλK∑k=0(2kR)αλ+3−3α ≈ 1ραλ(ρα)αλ+3−3α ≈ ρα(1−α)(3−λ).As R+ ρα ∼ ρα, this bound is the same as (2.3.2).We have shown the second inequality of (2.3.1) with C2 = C(ρR+ρα)α(3−λ).Remark. In particular, C2 ≲ ρα(1−α)(3−λ). If α = 1, we have C2 ≲ 1. Itis consistent with [4, Lemma 2.1]. As a result, for α ∈ [0, 1],ρα(λ−2)∥u0∥2F 2α,λ(ρ) ≲ ρα(λ−2)ρα(1−α)(3−λ) = (ρα)1−3α+αλ.Hence the vanishing condition (2.1.10) is automatic when 1− 3α+ αλ < 0,and has to be assumed otherwise, for global existence.Lemma 2.3.3 (Inclusion). Let 1 ≤ p <∞, 0 ≤ α ≤ 1 and 0 ≤ λ, µ < 3.Wehave(a) Lp(R3) ⊂ F pα,λ, and Lp(R3) ⊂ F qα,µ if 1− µ3 ≤ qp ≤ 1,(b) F pα,λ ⊂ F pα,µ if λ < µ, and(c) F pα,λ ⊂ F qα,µ if q < p and 3−µq = 3−λp .272.3. The intermediate spacesProof. First, we prove part (a). Let u ∈ Lp(R3). For any x ∈ R3 and R ≥ 1we have1Rλx∫B(x,Rx)|u(y)|p dy ≤ ∥u∥pp,R3where Rx =18{R, |x|}α. Therefore, u ∈ F pα,λ. The second half of (a) followsfrom (c) by taking λ = 0. Part (b) follows from basic algebraic inequality,for all 0 ≤ λ < µ and x ∈ R3, R ≥ 1:1Rµx∫B(x,Rx)|u(y)|p dy ≤ 1Rλx∫B(x,Rx)|u(y)|p dy.For part (c) we use Ho¨lder inequality: take any R ≥ 1, x ∈ R3 and λ, µ, p, qsatisfying 3−µq =3−λp , then( 1Rµx∫B(x,Rx)|u(y)|q dy) 1q ≤ |BRx(x)|p−qpq Rλ/qxRµ/qx( 1Rλx∫B(x,Rx)|u(y)|p dy) 1p= C( 1Rλx∫B(x,Rx)|u(y)|p dy) 1p,where C is independent of Rx.The following lemma is useful in localizing a divergence free vector fieldin a ball.Lemma 2.3.4. Fix a radial cut-off function χ ∈ C∞c (R3) with χ(x) = 1 for|x| < 34 and χ(x) = 0 for |x| > 78 . Let χR(x) = χ(x/R). For any v ∈ Lp(BR)with div v = 0, 1 ≤ p < ∞, R > 4, we have w ∈ W 1,p(R3;R3) such thatw(x) = 0 for |x| > R,divw = f := v · ∇χR, R ∥∇w∥Lp + ∥w∥Lp ≤ Cp ∥v · 1R∥Lp , (2.3.3)where 1R is the characteristic function of the support of ∇χR. Moreover,for any 0 ≤ α ≤ 1, 0 ≤ λ ≤ 2, we have∥w∥F pα,λ ≤ Cp,α,λ ∥v · 1R∥F pα,λ . (2.3.4)Above, the constants Cp and Cp,α,λ are independent of R and v.Proof. Let Φ ∈ C∞c (R3) be supported in |x| ≤ 12 with∫R3 Φ = 1. LetϕR(x) = R−3Φ( xR). We use the Bogovskii formula [23] for the ball BR:w(x) =∫BRN(x, y)f(y) dy, N(x, y) =x− y|x− y|3∫ ∞|x−y|ϕR(y + rx− y|x− y|)r2dr.(2.3.5)282.3. The intermediate spacesBy the properties of the Bogovskii formula, we have w ∈W 1,p0 (BR), divw =f and ∥∇w∥Lp(BR) ≤ C ∥f∥Lp(BR) with C independent of R. It shows (2.3.3).We now consider ∥w∥F pα,λ . For x, y ∈ BcR, we have|N(x, y)| ≲ 1|x− y|2∫ cR|x−y|R−3r2dr ≲ 1|x− y|2 .Also note the support of f is in AR = BR \BR/2, and for |k| ∼ R, we haverk ∼ Rα and∫B(k,rk)|f(y)| dy ≤(r−λk∫B(k,rk)|f(y)|p dy) 1prλ/pk(∫B(k,rk)1 dz)1− 1p≲ δRr3− 3−λpk , δ := ∥v · 1R∥F pα,λ .(2.3.6)Fix x0 ∈ R3 and r = rx0 . If B(x0, 2r) intersects AR, for any x ∈ B(x0, r),decomposew(x) =∫AR∩B(x0,2r)N(x, y)f(y) dy+∫AR\B(x0,2r)N(x, y)f(y) dy =: w1+w2.(2.3.7)By potential estimate,r−λ/p ∥w1∥Lp(B(x0,r)) ≲ r−λ/p r ∥f∥Lp(B(x0,2r)) ≲rδR.For w2, choosing one l0 ∈ Z3 closest to x0/r, we have|w2(x)| ≲∫AR\B(x0,2r)|f(y)||x− y|2 dy ≲∑k∈rZ3∩AR∫B(k,rk)|f(y)|r2 + |k − x0|2 dy≲∑l∈Z3∩AR/r1r2(1 + |l − l0|2)∫B(k,rk)|f(y)| dy (k = rl)≲∑l∈Z3∩AR/r1r2(1 + |l − l0|2)δRr3− 3−λp ≲∑l∈Z3∩A2R/r1r2(1 + |l|2)δRr3− 3−λp≲ Rr3δRr3− 3−λp = δ(Rα)− 3−λp . (2.3.8)Thus (r−λ∫B(x0,r)|w2|p)1/p≲ (Rα)3−λp δ(Rα)− 3−λp = δ.292.3. The intermediate spacesSumming the estimates for w1 and w2, we getr−λ/p ∥w∥Lp(B(x0,r)) ≲rδR+ δ ≲ δ.For B(x0, 2r) not intersecting AR, w(x) is nonzero only if |x0| < R/2.In this case, in the decomposition w = w1 + w2 in (2.3.7), we have w1 = 0,and the same upper bound (2.3.8) for w2 holds. Thus(r−λ∫B(x0,r)|w|p)1/p≲ r3−λp δ(Rα)− 3−λp ≲ δ.Taking sup in x0 ∈ R3, we get (2.3.4).In Theorem 2.1.3 we have used the spaces F˚ pα,λ and F˚pα,λ,σ, defined in Defi-nition 2.1.1, as the closure of C∞c (R3) and C∞c,σ(R3) = {ζ ∈ C∞c (R3;R3), div ζ =0} in F pα,λ-norm. They are needed for the initial data to get local existence.The following lemma will clarify the relation between them and the sub-spaces Fˆ pα,λ and Fˆpα,λ,σ for functions and vector fields vanishing at spatialinfinity:Fˆ pα,λ(R) ={f ∈ F pα,λ, lim|x|→∞1rλx∫B(x,rx)|f |p = 0},Fˆ pα,λ,σ(R) ={f ∈ F pα,λ,σ, lim|x|→∞1rλx∫B(x,rx)|f |p = 0},where rx =18 max{|x|, R}α. By Lemma 2.3.2, we may assume R = 1 andomit the dependence on R, using the notations Fˆ pα,λ = Fˆpα,λ(1) and Fˆpα,λ,σ =Fˆ pα,λ,σ(1).Now we can prove the following lemma that connects spaces F˚ and Fˆ :Lemma 2.3.5. Let 1 ≤ p <∞, 0 ≤ α ≤ 1 and 0 ≤ λ < 3. We have(a) F˚ pα,λ = Fˆpα,λ.(b) F˚ pα,λ,σ = Fˆpα,λ,σ.Proof. First we will prove that F˚ pα,λ ⊂ Fˆ pα,λ. Take u ∈ F˚ pα,λ and let ε > 0,then there exists uε ∈ C∞c such that ∥u−uε∥F pα,λ ≤ ε. Since uε is compactlysupported there exists ρ and supp{uε} ⊂ Bρ/2, then we have the followingboundsup|x|≥ρ(1Rλx∫B(x,Rx)|u|p dx)1/p≤ ∥u− uε∥F pα,λ ≤ ε (2.3.9)302.3. The intermediate spacesTherefore F˚ pα,λ ⊂ Fˆ pα,λ. Proof of inclusion F˚ pα,λ,σ ⊂ Fˆ pα,λ,σ follows the sameargument with additional assumptions that u, uε are divergence free.Next we will prove the other inclusion Fˆ pα,λ,σ ⊂ F˚ pα,λ,σ. Take v ∈ Fˆ pα,λ,σthen for any ε > 0 there exists ρ such thatsup|x|≥ρ(1Rλx∫B(x,Rx)|v|p dx)1/p≤ ε. (2.3.10)We apply Lemma 2.3.4 with R = 2ρ to find w satisfying (2.3.3) and (2.3.4).Extend w by zero to the whole space R3. We choose vε = vχR − w, whereχR is the same cut-off function as in Lemma 2.3.4. From (2.3.3) we get thatdiv vε = 0 and∥v − vε∥F pα,λ ≤ ∥v(1− χR)∥F pα,λ + ∥w∥F pα,λ ≤ ε+ Cp,α,λε.Here we used (2.3.10) and (2.3.4) for the last inequality. Take a suitablemollification vε ∗ ηt ∈ C∞c,σ with ∥vε − vε ∗ ηt∥Lp ≤ ε, we have ∥v − vε ∗ηt∥F pα,λ ≤ Cε. The inclusion Fˆpα,λ ⊂ F˚ pα,λ is similar but without the need ofw, and is skipped.Remark 2.3.1. In case α = 1, part (b) of Lemma 2.3.5 has an alternativeproof, similar to [3, 4], using a Bogovskii map in the annulus AR instead ofthe ball BR. Specifically, we havew ∈W 1,p0 (AR), divw = v · ∇χR, ∥∇w∥Lp(AR) ≤ Cp ∥v · ∇χR∥Lp .As A(R) can be covered by N balls Bk = B(xk, rxk), 1 ≤ k ≤ N , with Nindependent of R, if B(x, rx) intersects the support of w, then rx ∼ R and(1rλx∫B(x,rx)|w|p) 1p≲ R−λpR(∫A(R)|v · ∇χR|p) 1p≲ R−λp(∑k∫Bk|v|p) 1p≲ R−λp(∑kRλo(1)) 1p= o(1)where o(1)→ 0 as R →∞. We then define uR = fχR − w and smoothrizeit.This argument does not work for α < 1 mainly due to the differences inannulus covering Lemma 2.3.1 for α = 1 and α < 1. Specifically, we willneed N(R) = O(R3−3α) balls Bk = B(xk, rxk) to cover AR, with N(R)→∞as R→∞ if α < 1. □312.3. The intermediate spacesRemark 2.3.2. When α = 0, F p0,λ = Lpuloc. In this case, part (b) of Lemma2.3.5 is assumed in [18] and proved in [15, Appendix]. However, the proofof [15] has a gap: It claims Lemma 2.3.4 by applying Bogovskii map to theunit ball Bk = B(k, 1) for each k ∈ Z3 and get vk ∈ W 1,p0 (Bk;R3) for eachk ∈ N that is a solution of the equationdiv vk = (f · ∇χR)φk − ck in Bk, ck := 1|Bk|∫Bk(f · ∇χR)φk dx,where φk is a partition of unity such thatφk ∈ C∞c (R3), spt φk ⊂ Bk, φk ≥ 0,∑k φk = 1.Extending vk by zero to the whole space R3 and setting vR =∑k vk, we getdiv vR =∑k(f · ∇χR)φk −∑kck1Bk = f · ∇χR −∑kck1Bk .The proof of [15] overlooked the factor 1Bk and thought the last sum equaled∑k ck = 0 so that vR was our desired w. It is unclear how to fix this gap,and Lemma 2.3.5 provides a proof. □Example 2.3.6. Let χ ∈ C1c (B1) be a bump function. For k ∈ N, letxk = 2ke1, 1 ≤ Rk ≤ 2kα, |ak| ≤ 1, andf(x) =∑k∈Nak2kαλR−3k χ(x− xkRk).Then f ∈ F 1α,λ. If ak = o(1) as k → ∞, then f ∈ F˚ 1α,λ. If we takeak = Rk = 1, then f ̸∈ L2uloc if α > 0.Example 2.3.7. Letf(x) =1(1 + |x|)m , 0 ≤ m < 3.(If m ≥ 3, then f is localized and not so interesting.) If |x| > R, then1rλx∫B(x,rx)f ≈ |x|α(3−λ)−m.Its sup over |x| > R is finite if α(3− λ) ≤ m. If |x| < R, then1rλx∫B(x,rx)f ≲ R−αλ(Rα)3−m = Rα(3−λ−m).322.3. The intermediate spacesThus∥f∥F 1α,λ(R) ∼ Rα(3−λ−m), if α(3− λ) ≤ m,and f ∈ F˚ 1α,λ if α(3−λ) < m. However, if α(3−λ) < m < 3−λ, (for whichwe need α < 1), then f ∈ F˚ 1α,λ, f ̸∈ F 1α=1,λ, and limR→∞ ∥f∥F 1α,λ(R) = ∞.Moreover,Rα(λ−2) ∥f∥F 1α,λ(R) ∼ Rα(1−m)still goes to infinity as R→∞ if m < 1. This shows that, if f = |u0|2, thenTn defined by (2.5.8) may not go to ∞ if u0 ∈ F˚ 2α,λ with α(3 − λ) < 1 bychoosing m ∈ (α(3 − λ), 1). This example does not exist when α = 1 sincem cannot be chosen. □The following lemma is concerned with the borderline case (3− λ)α = 1of (2.1.10). We will let f = |u0|2, so ∥f∥F 1α,λ(ρ) = ∥u0∥2F 2α,λ(ρ), to show thatu0 ∈ F˚ 2α,λ implies (2.1.10). This result is already known for (α, λ) = (1, 2)by [4, Lemma 2.1].Lemma 2.3.8. Let 0 ≤ α ≤ 1, 0 ≤ λ ≤ 2, and (3 − λ)α = 1. Supposef ∈ F˚ 1α,λ, thenlimρ→∞ ρα(λ−2)∥f∥F 1α,λ(ρ) = 0.Proof. We may assume f ≥ 0. Noteρα(λ−2)∥f∥F 1α,λ(ρ) = 8λmax(I, J),whereI = sup|x|<ρρ−2α∫B(x, 18ρα)f, J = sup|x|≥ρρα(λ−2)|x|−αλ∫B(x, 18|x|α)f.Since f ∈ F˚ 1α,λ, for any ε > 0, there is R > 1 such thatsup|x|>R/2|x|−αλ∫B(x, 18|x|α)f ≤ ε. (2.3.11)Consider ρ≫ R so that ρα > 8R. As ρα(λ−2) ≤ 1, we have J ≤ ε.We next focus on I. Denote Rx =18 max(R, |x|)α and ρx = 18ρα for|x| < ρ.332.3. The intermediate spacesIf 2ρx < |x| < ρ, all y ∈ B(x, ρx) has |y| ∼ |x| and hence Ry ∼Rx =18 |x|α since R < ρx < |x|. Hence B(x, ρx) can be covered by Kballs B(yk, Ryk) with K ≲ (ρx/Rx)3 ≈ (ρ/|x|)3α, and, using (2.3.11),1ρ2α∫B(x,ρx)f ≤ 1ρ2αK∑k=1∫B(yk,Ryk )f≲ 1ρ2αK∑k=1Rλykε ≲1ρ2α(ρ/|x|)3α|x|αλε= ρα|x|−(3−λ)αε = ρα|x|−1ε ≤ Cε.The second equality uses α(3− λ) = 1. The last inequality uses 2ρx < |x|.If |x| < 2ρx, then B(x, ρx) ⊂ B(0, ρα). We cover B(0, ρα) by BR(0) andK annulusAk = {y ∈ R3 : 2k−1R ≤ |y| < 2kR}, k = 1, . . . ,K,with 2K−1R ≤ ρα < 2KR. Each Ak can be covered by Nk balls Bk,j =B(yk,j , Ryk,j ) with yk,j ∈ Ak, Ryk,j ∼ (2kR)α, and Nk ≲ (2kR)3(1−α). De-noting CR =∫BR(0)f and using (2.3.11),1ρ2α∫B(x,ρx)f ≤ 1ρ2α(CR +K∑k=1Nk∑j=1∫Bk,jf)≤ CRρ2α+Cρ2αK∑k=1(2kR)3(1−α)+αλε≤ CRρ2α+Cρ2α(ρα)3−3α+αλε =CRρ2α+ Cε.The last equality uses α(3− λ) = 1. The first term is bounded by ε when ρis sufficiently large, using α > 0.Combining the 2 cases we have shown I ≤ Cε for ρ sufficiently large.Since ε > 0 is arbitrary, we have shown the lemma.We include a second proof of Lemma 2.3.8 which uses an auxiliary func-tion g to avoid the series appearing in the preceding proof.Second proof of Lemma 2.3.8. Let g(x) = (1+|x|3α−αλ)−1 and rx = 18 max(ρ, |x|)α.For the moment we do not assume 3α − αλ = 1. We claim there exists a342.3. The intermediate spacesconstant c˜ > 0 independent of ρ so thatc˜−1 ≤ infx∈R31rλx∫B(x,rx)g. (2.3.12)Indeed,1rλx∫B(x,rx)g ≥ r3−λx minB(x,rx)g =r3−λx1 + (|x|+ rx)(3−λ)α≳ r3−λx1 + max(ρ, |x|)(3−λ)α ≈ 1.It is bounded from below by a constant independent of x and ρ.Observe also that, for ρ ≥ 1, and under the requirement that 3α−αλ = 1,1ρ2α∫B(0,3ρα)g < C˜,where C˜ is independent of ρ.For |x| ≤ ρ, let C be a covering of B(x, ρα/8) by balls of the form B(y, ry)which are taken so that there is a uniform bound on the number of coverelements that intersect any single cover element. The following bound is thekey tool in this proof,∑xk∈Crλxkρ2α≤∑xk∈Cc˜ρ2α∫B(xk,rxk )g ≲ cρ2α∫B(x,3ρα)g ≲ cρ2α∫B(0,3ρα)g < C,(2.3.13)where, abusing notation, we are indexing cover elements by their centers xk.Note that in the second last inequality we were able to pass from B(x, 3ρα)to B(0, 3ρα) because g is radial and non-increasing in |x|. The uniformbound above is independent of ρ and x.Suppose f ∈ F˚ 1α,λ and, without loss of generality, ∥f∥F 1α,λ = 1. Letϕ(r) = sup|x|≥r1rλx∫B(x,rx)f.Then ϕ is decreasing in r and bounded by 1. Suppose |x| ≤ ρ. Let C be acovering of B(x, ρα/8) as above. Then1ρ2α∫B(x,ρα/8)f ≲∑xk∈C1ρ2α∫B(xk,rxk )f ≤∑xk∈Cϕ(|xk|)rλxkρ2α. (2.3.14)352.3. The intermediate spacesLet ε > 0 be given. The above is equal to=∑xk∈C;|xk|≤εραϕ(|xk|)rλxkρ2α+∑xk∈C;|xk|>εραϕ(|xk|)rλxkρ2α≤∑xk∈C;|xk|≤εραε2rλxkε2ρ2α+ ϕ(ερα) ≲ ε2 + ϕ(ερα),where we have used (2.3.13) applied to the ball B(0, 2ερα) to get the ε2 termon the right-hand side, noting that the collection of balls with |xk| ≤ ερα iscontained in a suitable cover of B(0, 2ερα). With ε fixed, by taking ρ largeϕ(ερα) can be made small.To conclude the proof note thatsup|x|≥ρραλ−2α1rλx∫B(x,rx)f ≤ ραλ−2αϕ(ρ)→ 0,as ρ→∞.The following lemma will be useful to Lemma 2.3.10 and Remark 2.3.3.It is given in [4, Lemma 2.1] forM2,2, but the result for generalMp,λ, λ > 0,is proven identically.Lemma 2.3.9 (Centered definition). Let p,R ≥ 1 and λ > 0. The Herzspace Mp,λ = F p1,λ defined in (2.1.4) has three equivalent norms:∥f∥pMp,λ(R)=1Rλ∫BR(0)|f |p + sup|x|>R1|x|λ∫B(x,|x|2)|f |p,∥f∥pMp,λ∗ (R)= supr≥R1rλ∫B(0,r)|f |p dx,and ∥f∥F p1,λ(R). The constant c1 ∈ (0, 1) with ∥f∥Mp,λ∗ (R), ∥f∥F p1,λ(R) ∈(c1, c−11 )∥f∥Mp,λ(R) is independent of R ≥ 1. For f ∈ Mp,λ, the following 3statements are equivalent:(a) f ∈ M˚p,λ defined in (2.1.5).(b) limR→∞ ∥f∥Mp,λ(R) = 0.(c) limr→∞ 1rλ∫B(0,r) |f |p dx = 0.Sketch of proof. For the equivalence of norms, since the proof is similar tothat in [4, Lemma 2.1], we only show as an example that∥f∥pF p1,λ(R)∼ supr>R1rλ∫B(0,r)|f |p dx.362.3. The intermediate spacesBy Lemma 2.3.1 when α = 1, shells of the form B(0, 2r) \ B(0, r) and theball B(0, R), can be covered by a bounded number (independent of r,R) ofelements of {B(x, rx)} where rx = 18 max{R, |x|} and R ≤ r. This implies,for instance, that, taking R ≤ r and k ≥ 0 so that 2kR ≤ r < 2k+1R,1rλ∫B(0,r)|f |p dx ≲ 1rλk∑i=0(2iR)λ1(2iR)λ∫2iR≤|x|≤2i+1R|f |p dx+1Rλ∫B(0,R)|f |p dx ≲ ∥f∥pF p1,λ(R),where we used a finite cover of each shell or ball to pass from the centeredintegrals to those over elements of {B(x, rx)}. The reverse direction of thenorm equivalence is straightforward.For the equivalence of vanishing statements, (b) and (c) are clearly equiv-alent and imply (a) by the equivalence of 3 norms. Suppose (a) holds. Forany ε > 0 there is R > 1 so that sup|x|>R1|x|λ∫B(x,|x|2)|f |p < ε. For r ≫ R,2kR ≤ r < 2k+1R, we have1rλ∫B(0,r)|f |p dx ≤ 1rλk∑i=0(2iR)λ1(2iR)λ∫2iR≤|x|≤2i+1R|f |p dx+1rλ∫B(0,R)|f |p dx ≤ 1rλk−1∑i=0(2iR)λCε+1rλCR ≤ Cε+ 1rλCRwhich is less than Cε for r sufficiently large. Hence (a) implies (c).The following lemma confirms the embeddings illustrated in Figure 2.2.Recall F˚2α,λ is defined in (2.2.3).Lemma 2.3.10. If α1λ1 = α2λ2 = q ∈ (0, 1] with λ1 < λ2 and 0 < α1, α2 ≤1, thenF˚2α1,λ1 ⊊ F˚2α2,λ2 .Proof. Let α1λ1 = α2λ2 = q with λ1 < λ2. Thensupx∈R31max(R, |x|)α2λ2∫B(x,max(R,|x|)α2/8)|f |2 dx≤ supx∈R31max(R, |x|)α1λ1∫B(x,max(R,|x|)α1/8)|f |2 dx.372.3. The intermediate spacesIt implies∥f∥F 2α2,λ2 (R) ≤ ∥f∥F 2α1,λ1 (R),and soF˚2α1,λ1 ⊂ F˚2α2,λ2 .We now show this inclusion is strict. Letf(x) =1√|x|3−λ12 −λ22 + 1.We have that1Rα1λ1∫B(0,Rα1 )|f |2 dx ∼ Rα1(λ2−λ1)/2,which is not bounded in R. Hence, f /∈ F2α1,λ1 .On the other hand, since f is decreasing in |x|, we havesup|x|≤R∫B(x,Rα2 )|f |2 dx ≤∫B(0,Rα2 )|f |2 dx,and1Rα2λ2∫B(0,Rα2 )|f |2 dx ∼ Rα2(λ1−λ2)/2.This vanishes as R→∞. Additionally note that1|x|α2λ2∫B(x,|x|α2/8)|f |2 dx ≲ |x|3α2−α2λ2(1|x|α2(|x|1−α2 − 18))3−λ12−λ22= |x|α2(λ1−λ2)/2(1|x|1−α2 − 18)3−λ12−λ22.After applying sup|x|>R to the right-hand side above we get something thatvanishes as R→∞. It follows that f ∈ F˚2α2,λ2 .Remark 2.3.3. In the preceding proof a centered definition of the norm ofM2,q, as in Lemma 2.3.9, is used. In this remark we show that there is nocentered definition of ∥ · ∥F pα,λ when α < 1. We will not use this observationanywhere, but hope that it is illuminating to the reader. By this we meanthat there exists no function Φ so that∥f∥pF pα,λ(R=1)∼ supr≥11Φ(r)∫B(0,r)|f |p dx. (2.3.15)382.3. The intermediate spacesWithout loss of generality we only consider F 1α,λ with p = 1. Fix α and λ.Let Cr denote the collection of all covers of B(0, r) by elements of B1. LetΨ(r) = infC¯∈Cr∑B(x,rx)∈C¯rλx , rx =18max(1, |x|)α.Then,supr≥11Ψ(r)∫B(0,r)|f | ≲ ∥f∥F 1α,λ . (2.3.16)We may approximate Ψ(r) using the function g in the second proof of Lemma2.3.8, defined above (2.3.12). Note that we may choose c˜ in (2.3.12) so thatc˜−1 ≤ supx∈R31rλx∫B(x,rx)g(y) dy ≤ c˜. (2.3.17)The upper bound is new compared to what is written in the second proofof Lemma 2.3.8, but the supporting estimate is similar to the lower bound,using R = 1. Choosing a particular cover C of B(0, r) in which each coverelement is overlapped by a bounded number N¯ of other cover elements, weobtainΨ(r) ≤∑B(y,ry)∈Crλy ≤ c˜∑B(y,ry)∈C∫B(y,ry)g(x) dx ≤c˜N¯∫B(0,2r)g(x) dx ∼ r3−3α+αλ.On the other hand, for any given cover C ′ of B(0, 2r), we haver3−3α+αλ ∼∫B(0,2r)g(x) dx ≤∑B(y,ry)∈C ′∫B(y,ry)g dx ≤ c˜∑B(y,ry)∈C ′rλy ,using (2.3.17) in the last inequality. This impliesΨ(r) ∼ r3−3α+αλ.We now argue that if Φ(r) satisfiessupr≥11Φ(r)∫B(0,r)|f | ≤ c¯∥f∥F 1α,λ , (2.3.18)for some c¯ > 0 and all f ∈ F 1α,λ, then Ψ(r) ≲ Φ(r). Using the function gagain, we haveΨ(r)∥g∥F 1α,λ ≲∫B(0,r)g dx.392.4. Pressure estimateSince g ∈ F 1α,λ we have1Φ(r)∫B(0,r)g dx ≤ c¯∥g∥F 1α,λ ≲1Ψ(r)∫B(0,r)g dx,implying Ψ(r) ≲ Φ(r). This and (2.3.16) imply for all f ∈ F 1α,λsupr≥11Φ(r)∫B(0,r)|f | ≲ supr≥11Ψ(r)∫B(0,r)|f | ≤ ∥f∥F 1α,λ . (2.3.19)We conclude with an argument by contradiction. If there exists a weighteddefinition of the norm of F 1α,λ with weight Φ so that (2.3.15) holds, then theleft term of (2.3.19) is equivalent to ∥f∥F 1α,λ , and the middle term of (2.3.19)therefore provides an equivalent norm to that of F 1α,λ. But, the middle termis exactly the norm of M1,3−3α+αλ because of Ψ(r) ∼ r3−3α+αλ and Lemma2.3.9. So, if there exists a weighted definition of the norm of F 1α,λ, thenF 1α,λ =M1,3−3α+αλ as normed spaces. This leads to a contradiction if α < 1,as evidenced by the functionh(x) = |x|2−3α+αλχ{|x−(x1,0,0)|<1}(x) ∈M1,3−3α+αλ \ F 1α,λ.Here the region |x− (x1, 0, 0)| < 1 is the cylinder x22 + x23 < 1. Hence, thereis no centered definition of ∥ · ∥F 1α,λ when α < 1. □2.4 Pressure estimateIn this Section we prove the pressure estimate in terms of the global F pα,λ(R3)norm of u with p = 2, defined in (2.1.6).We will use the following function (defined in (2.1.12)) in the pressureestimate:an(t) = ∥u(t)∥2F 2α,λ(R=2n) = supx∈R31rλx∫B(x,rx)|u(·, t)|2, rx = 18max(2n, |x|)α.(2.4.1)Lemma 2.4.1 (Annulus integral). Fix n ∈ N0 and let an(t) be defined as(2.4.1). For integer l ≥ n, we have∫2l<|y|<2l+1|uiuj(y, t)| dy ≤ C2l(αλ+3−3α)an(t), (2.4.2)where C is independent of n and l. Moreover, we have∫B(0,2n)|uiuj(y, t)| dy ≤ C2n(αλ+3−3α)an(t). (2.4.3)402.4. Pressure estimateProof. By Lemma 2.3.1, we can coverA2l byN(2l) ballsBl,k, k = 1, . . . , N(2l),with center xl,k insideA2l and radius rxl,k ∼ 2lα. Hence the integral in (2.4.2)is bounded byN(2l)∑k=1∫Bl,k|uiuj(y, t)| dy ≲N(2l)∑k=12lαλan(t) ≲ 2l(3−3α)2lαλan(t).The integral in (2.4.3) is bounded similarly.For a given ball B of radius r, let B∗ and B∗∗ denote concentric balls ofradii 98r and54r, respectively. Recall the set Bn of balls defined for n ∈ N0in (2.1.11),Bn ={B(z, rz) : z ∈ R3}, rx =18max(2n, |x|)α.Lemma 2.4.2 (Pressure estimate). Fix n ∈ N0. Assume u is a local energysolution to (2.1.1) on R3 × [0, T ) with an associated pressure p. Then, fort ∈ (0, T ) and any ball B ∈ Bn,1|B|1/3∫ t0∫B∗|p− pB(s)|3/2 dx ds≤ C supB′∈Bn;B′∩B∗∗ ̸=∅(1|B|1/3∫ t0∫B′|u|3 dx ds)+ C|B| 12λ− 56∫ t0an(s)3/2 ds,(2.4.4)where C is independent of n and B.Remark. The factor of the last term in [4, Lemma 3.1] is approximately|Q|λ/2−5/6(log ⟨|Q|1/3/2n⟩)3/2 which agrees with |B| 12λ− 56 when α = 1 up toa log factor. In fact, the log factor in [4] can be avoided, as shown by ourproof. The log factor in [4] appears because its proof replaces |Q′|λ/3 by|Q|λ/3 and count all cubes together. Our proof counts balls in each annulusand then takes the sum. The size of |Q′|λ/3 in each annulus is differentand hence the sum is a geometric series. The idea of using a convergence(non-geometric) series also helps to improve the J2b estimate (2.4.6) below.Proof. For a given ball B = B(z, rz) ∈ Bn, we have the local pressure412.4. Pressure estimatedecomposition for (x, t) ∈ B∗ × (0, T ),p(x, t)− pB(t) = −13|u|2(x, t) + p.v.∫y∈B∗∗Kij(x− y)uiuj(y, t) dy+∫y/∈B∗∗[Kij(x− y)−Kij(z − y)]uiuj(y, t) dy=: pnear(x, t) + pfar(x, t).(2.4.5)For pnear, the Caldero´n-Zygmund inequality implies1|B|1/3∫ t0∫B∗|pnear|3/2 dx ds ≤ C|B|1/3∫ t0∫B∗∗|u|3 dx ds.Since B∗∗ can be covered by N balls B′ ⊂ Bn with N independent of n andB, no matter whether |z| > 2n+1 or not, we have1|B|1/3∫ t0∫B∗∗|u|3 dx ds ≤ C supB′∈Bn,B′∩B∗∗ ̸=∅1|B|1/3∫ t0∫B′|u|3 dx ds.For pfar, we have|pfar(x, t)| ≲∫y/∈B∗∗rz|z − y|4 |uiuj(y, t)| dy.We first consider B = B(z, rz) with |z| > 2n+1. Choose integer m > n sothat 2m < |z| ≤ 2m+1. We have|pfar(x, t)| ≲{∫I+∫II+∫III}rz|z − y|4 |uiuj(y, t)| dy = J1 + J2 + J3.whereI = {y : |y| < 2m−1}II = {y : 2m−1 ≤ |y| ≤ 2m+2, y /∈ B∗∗}III = {y : |y| > 2m+2}.For J1, we haveJ1 ≲rz|z|4∫I|uiuj(y, t)| dy.If m ≥ n+ 2,J1 ≲rz|z|4(∫B(0,2n)+m−2∑l=n∫2l<|y|<2l+1)|uiuj(y, t)| dy.422.4. Pressure estimateBy Lemma 2.4.1,J1 ≲ 2m(α−4)(2n(3−3α+αλ)an(t) +m−2∑l=n2l(αλ+3−3α)an(t))≲ 2m(αλ−2α−1)an(t).If m = n+ 1, we do not need the sum and have the same bound.For J3, by Lemma 2.4.1,J3 ≲∫IIIrz|y|4 |uiuj(y, t)| dy≲∞∑l=m+2∫2l<|y|<2l+12mα24l|uiuj(y, t)| dy≲∞∑l=m+22mα24l2l(αλ+3−3α)an(t) =∞∑l=m+22mα2l(αλ−1−3α)an(t).The sum converges since αλ < 1 + 3α (which is always valid if λ ≤ 3), andwe getJ3 ≲ 2m(αλ−2α−1)an(t).For J2, we further divide the region II to two parts:IIa = {y ∈ II, |y − z| ≥ 2m−1}, IIb = {y ∈ II, rz < |y − z| < 2m−1},and decompose J2 = J2a + J2b over these two subregions. The estimate ofJ2a is the same as the integral over the annulus A2m+1 which is part of J3.For J2b, we cover IIb by K annuliIIb ⊂ ∪Kk=1Ak, Ak = {y : krz < |y − z| ≤ (k + 1)rz} ,with K = ⌈2m−1/rz⌉ ≤ C2m(1−α). Each annulus Ak can be covered by Nkballs Bk,j = B(xk,j , rxk,j ), j = 1, . . . , Nk with Nk ≲ k2 since |xk,j − z| ∼ krzand rxk,j ∼ rz.By the definition (2.4.1) of an(t), we haveJ2b ≲K∑k=1Nk∑j=1rz(krz)4∫Bk,j|uiuj(y, t)| dy≲K∑k=1k2rz(krz)4rλz an(t) ≲ rλ−3z an(t).(2.4.6)432.5. A priori boundsCombining all together the final estimates for J1, J2a, J2b, and J3,J1 + J2a + J3 ≲ 2m(αλ−2α−1)an(t), J2b ≲ rλ−3z an(t).We see that J2b is the dominant term, and we conclude, when |z| > 2n+1,|pfar(x, t)| ≲ rλ−3z an(t). (2.4.7)For the case B = B(z, rz), |z| ≤ 2n+1, we have|pfar(x, t)| ≲{∫I+∫II}rz|z − y|4 |uiuj(y, t)| dy = I1 + I2,whereI = {y : y /∈ B∗∗, |y| < 2n+2}, II = {y : y /∈ B∗∗, |y| > 2n+2}.Term I2 is estimated similarly to J3 and has the same boundI2 ≲ 2n(αλ−1−2α)an(t).Term I1 is estimated similarly to J2 and has the same bound (2.4.6). Thusfor all cases, we have estimate (2.4.7) for pfar(x, t), and1|B|1/3∫ t0∫B∗|pfar|3/2 dx ds ≲ 1|B|1/3∫ t0|B| · (r(λ−3)z an(s))3/2 ds= |B|2/3|B| 12 (λ−3)∫ t0an(s)3/2 ds.(2.4.8)Combining these estimates yields (2.4.4), and the proof is concluded.2.5 A priori boundsThe next is just [4, Lemma 3.2].Lemma 2.5.1. Let u : R3 × (0, T ) and ε > 0 be given. Then, there exists aconstant C(ε) so that, for any ball B ⊂ R3,1|B|1/3∫ t0∫B|u|3 dx ds ≤ C(ε)|B|λ−4/3∫ t0(1|B|λ/3∫B|u|2 dx)3ds+ ε∫ t0∫B|∇u|2 dx ds+ C|B|λ/2−5/6∫ t0(1|B|λ/3∫B|u|2)3/2ds.442.5. A priori boundsFor a ball B = B(x, rx) we introduce a cut-off function ϕB which issupported on B∗ and identically 1 on B. All such cut-off can be obtainedby scaling and translating a single cut-off, in which case we have |DkϕB| ≲r−kx ∼ |B|−k/3 where suppressed constants are independent of x.Recall (2.1.12) and (2.1.13) thatan(t) = supB∈Bn1|B|λ/3∫B|u(x, t)|2 dx,bn(t) = supB∈Bn1|B|λ/3∫ t0∫B|∇u(x, s)|2 dx ds.Lemma 2.5.2 (A priori inequalities). Assume u0 ∈ F 2α,λ(Rn) is divergencefree and let (u, p) be a local energy solution with initial data u0 on R3×(0, T ).Let it also satisfy the following:ess sup0<s<tan(s) + bn(t) <∞, ∀t < T. (2.5.1)Then for all t ∈ (0, T ) we have the following a priori estimatean(t) + bn(t) ≤ Can(0) + C2−2nα∫ t0an ds+ C22nα(λ−2)∫ t0a3n ds. (2.5.2)Proof. Fix B ∈ Bn. The local energy inequality reads∫|u(t)|2ϕB + 2∫ t0∫|∇u|2ϕB ≤∫|u0|2ϕB+∫ t0∫|u|2∆ϕB +∫ t0∫ (|u|2u · ∇ϕB + 2(p− pB)u · ∇ϕB).Observe that ∫ t0∫|u|2∆ϕB ≲ |B|−2/3∫ t0∫B∗|u|2.But, B∗ can be covered by N balls from Bn with N independent of n,B,and so∫ t0∫|u|2∆ϕB ≲ N |B|−2/3 supB′∈Bn;B′∩B∗ ̸=∅∫ t0∫B′|u|2 ≲ |B|(λ−2)/3∫ t0an.452.5. A priori boundsCombining this with Lemmas 2.4.2 and 2.5.1 we obtain|B|−λ/3∫|u(t)|2ϕB + 2|B|−λ/3∫ t0∫|∇u|2ϕB≤ |B|−λ/3∫|u0|2ϕB + C|B|−2/3∫ t0an ds+ Cε|B|2λ/3−4/3∫ t0a3n ds+ C|B|λ/6−5/6∫ t0a3/2n ds+ Cεbn(t).(2.5.3)By Young’s inequality for products,|B|λ−56 a3/2n = (|B|−12a3/4n ) · (|B|λ−26 a3/4n ) ≤ (|B|−12a3/4n )4/3 + (|B|λ−26 a3/4n )4.(2.5.4)Hence the second last term in (2.5.3) can be dropped.Thanks to the assumption λ ≤ 2, all exponents of |B| in (2.5.3) arenonpositive. Observe that the smallest ball in Bn has radius ∼ 2nα. Hence,|B|−1 ≲ 2−3nα.Taking sup of (2.5.3) over B ∈ Bn, we obtainan(t)+2bn(t) ≤ Can(0)+C2−2nα∫ t0an ds+Cε22nα(λ−2)∫ t0a3n ds+Cεbn(t).Choosing ε = 1/C, we can absorb the last term by the left side and get(2.5.2).Remark 2.5.1. The condition λ ≤ 2 is essential in taking the sup of (2.5.3).If it fails, we cannot not have the a priori bound (2.5.2).Next we will use the foollowing version of Gro¨nwall lemma.Lemma 2.5.3. Suppose f(t) ∈ L∞([0, T ]; [0,∞)) satisfies, for some m ≥ 1,f(t) ≤ a+∫ t0(c1f(s) + c2f(s)m) ds for 0 < t < T, (2.5.5)where a, c1, c2 > 0 then for T0 = min(T, T1), withT1 =a2ac1 + (2a)mc2we have f(t) ≤ 2a for t ∈ (0, T0).462.6. Local and global existence of weak solutionsThe proof is essentially that of [7, Lemma 2.2] although we do not takec1 = c2. Note that f(t) may be discontinuous.Our goal is to get uniform a priori estimates for local energy solutionsin classes F 2α,λ:Theorem 2.5.4 (A priori estimates). Assume the condition (2.1.9) for αand λ. Assume u0 ∈ F 2α,λ(Rn) is divergence free and let (u, p) be a localenergy solution with initial data u0 on R3 × (0, T ), withess sup0<s<tan(s) + bn(t) <∞ (2.5.6)for all t < T , for one fixed n ∈ N0. Then there exist C0, C1 independent ofn such thatess sup0<s<T ∗nan(s) + bn(T∗n) ≤ C0∥u0∥2F 2α,λ(R=2n), (2.5.7)where T ∗n = min(T, Tn) andTn = C1(2−2nα + 22nα(λ−2)∥u0∥4F 2α,λ(R=2n))−1. (2.5.8)Proof. We will apply Lemma 2.5.3 to the apriori estimate we got in Lemma2.5.2:an(t) ≤ Can(0) +∫ t0(c1(n)an(s) + c2(n)an(s)3) ds,where c1 = C2−2nα and c2 = C22nα(λ−2). Therefore due to Lemma 2.5.3 weget thatan(t) ≤ 2Can(0), ∀0 < t < min(T, Tn)where Tn = [2c1 + 2(2Can(0))2c2]−1, with C independent of n, u0. Thebound for bn(t) then follows from (2.5.2) of Lemma 2.5.2.2.6 Local and global existence of weak solutionsWe now prove our main theorem, Theorem 2.1.3, on the local and globalexistence of weak solutions with initial data from F˚ 2α,λ.Proof of Theorem 2.1.3. If α = 0, then we are back to local energy solutionsin E2 constructed by [18], with details given by [15]. In the following weassume α > 0.472.6. Local and global existence of weak solutionsWe first show local existence. Take j ∈ N. Since u0 ∈ F˚ 2α,λ,σ there existsuj0 ∈ L2σ(R3) such that ∥u0 − uj0∥F 2α,λ(R=1) ≤1j . Let (uj , p¯j) be a globalLeray solution for initial data uj0 in the energy class. By Theorem 2.5.4, forn ∈ N0, uj is bounded uniformly in L∞(0, Tn;F 2α,λ(2n)) and satisfiesess sup0<s<TnsupB∈Bn1|B|λ3∫B|uj(s)|2+ supB∈Bn1|B|λ3∫ Tn0∫B|∇uj |2 ≤ C0∥u0∥2F 2α,λ(2n)+εn,j ,(2.6.1)where Tn given by (2.5.8), and εn,j = C0∥u0− uj0∥2F 2α,λ(2n) → 0 as j →∞ byLemma 2.3.2.Now let T ∗n = maxnj=1 Tj = Tj(n) for some j(n) ≤ n. Let T+ = supn∈N Tn.We have limn→∞ T ∗n = T+. By the uniform bound (2.6.1), the sequence ujis also bounded locally in energy space on B× (0, T ∗n) for every ball B ⊂ R3,since B can be covered with finite number of balls Brx(x) correspondingto the space F 2α,λ(2j(n)). Denote u0,k = uk. By the uniform bound andinduction, for any n ∈ N we can construct a subsequence {un,k}k∈N from{un−1,k}k∈N which converges to a vector field un on B2n × (0, T ) as k →∞in the following sense:un,k∗⇀ un in L∞(0, T ∗n ;L2(B2n)),un,k ⇀ un in L2(0, T ∗n ;H1(B2n)),un,k → un in L3(0, T ∗n ;L3(B2n)).(2.6.2)Denote u˜n as a 0 extension of un to R3 × (0,∞). From our constructionu˜n = u˜n−1 on B2n−1 × (0, T ). Let u = limn→∞ u˜n. Then, u = un onB2n × (0, T ∗n) for every n ∈ N. Let u(k) = uk,k on B2k × (0, T ∗n) and 0elsewhere. Then for any n ∈ N we have the following convergence as k →∞:u(k)∗⇀ u in L∞(0, T ∗n ;L2(B2n)),u(k) ⇀ u in L2(0, T ∗n ;H1(B2n)),u(k) → u in L3(0, T ∗n ;L3(B2n)).(2.6.3)Notice that our sequence {u(k)} still satisfies (2.6.1) for all n up to timeTn, therefore due to convergence (2.6.3) the limit function u also enjoys thesame bound for every n ∈ N0ess sup0<s<TnsupB∈Bn1|B|λ/3∫B|u(s)|2 + supB∈Bn1|B|λ/3∫ Tn0∫B|∇u|2 ≤ C0∥u0∥2F 2α,λ(2n).(2.6.4)482.6. Local and global existence of weak solutionsThe pressure is dealt similarly to [16, Section 3] and [4, Section 4]. Frompressure estimate Lemma 2.4.2, as well as (2.6.3) we know that associatedpressure sequence p(k) converge in the following sense for any np(k) ⇀ p in L3/2((0, T ∗n)×B2n), (2.6.5)and the limit satisfies the Navier-Stokes equation (2.1.1). Next we will es-tablish pressure decomposition and its convergence. We follow the sameargument as in [4]: For any B ⊂ R3, T < T+, and any k ∈ N sufficientlylarge, there exists p(k)B (t) ∈ L3/2(0, T ) such that ∀t < T we have the follow-ing:p(k)(x, t)− p(k)B (t) = GBij(u(k)i u(k)j ), (2.6.6)whereGBij(u(k)i u(k)j ) = −13|u(k)(x)|2 + p.v.∫y∈B∗∗Kij(x− y)(u(k)i (y, s)u(k)j (y, s)) dy+∫y/∈B∗∗(Kij(x− y)−Kij(xB − y))(u(k)i (y, s)u(k)j (y, s)) dy.(2.6.7)Here we apply the convergence of u(k) in energy space, properties of singularintegrals, and the a priori bounds (2.6.1) and (2.6.4) to prove thatGBij(u(k)i u(k)j )→ GBij(uiuj) in L32 (0, T ;L32 (B)). (2.6.8)Indeed, choose n so that T ≤ Tn, choose N ≥ n large enough so thatB∗∗ ⊂ B(0, 2n), and split GBij(u(k)i u(k)j ) =M (k)1 +M (k)2 +M (k)3 , whereM(k)1 = −13|u(k)(x)|2 + p.v.∫y∈B∗∗Kij(x− y)(u(k)i u(k)j (y, s))M(k)2 =∫y∈B(0,2N )\B∗∗(Kij(x− y)−Kij(xB − y))(u(k)i u(k)j (y, s)) dyM(k)3 =∞∑m=N∫y∈B(0,2m+1)\B(0,2m)(Kij(x− y)−Kij(xB − y))(u(k)i u(k)j (y, s)) dy,where N ∈ N and N ≥ n. We will use similar notations M1,M2,M3 butfor function u instead of u(k). First we will prove smallness of |M (k)3 −M3|.After applying annulus covering Lemma 2.4.1 using balls from Bn, we get492.6. Local and global existence of weak solutionsthat∞∑m=N∫y∈B(0,2m+1)\B(0,2m)(Kij(x− y)−Kij(xB − y))∣∣u(k)i u(k)j (y, s)− uiuj(y, s)∣∣ dy≲∞∑m=N∫y∈B(0,2m+1)\B(0,2m)|B|1/324m∣∣u(k)i u(k)j (y, s)− uiuj(y, s)∣∣ dy≲∞∑m=N|B|1/324m2m(αλ+3−3α)(∥u∥2F 2α,λ(2n) + ∥u(k)∥2F 2α,λ(2n)) ≲|B|1/32N(αλ−1−3α)∥u0∥2F 2α,λ(2n).In the last inequality we have used s ≤ T ≤ Tn, the a priori bounds (2.6.1)and (2.6.4), and taking k sufficiently large (independent of ε) so that εn,k in(2.6.1) is bounded using Lemma 2.3.2,εn,k = C0∥u0 − uk0∥2F 2α,λ(2n) ≤ C0∥u0 − uk0∥2F 2α,λ(1)2n(1−α)α(3−λ)≤ C0k−22n(1−α)α(3−λ) ≤ ∥u0∥2F 2α,λ(2n).Therefore, for any ε > 0 we can find large enough n so that∥M (k)3 −M3∥L3/2((0,T )×B) ≤ C(B, T )2N(αλ−1−3α)∥u0∥2F 2α,λ(2n)≤ C(B, T )2N(αλ−1−3α)∥u0∥2F 2α,λ(1)C2n(1−α)α(3−λ) ≤ ε(2.6.9)using Lemma 2.3.2 again, N ≥ n, and (αλ− 1− 3α) + (1− α)α(3− λ) < 0.After choosing N we apply (2.6.3) for M(k)1 ,M(k)2 to find k large enough sothat∥M (k)1 −M1∥L3/2((0,T )×B) + ∥M (k)2 −M2∥L3/2((0,T )×B) ≤ ε. (2.6.10)This proves convergence (2.6.8). Lastly, in any fixed domain B × [0, T ] pair(u,GBij(uiuj)) solves (2.1.1). Therefore, ∇p = ∇GBij(uiuj) in D′(R3) at everytime t ≤ T and so there exists a function of time pB(t) such thatp(x, t) = GBij(uiuj)(x) + pB(t), for (x, t) ∈ B × [0, T ].Also from this identity we get pB(t) ∈ L 32 (0, T ). This finishes proof ofpressure decomposition and convergence.Next we need to establish local energy estimate using the above conver-gence (2.6.3) for u(k) → u and (2.6.5) for p(k) → p in L3/2loc . Since {u(k)} is a502.7. Eventual regularitysubsequence of {uj} which are global Leray solutions we have local energyinequalities for (u(k), p(k)). For any ϕ ∈ C∞c (R3 × (0, T+)), ϕ ≥ 0,∫|u(k)(x, t)|2ϕ(x, t) dx+ 2∫ t0∫|∇u(k)(x, s)|2ϕ(x, s) dx ds≤∫ t0∫|u(k)(x, s)|2(∂tϕ(x, s) dx ds+∆ϕ(x, s)) dx ds+∫ t0∫(|u(k)(x, s)|2 + 2p(k)(x, s))(u(k)(x, s) · ∇ϕ(x, s)) dx ds.(2.6.11)We can find sufficiently large n so that supp{ϕ} ⊂ B2n×[0, T ∗n ] and applyingthe convergence of u(k) in energy space and p(k) in L3/2 on B2n we get thelocal energy inequality for (u, p) which finishes the proof of local existence.For global existence, we want to ensure that T+ = lim supn→∞ Tn =∞.By the definition (2.5.8) of Tn, it is equivalent tolim infn→∞ 2nα(λ−2)∥u0∥2F 2α,λ(R=2n) = 0.It suffices to have (2.1.10),limρ→∞ ρα(λ−2)∥u0∥2F 2α,λ(ρ) = 0.This condition is assumed when α(3 − λ) < 1 by Theorem 2.1.3. Whenα(3 − λ) > 1, by Lemma 2.3.2 with C2 ≲ ρα(1−α)(3−λ), for u0 ∈ F 2α,λ with∥u0∥2F 2α,λ(R=1) = A,ρα(λ−2)∥u0∥2F 2α,λ(ρ) ≲ ρα(λ−2)ρα(1−α)(3−λ)A = (ρα)1−3α+αλA,which vanishes as ρ → ∞. When α(3 − λ) = 1, This condition is moredelicate, and is proved by Lemma 2.3.8 for u0 ∈ F˚ 2α,λ. This finishes theproof of global existence.2.7 Eventual regularityIn this Section we consider eventual regularity of our weak solutions, andgive a proof of Theorem 2.2.1. The proof requires a variant of the Caffarelli-Kohn-Nirenberg regularity criteria [10] due to Lin [22]; see also [17].512.7. Eventual regularityLemma 2.7.1 (ε-regularity criteria). For any σ ∈ (0, 1), there exists auniversal constant ε∗ = ε∗(σ) > 0 such that, if a pair (u, p) is a suitableweak solutions of (2.1.1) in Zr = Zr(x0, t0) = Br(x0)× (t0 − r2, t0), andε3 =1r2∫Zr(|u|3 + |p|3/2) dx dt < ε∗,then u ∈ L∞(Zσr). Moreover,∥∇ku∥L∞(Zσr) ≤ Ckε r−k−1, k ∈ N0for universal constants Ck = Ck(σ).There are many variants of Lemma 2.7.1, which involve different localspace-time norms. The following lemma demonstrates how to interpolatesuch space-time norms.Lemma 2.7.2. Fix x and t and suppose B(x,√t) ⊂ B(x, rx) ∈ Bn. Letθ = 3(1/2− 1/p) ∈ [0, 1] and θq ≤ 2. Then,∫ t0(∫B(x,√t)|u|p dx) qpds ≲ rλq/2x t1−θq/2(an(t)(1−θ)q/2 bn(t)θq/2 + an(t)q/2).Proof. Let B = B(x,√t). By the Gagliardo-Nirenberg and Ho¨lder inequal-ities we have∫ t0(∫B(x,√t)|u|p dx) qpds ≲ t1−θq/2 sup0<τ<t∥u(τ)∥(1−θ)q/2L2(B)(∫ t0∥∇u∥2L2(B))θq/2+ |B|q(1/p−1/2)∫ t0∥u∥qL2(B)ds.Inserting the correct powers of rx and noting B(x,√t) ⊂ B(x, rx) ∈ Bncompletes the proof.Our strategy will be to divide the estimate in Lemma 2.7.2 by t1+q(3/p−1)/2,which makes the left-hand side dimensionless. On the right-hand side, thisleads to the following pre-factor on the leading order term when |x| ≥ 2n,|x|αλq/2 t−q(3/p−1)/2−θq/2.To close our argument, this will need to be bounded by 1. Working out thecancellations results in the requirement that|x|2αλ ≤ t.522.7. Eventual regularityThis condition is independent of p and q. We will therefore choose p = q = 3in what follows.The remainder of this Section is entirely devoted to the proof of Theorem2.2.1.Proof of Theorem 2.2.1. Fix x and t and suppose B(x,√t) ⊂ B(x, rx) ∈ Bn.This holds, e.g., if√t ≤ 2nα−3, which we assume to be true. We have byLemma 2.7.2 and Theorem 2.5.4 that1t∫ t0∫B(x,√t)|u|3 dx ds ≲ r3λ/2x t−3/4an(t)3/4bn(t)3/4 + r3λ/2x t−3/4an(t)3/2≲ r3λ/2x t−3/4an(0)3/2.(2.7.1)In the preceding application of Theorem 2.5.4, we need to have Tn ≥ t. Thisis true if Tn ≥ 22nα−6, at least for large n. This holds provided u0 ∈ F˚2α,λ,λ ≤ 1, as can be seen by the estimate64C1 ≥ 22nα(2−2nα + 22nα(λ−2)∥u0∥4F 2α,λ(R=2n)) = 1 + 22nα(λ−1)∥u0∥4F 2α,λ(R=2n).For the vanishing of (2.7.1), we requirer3λ2x t−3/4 ≤ 1, i.e., r2λx ≤ t.Note that an(0)3/2 → 0 because u0 ∈ F˚2α,λ.Before we apply ϵ-regularity we must similarly bound the pressure. Thepressure is controlled using Lemma 2.4.2 noting that this implies, for B¯ =B(x, rx), that1t∫ t0∫B(x,√t)|p− pB(s)|3/2 dy ds≲ supB′∈Bn;B′∩B¯∗∗ ̸=∅1t∫ t0∫B′|u|3 dy ds+ 1t|B¯|λ2− 12an(0)3/2.Note that we replaced p in Lemma 2.7.1 with p−pB(s) because ∇p in (2.1.1)is equal to ∇(p − pB(s)). Observe that if B′ ∈ Bn, B′ ∩ B¯∗∗ ̸= ∅, and yis the center of such a ball, then |y| ≲ |x| with the suppressed constantindependent of n. Applying the Gagliardo-Nirenberg inequality to B′, wehave for t < Tn that∫ t0∫B′|u|3 dx ds ≲ |B′|λ2 t1/4an(t)3/4bn(t)3/4 + |B′|λ−12 tan(t)3/2.532.7. Eventual regularityHence,1t∫ t0∫B′|u|3 dy ds ≲ r3λ2x t−3/4an(0)3/2 + r3λ−12x an(0)3/2.Because r2λx ≤ t, λ ≤ 1 and u0 ∈ F˚2α,λ, the preceding terms can be madesmall by taking n large.Combining these estimates we see that1t∫ t0∫B(x,√t)|u|3 dy ds+1t∫ t0∫B(x,√t)|p−pB(s)|3/2 dy ds < Can(0)3/2 < ε∗,provided n is large enough and r2λx ≤ t ≤ 22nα−6. Let n∗ ∈ N be thesmallest such value of n so that the second inequality holds. Then, applyingTheorem 2.7.1 in B(x,√t) × [0, t] for |x|αλ ≤ √t implies u is regular inB(x, σ√t)× [(1− σ2)t, t], where we will specify σ momentarily, with∥u(x, t)∥L∞(B(x,σ√t)×[(1−σ2)t,t]) ≲ (σ√t)−1.This holds in particular for t = 22n∗α−6. Repeating this for n ≥ n∗ we findthat u is regular in⋃n≥n∗⋃|x|2λα≤22nα−6B(x, σ2nα−3)× [(1− σ2)22nα−6, 22nα−6].Choose σ so that(1− σ2)22nα−6 = 22(n−1)α−6,which ensures there are no time-gaps in the preceding double union. If (x, t)satisfies |x|2αλ ≤ t, t > M and n is chosen so that (1 − σ2)22nα−6 ≤ t ≤22nα−6, then (x, t) ∈ ∪|x|2λα≤22nα−6B(x, σ2nα−3) × [(1 − σ2)22nα−6, 22nα−6].Hence u is regular within the set{(x, t) : |x|2αλ ≤ t; t ≥M},whereM = (1−σ2)22n∗α−6 and additionally satisfies |u(x, t)| ≲ √t−1. Thisfinishes the proof of Theorem 2.2.1.Remark 2.7.1. When 1 < λ ≤ 2, observe that in the argument above werequired the following:limn→∞ 2nα(λ−1)an(0) = 0; limn→∞ rλxt−1/2an(0) = 0; limn→∞ rλ−1x an(0) = 0.(2.7.2)542.7. Eventual regularityIfan(0) ≤ o(1)2nα(1−λ),then we have smallness whenrx ≤ 2nα, rλxt−1/2 ≲ 2nα(λ−1).This leads to an algebraic region of the form, for example,{(x, t); |x|2α ≲ t; 1 ≲ t}.However, this does not give a new result compared to Theorem 2.2.1 becausethe conditionlimn→∞ 2nα(λ−1)an(0) = 0,implies u0 ∈ F˚2α,1, as we presently explain. Note that an is defined withrespect to F 2α,λ. Suppose f satisfies the preceding condition. If |x| ≤ 2n,then12nα∫B(x,2αn/8)|f |2 dx = 2nα(λ−1)2nαλ∫B(x,2αn/8)|f |2 dx ≤ 2nα(λ−1)an(0).On the other hand, if |x| > 2n, then let k ≥ n satisfy 2k ≤ |x| < 2k+1. Then,1|x|α∫B(x,|x|α/8)|f |2 dx ≲ 12αk∫B(x,|x|α/8)|f |2 dx.We know supk≥n 2kα(λ−1)ak(0) <∞. So,12αk∫B(x,|x|α/8)|f |2 dx ≲ 2kα(λ−1)2kαλ∫B(x,2α(k+1)/8)|f |2 dx ≤ supk≥n2kα(λ−1)ak(0) <∞.Hence, f ∈ F 2α,1(2n) with∥f∥2F 2α,1(2n) ≤ supk≥n2kα(λ−1)ak(0). (2.7.3)Observing that the right-hand side above vanishes as n → ∞, we concludef ∈ F˚2α,1.55Chapter 3Global Navier-Stokes flows inweighted spacesIn this chapter we obtain a global existence result for the three-dimensionalNavier-Stokes equations with a large class of data allowing growth at spatialinfinity. Our work is a continuation of the results by T.-P. Tsai, Z. Bradshaw,I. Kukavica [3] and [4] and proves global existence of local energy solutionswith initial data in different weighted spaces as well as eventual regularity.3.1 IntroductionLet (u, p) be a solution the three dimensional Navier-Stokes equations in thesense of distributions:{∂tu−∆u+ (u · ∇)u+∇p = 0,div u = 0in R3 × (0, T ) (3.1.1)For T > 0 and initial data u(x, 0) = u0(x). One of the first results aboutglobal in time existence of weak solutions was done by J.Leray in [21] withdivergence free initial data u0 ∈ L2(R3). Later in [18] Lemarie´-Rieussetproved the similar result for the local initial data u0 ∈ L2uloc. Where thelocal space is defined as following:∥u0∥L2uloc = supx∈R3∥u0∥L2(B(x,1)) <∞. (3.1.2)The focus of this work is on finding a different class for initial data, for whichwe can prove global in time existence, for that we need to define local-energysolutions. We will use the following notations, for any cube Q ∈ R3 withradius r we will denote Q∗ a slightly bigger cube with the same center andradius 5r/4 and Q∗∗ is similarly a small extension of Q∗. Next we introducethe definition of local-energy solutions:Definition 3.1.1. A vector field u ∈ L2loc(R3 × (0, T )), where T > 0 is alocal energy solution to (3.1.1) with initial data u0 ∈ L2loc(R3) if the followingholds:563.1. Introduction u ∈ ∩R>0L∞(0, T ;L2(BR(0))), ∇u ∈ L2loc(R3 × [0, T )), there is p ∈ L32loc(R3×[0, T )) such that (u, p) is NSE solution in a senseof distributions, For all compacts K ⊂ R3 we have u(t)→t→0 u0 in L2(K), u satisfies local energy inequality, for all ϕ ∈ C∞0 (R3 × (0, T )), ϕ ≥ 0,2∫ ∫|∇u|2ϕ dx dt ≤∫ ∫|u|2(∂tϕ+∆ϕ) dx dt+∫ ∫(|u|2 + 2p)(u · ∇ϕ) dx dt,(3.1.3) the function t 7→ ∫ u(x, t)w(x) dx is continuous on [0, T ) for any w ∈L2(R3) with a compact support, for every cube Q ⊂ R3, there exists pQ(t) ∈ L 32 (0, T ) such that forx ∈ Q∗ and 0 < t, T ,p(x, t)− pQ(t) = −13|u(x)|2 + p.v.∫y∈Q∗∗Kij(x− y)(ui(y, s)uj(y, s)) dy+∫y/∈Q∗∗(Kij(x− y)−Kij(xQ − y))(ui(y, s)vj(y, s)) dy,(3.1.4)where xQ is the center of Q and Kij = ∂i∂j(4π|y|)−1.This definition was used in [4] and differs from previous definitions in[7, 14, 15, 18, 26] since we dont require u0 ∈ L2uloc. Nevertheless local energysolutions first were proven to exist locally in time for initial data in L2uloc(Ω)for Ω = R3 or R3+. Most of results for global existence included a decayassumption on the data, see [15, 18] for u0 ∈ E2. One of the examples ofsuch conditions is:limR→∞supx0∈R31R2∫BR(x0)|v0|2 dx = 0. (3.1.5)Later global existence for non-decaying data in uniform space L2ulocwas done in two dimensional case by Bason [1]. For initial data in L2loc \L2uloc Fernandez-Dalgo and Lemarie´-Rieusset [13] constructed global solu-tions with the following condition:∫R3|u0(x)|2(1 + |x|)2 dx <∞. (3.1.6)573.1. IntroductionBradshaw and Kukavica [3] constructed local in time solutions for initialdata in the space M˚p,λC which is related to our project and will be definedlater. In [4] with Tai-Peng Tsai they proved global existence in class M˚p,λC, this class strictly includes the initial data from [7, 13].For n ∈ N, let S0 = {x : |xi| ≤ 2; i = 1, 2, 3} and Rn = {x : |xi| ≤2n; i = 1, 2, 3}. Denote Sn = ¯Rn+1 \Rn for n ∈ N \ {1} and S1 = ¯R2 \ S0.Then |Sn| = 56 · 23n. We separate S0 = ∪641 Qk00 into 64 cubes with side-length 1 and Sn = ∪56kn=1Qknn into 56 cubes with side-length 2n, the set ofall these cubes we denote as C = ∪∞n=0 ∪kn Qknn . This set will satisfy thefollowing properties Side-length of each cube is proportional to the distance to the origin. Adjacent cubes have proportional volume. The distance between cubes Q,Q′ is proportional to max{|Q| 13 , |Q′| 13 }. The amount of cubes Q′, such that |Q′| < |Q| is bounded above by|Q| 13Using set C let us introduce the definition of Mp,λC and M˚p,λC :Definition 3.1.2. Let p ∈ [1,∞), λ ≥ 0. We have f ∈Mp,λC if∥f∥pMp,λC:= supQ∈C1|Q|λ3∫Q|f(x)|p dx <∞. (3.1.7)If f ∈Mp,λC and satisfies the following condition then f ∈ M˚p,λC :1|Q|λ3∫Q|f |p dx→ 0, as |Q| → ∞, Q ∈ C (3.1.8)Note that global existence [4] that is proven in M˚2,2C improves the con-structions in [3, 7, 13]. Moreover, using the a priori estimate there was alsoproven eventual regularity property. The goal of our project is to furtherimprove the result of Bradshaw, Kukavica and Tsai [4] for another set ofinitial data, using the set of cubes C let us introduce the following spaces:Definition 3.1.3. Let q ≤ 2 and r > 2, we define spaces F qr :F qr ={f ∈ L2loc(R3)∣∣∣ ∑n,kn( 1|Qknn | q3∫Qknn|f |2 dx) r2<∞}(3.1.9)With the norm ∥f∥rF qr=∑∞n,kn(1|Qknn |q3∫Qknn|f |2 dx) r2583.1. IntroductionThe following is our main theorem on global existence of the solutionswith initial data from F qr :Theorem 3.1.1. Let q ≤ 2, r > 2 and assume u0 ∈ F qr is divergence free.Then there exists u : R3×(0,∞)→ R3 and p : R3×(0,∞)→ R so that (u, p)is a local energy solution to the Navier-Stokes equations on R3 × (0,∞).Both spaces Mp,λC and Fqr are equivalent to certain Herz spaces. LetAk = B2k+1 \B2k where B2k is a ball in R3 with radius 2k. For n ∈ N, s ∈ Rand p, q ∈ (0,∞] the non-homogeneous Herz space Ksp,q(Rn) is the space offunctions f ∈ Lploc(Rn \ {0}) with finite norm:∥f∥Ksp,q =( ∑k∈N02ksq∥f∥qLp(Ak)) 1qif q <∞,supk∈N02ks∥f∥Lp(Ak) if q =∞.(3.1.10)Then the space Mp,λC is equivalent to Herz space K−λp,∞ and Fqr is equivalenttoK−q2,r . Local in time existence of mild solutions with initial data from someHerz spaces was proven by Tsutsui [25] (in case p > 3). Global existence forthe case K−22,∞ follows from [4].We will use the following sequence of norms F q,mr using set of cubes C.Definition 3.1.4. For m ∈ N we denote Q0,m = Rm and for any n ≥1, kn = 1...56, Qknn,m = Qknm+n, where Qknn are cubes of family C. Withrespect to these sets we can denote norms∥f∥rF q,mr =∑n,kn( 1|Qknn,m| q3∫Qknn,m|f |2 dx) r2.We will denote Cm family of cubes in this defintion that corresponds tospace F q,mr . The following lemma shows the relation between spaces Fqr andF q,nr :Lemma 3.1.2. Let u ∈ F qr , then ∥u∥F q,nr →n→∞ 0Proof. We only have to prove the convergence of the first term, correspond-ing to Q0,n, since we know that u ∈ F qr and the rest of the sum convergesto 0:12nq∫B2n|u|2 dx →n→∞ 0.593.1. IntroductionWe denote am,km =(12mq∫Qkmm|u|2 dx) r2, for m ≤ n, km ≤ 64, here Qkmm arecubes in the family C with radius 2m, the following estimate holds12nq∫B2n|u|2 dx = 12nqn−1∑m=0∑km∫Qkmm|u|2 dx =12nqn−1∑m=0∑km2mqam,km2r =n−1∑m=0∑km12(n−m)qam,km2r ≤l−1∑m=0∑km12(n−m)qam,km2r +n−1∑m=l−1∑km12(n−m)qam,km2r .(3.1.11)Where l ≤ n − 1 is an integer that we will choose later. Next we willestimates last two terms of (3.1.11) separatelyl−1∑m=0∑km12(n−m)qam,km2r ≤ C2(n−l)qsupn,kn|an,kn |2r (3.1.12)For the last term we will use Holder inequality:n−1∑m=l−1∑km12(n−m)qam,km2r ≤( n−1∑m=l−1∑km12r(n−m)qr−2) r−2r( ∞∑m=l−1∑kmam,km) 2r ≤≤ C(q, r)( ∞∑m=l−1∑kmam,km) 2r(3.1.13)Next we choose n sufficiently large and l ≈ n2 , with this both terms will besufficiently small, which implies the convergence:12nq∫B2n|u|2 dx →n→∞ 0Using Lemma (3.1.2) and a priori bounds we can establish the followingregularity result:Theorem 3.1.3. (Eventual Regularity) Assume q < 1 and u0 ∈ F qr isdivergence-free, and let (u, p) be a local energy solution on R3 × (0,∞) withinitial data u0. Assume additionally that:∥u(·, t)∥rF qr +∞∑m=1( 1|Qm|q3∫ t0∫Qm|∇u|2 dx dt) r2<∞, (3.1.14)603.2. Pressure estimatefor all t <∞. Then for any δ > 0, there exists a time τ depending on u0, δso that u is smooth on{(x, t), t > max{δ|x|2, τ}} (3.1.15)3.2 Pressure estimateWe use a similar representation of pressure as in [4], in this Section we fixsome integer N and family of cubes corresponding to the space F q,Nr , takeQ as one of the cubes in this family:p(x, t)− pQ(t) = (GQijuiuj)(x, t) , ∀x ∈ Q∗,WhereGijf(x) = −13δijf(x) + p.v.∫y∈Q∗∗Kij(x− y)f(y) dy++∫y/∈Q∗∗(Kij(x− y)−Kij(xQ − y))f(y) dy(3.2.1)The following lemma will be the estimate of pressure in spaces F q,NrLemma 3.2.1. Let q ≤ 2 and (u, p) be a local energy solution and associatedpressure on R3 × [0, T ]. Then for any integer N and cube Q in a family ofcubes corresponding to the space F q,Nr we have the following estimate:1|Q| q+13∫ t0∫Q|p(x, t)− (p)Q(t)|32 dx ds ≤C|Q| 5−q6∫ t0( ∑Q′⊂(Q∗∗)c( 1|Q′| q3∫Q′|u|2 dy) r2) 3rds+C|Q| q+13∫ t0∫Q∗∗|u|3 dx ds,(3.2.2)where Q∗∗ is a cube of radius 54 |Q|13 and the same center as Q.Proof. We separate Gij into two terms:Inear = −13δijf(x) + p.v.∫y∈Q∗∗Kij(x− y)f(y) dy,Ifar =∫y/∈Q∗∗(Kij(x− y)−Kij(xQ − y))f(y) dy.613.2. Pressure estimateNext we apply Calderon-Zigmund estimates for the term Inear:1|Q| 13∫ t0∫Q∗|Inear| 32 dx ds ≤ C|Q| 13∫ t0∫Q∗∗|u|3 dx ds.This term is similar to the one we will have in local energy inequality andwill be estimated in the next Section. Now we will estimate the other termIfar:Ifar ≤ C|Q|13∫R3\Q∗∗1|x− y|4 |u|2 dy ≤≤ C∑Q′∈S1|Q| 13∫Q′∩(Q∗∗)c1|x− y|4 |u|2 dy+C∑Q′∈S2|Q| 13∫Q′∩(Q∗∗)c1|x− y|4 |u|2 dy(3.2.3)Where S1 is the subset of cubes with |Q′| ≤ 8|Q| and S2 are cubes with|Q′| > 8|Q|. There is only finite amount of cubes in S1 and we can alsoassume for them that |Q| 13 ≈ |xQ−xQ′ | due to construction of our family ofcubes C. Moreover, if x ∈ Q and y ∈ Q′, Q′ ∈ S1 then |x−y| ≈ |xQ−xQ′ | ≈|Q| 13 . Hence,∑Q′∈S1|Q| 13∫Q′∩(Q∗∗)c1|x− y|4 |u|2 dy ≤∑Q′∈S11|Q|∫Q′∩(Q∗∗)c|u|2 dy ≤≤ 1|Q|( ∑Q′∈S1( 1|Q′| q3∫Q′|u|2 dy) r2) 2r( ∑Q′∈S1|Q′|qr3(r−2)) r−2r ≤≤ C|Q|1− q3( ∑Q′∈S1( 1|Q′| q3∫Q′|u|2 dy) r2) 2r(3.2.4)On the other hand for Q′ ∈ S2 we have a different relation for x ∈ Q,y ∈ Q′ : |x− y| ≈ |Q′| 13∑Q′∈S2|Q| 13∫Q′1|x− y|4 |u|2 dy ≤ C|Q| 13∑Q′∈S21|Q′| 43∫Q′|u|2 dy ≤≤ C|Q| 13( ∑Q′∈S2( 1|Q′| q3∫Q′|u|2 dy) r2) 2r ·( ∑Q′∈S2( 1|Q′| 4−q3) rr−2) r−2r(3.2.5)The last term in the product is finite only for r > 2 and q < 4 , since allof our cubes have radius greater than |Q| 13 we can calculate the multiplier,623.3. Local Energy Inequalitydenote M = log2(|Q|13 ):∑Q′∈S2( 1|Q′| 4−q3) rr−2= C∑n≥M2−n(4−q)r3(r−2) = C2−M(4−q)r3(r−2)1− 2−(4−q)r3(r−2)= C(q, r)|Q|−(4−q)r3(r−2)(3.2.6)Therefore, we get the estimate for IfarIfar ≤ C|Q|13− (4−q)3( ∑Q′∈S2( 1|Q′| q3∫Q′|u|2 dy) r2) 2r+C|Q|1− q3( ∑Q′∈S1( 1|Q′| q3∫Q′|u|2 dy) r2) 2r ≤C|Q| q3−1( ∑Q′⊂(Q∗∗)c( 1|Q′| q3∫Q′|u|2 dy) r2) 2r.(3.2.7)Take any cube Q in family of cubes corresponding to space F q,Nr , and weget the following estimate for Ifar1|Q| q+13∫ t0∫Q|Ifar|32 dx ds ≤C|Q| q+13 + 3−q2 −1∫ t0( ∑Q′⊂(Q∗∗)c( 1|Q′| q3∫Q′|u|2 dy) r2) 3rds =C|Q| 5−q6∫ t0( ∑Q′⊂(Q∗∗)c( 1|Q′| q3∫Q′|u|2 dy) r2) 3rds.(3.2.8)Combining the estimates for Ifar and Inear we finish the proof of Lemma(3.2.1).3.3 Local Energy InequalityIn the Section we will prove a priori bound for solutions of (3.1.1) in spacesF q,Nr .Lemma 3.3.1. Assume u0 ∈ F qr is divergence free and let (u, p) be a localenergy solution with initial data u0 on R3 × (0, T ). Let us assume that forsome N we have the following∥u(·, t)∥rF q,Nr+∞∑n=1( 1|Qn|q3∫ t0∫Qn|∇u|2 dx dt) r2<∞. ∀t < T. (3.3.1)633.3. Local Energy InequalityThen for all t ∈ (0, T ) we have an a priori estimate∥u(·, t)∥rF q,Nr+∞∑n=1( 1|Qn|q3∫ t0∫Qn|∇u|2 dx dt) r2 ≤ ∥u(·, 0)∥rF q,Nr+C0tr−222rN∫ t0∥u(·, s)∥rF q,Nrds+C1tr−222rN(2−q)∫ t0∥u(·, s)∥3rF q,Nrds(3.3.2)Where C0, C1 are global constants that dont depend on N and Qn are cubesof the family C corresponding to the space F q,Nr .Proof. Similarly to previous Section we choose arbitrary integer N and de-note Qn as one of the cubes in the set corresponding to the space Fq,Nr . Thesolution satisfies the local energy inequality for any compactly supportedpositive test function ϕ ≤ 1 such that ϕ = 1 on Qn, supp{ϕ} ⊂ Q∗n andt ≤ T :∫Qn|u|2(x, t) dx+∫ t0∫Qn|∇u|2 dx dt ≤∫ϕ(x, 0)|u(x, 0)|2 dx+∫ ∫Qn|u|2∆ϕ dx dt+∫ ∫|u|2u · ∇ϕ+ 2(p− (p))u · ∇ϕ dx dt.(3.3.3)Dividing by |Qn|q3 and taking to the power r2 we proceed to:( 1|Qn|q3∫Qn|u|2(x, t) dx) r2+( 1|Qn|q3∫ t0∫Qn|∇u|2 dx dt) r2 ≤C( 1|Qn|q3∫ϕ(x, 0)|u(x, 0)|2 dx) r2+ C( 1|Qn|q3∫ t0∫|u|2∆ϕ dx dt) r2+C( 1|Qn|q3∫ t0∫|u|2u · ∇ϕ+ 2(p− (p))u · ∇ϕ dx dt) r2 ≤C( 1|Qn|q3∫Q∗n|u(x, 0)|2 dx) r2+ C( 1|Qn|q+23∫ t0∫Q∗n|u|2 dx dt) r2+C( 1|Qn|q+13∫ t0∫Q∗n|u|3 dx dt) r2+ C( 1|Qn|q+13∫ t0∫|p− (p)| 32 dx dt) r2.(3.3.4)Here C = C(r) is a uniform constant. For the non-linear term we apply643.3. Local Energy InequalityGagliardo-Nirenberg, Ho¨lder and Young inequalities:1|Qn| 13∫ t0∫Qn|u|3 dx dt ≤ C(ε)|Qn|q− 43∫ t0( 1|Qn|q3∫Qn|u|2 dx)3dt+ε∫ t0∫Qn|∇u|2 dx ds+ C|Qn|q2− 56∫ t0( 1|Qn|q3∫Qn|u|2 dx) 32ds(3.3.5)Plugging the above estimate together with pressure estimate in the equation(3.3.3) we get( 1|Qn|q3∫Qn|u|2(x, t) dx) r2+( 1|Qn|q3∫ t0∫Qn|∇u|2 dx dt) r2 ≤≤( 1|Qn|q3∫Q∗n|u(x, 0)|2 dx) r2+( 1|Qn|q+23∫ t0∫Q∗n|u|2 dx ds) r2++(C(ε)|Qn|2q−43∫ t0( 1|Qn|q3∫Q∗∗n|u|2 dx)3ds) r2+εr2( 1|Qn|q3∫ t0∫Q∗∗n|∇u|2 dx ds) r2++(|Qn|q−56∫ t0( 1|Qn|q3∫Q∗∗n|u|2 dx) 32ds) r2+( C|Qn|5−q6∫ t0( ∑Q′⊂(Q∗∗)c( 1|Q′| q3∫Q′|u|2 dy) r2) 3rds) r2(3.3.6)653.3. Local Energy InequalityNext we use Holder inequality in time:( 1|Qn|q3∫Qn|u|2(x, t) dx) r2+( 1|Qn|q3∫ t0∫Qn|∇u|2 dx dt) r2 ≤≤( 1|Qn|q3∫Q∗n|u(x, 0)|2 dx) r2+ tr−22∫ t0( 1|Qn|q+23∫Q∗n|u|2 dx) r2ds++C(ε)|Qn|r(q−2)3 tr−22∫ t0( 1|Qn|q3∫Q∗∗n|u|2 dx) 3r2ds+εr2( 1|Qn|q3∫ t0∫Q∗∗n|∇u|2 dx ds) r2++|Qn|r(q−5)12 tr−22∫ t0( 1|Qn|q3∫Q∗∗n|u|2 dx) 3r4ds+C|Qn|r(5−q)12tr−22∫ t0( ∑Q′∈C( 1|Q′| q3∫Q′|u|2 dy) r2) 32ds(3.3.7)All of the cubes Qn have radius at least 2N and we substitute |Qn| by 23Nin the following terms:( 1|Qn|q+23∫Q∗n|u|2 dx) r2 ≤ C2rN( 1|Qn|q3∫Q∗n|u|2 dx) r2,|Qn|r(q−2)3 tr−22∫ t0( 1|Qn|q3∫Q∗∗n|u|2 dx) 3r2ds ≤C2rN(2−q)tr−22∫ t0( 1|Qn|q3∫Q∗∗n|u|2 dx) 3r2ds,|Qn|r(q−5)12 tr−22∫ t0( 1|Qn|q3∫Q∗∗n|u|2 dx) 3r4ds ≤C2rN(5−q)4tr−22∫ t0( 1|Qn|q3∫Q∗∗n|u|2 dx) 3r4ds.(3.3.8)Every Q∗∗n intersects with a fixed amount of cubes from the family CN , whichis independent of n, therefore we can substitute integrals over Q∗∗n with sumof integrals over Q′n such that Q′n ∩ Q∗∗n ̸= ∅. Therefore after taking sum663.3. Local Energy Inequalityover all Qn we get the following estimate:∥u(·, t)∥rF qr +∞∑n=1( 1|Qn|q3∫ t0∫Qn|∇u|2 dx dt) r2 ≤∥u(·, 0)∥rF qr +C2rNtr−22∫ t0∥u(·, s)∥rF qr ds+C(ε)C2rN(2−q)tr−22∫ t0∞∑n=1( 1|Qn|q3∫Qn|u|2 dx) 3r2ds++Cεr2∞∑n=1( 1|Qn|q3∫ t0∫Qn|∇u|2 dx ds) r2+C2rN(5−q)4∞∑n=1tr−22∫ t0( 1|Qn|q3∫Q∗∗n|u|2 dx) 3r4ds+∞∑n=1C|Qn|r(5−q)12tr−22∫ t0( ∑Q′∈C( 1|Q′| q3∫Q′|u|2 dy) r2) 32ds.(3.3.9)Next we choose ε sufficiently small to cancel the gradient term and applythe following algebraic inequalities:∞∑n=1( 1|Qn|q3∫Qn|u|2 dx)αr2 ≤( ∞∑n=1( 1|Qn|q3∫Qn|u|2 dx) r2)α(3.3.10)Where α equals to 3 or 32 , we can also calculate the multiplier∑∞n=1C|Qn|r(5−q)12≈C2rN(5−q)4and plugging it into (3.3.9) we get:∥u(·, t)∥rF q,Nr+∞∑n=1( 1|Qn|q3∫ t0∫Qn|∇u|2 dx dt) r2 ≤∥u(·, 0)∥rF q,Nr+C2rNtr−22∫ t0∥u(·, s)∥rF q,Nrds+Ctr−22∫ t012rN(2−q)∥u(·, s)∥3rF q,Nr+12rN(5−q)4∥u(·, s)∥3r2F qrds ≤ ∥u(·, 0)∥rF q,Nr+C0tr−222rN∫ t0∥u(·, s)∥rF q,Nrds+C1tr−222rN(2−q)∫ t0∥u(·, s)∥3rF q,Nrds.(3.3.11)In the last line we also used Young inequality and C0, C1 are global constants.673.4. Global Existence3.4 Global ExistenceThe idea of the proof relies on getting a-priori estimates with help of avariation of Gronwall which we will apply to the norm of u in spaces F q,mr .We use the same lemma as in [4]:Lemma 3.4.1. Suppose f(t) ∈ L∞([0, T ]; [0,∞)) satisfies, for some m ≥ 1,f(t) ≤ a+∫ t0(b1f(s) + b2f(s)m) ds for 0 < t < T, (3.4.1)where a, b1, b2 ≥ 0 then for T0 = min(T, T1), withT1 =a2ab1 + (2a)mb2we have f(t) ≤ 2a for t ∈ (0, T0).Remark 3.4.2. In our case we will apply this lemma for arbitrary largeT ,m = 3 and f(t) = ∥u(·, t)∥rF q,Nr, a = f(0), b1 =C0Tr−222rN, b2 =C1Tr−222rN(2−q) ,which will guarantee the uniform apriori bound up to the time min{T1, T},whereT1 =12C0Tr−222rN+ (2∥u0∥rF q,Nr)2 C1Tr−222rN(2−q)We can take T = n for some large integer n and find N such that T1 > n,as will be shown in the next lemma. For eventual regularity Theorem 3.1.3we will take T = 22N and thereforeT1 =1C02−2N + C12rN(q−1)−2N (2∥u0∥rF q,Nr)2≈ 22N , if q ≤ 1Lemma 3.4.3. Assume u0 ∈ F qr is divergence-free, and let (u, p) be a localenergy solution with initial data u0 on R3 × (0,∞), we also assume that(u, p) satisfies the following for all N :∥u(·, t)∥rF q,Nr+∞∑m=1( 1|Qm|q3∫ t0∫Qm|∇u|2 dx dt) r2 ≤ ∞ ∀t <∞ (3.4.2)Where Qm are cubes in CN . Then for any n ∈ N there exists N and suchthat∥u(·, t)∥rF q,Nr+∞∑m=1( 1|Qm|q3∫ t0∫Qm|∇u|2 dx dt) r2 ≤ c0∥u0∥rF q,Nr ∀t < n(3.4.3)Here c0 is a global constant that does not depend on n.683.4. Global ExistenceProof. We will combine Lemma 3.3.1 and Lemma 3.4.1, take some integern and T = n in Lemma 3.3.1. From local energy estimate for all N we havethe following∥u(·, t)∥rF q,Nr+∞∑n=1( 1|Qn|q3∫ t0∫Qn|∇u|2 dx dt) r2 ≤ ∥u(·, 0)∥rF q,Nr+C0nr−222rN∫ t0∥u(·, s)∥rF q,Nrds+C1nr−222rN(2−q)∫ t0∥u(·, s)∥3rF q,Nrds(3.4.4)For all t ≤ n, therefore we can apply Gronwall lemma with f = ∥u(·, t)∥F q,Nr.Since we know that u0 ∈ F qr and limN→∞∥u0∥F q,Nr = 0 we can find N such thatT1 =12C0nr−222rN+ (2∥u0∥rF q,Nr)2 C1nr−222rN(2−q)> nHere T1 is given to us by Lemma 3.4.1 and therefore we have a uniformbound (3.4.3)Lemma 3.4.4. Assume f ∈ F qr is divergence free then for any ε > 0 thereexists a divergence free g ∈ L2 such that ∥f − g∥F qr ≤ ε and ∥f − g∥F q,mr ≤ εfor any m ∈ NProof. Let ε > 0 and f ∈ F qr is divergence free, we choose R > 0 such that∥fχR3\BR∥F qr ≤ ε. Then we can find smooth cut-off radial function ϕ ≤ 1such that suppϕ ⊂ B2R and ϕ = 1 on BR. Then we can use Bogovskii map[23] to construct function h such thatdiv h = −f∇ϕ, supph ⊂ B2R \BR, ∥h∥W 1,2 ≤ ∥f∇ϕ∥2,B2R\BR . (3.4.5)Moreover, we can estimate the following norm:∫B2R\BR|h|2 dx ≤ C0R2∫B2R\BR|f |2|∇ϕ|2 dx ≤ C0∫B2R\BR|f |2 dx (3.4.6)For some universe constant C0. Next we choose g = fϕ+ h, from (3.4.5)weget that div g = 0. Now we only need to check that ∥f − g∥F qr is small,indeed∥f − g∥F qr ≤ ∥fχR3\BR∥F qr + ∥h∥F qr ≤∥fχR3\BR∥F qr +( ∑Q∈C| Q∩B2R\BR ̸=∅(1|Q|q/3∫Q|h|2 dx)r/2) 1r . (3.4.7)693.4. Global ExistenceNotice that there are only finite amount of cubes in C intersecting B2R \BRand the amount does not depend on R due to properties of C moreoverall such cubes have comparable radius to R, therefore for some universeconstant C we have the following∑Q∈C| Q∩B2R\BR ̸=∅(1|Q|q/3∫Q|h|2 dx)r/2 ≤ ( C|BR|q/3∫B2R\BR|h|2 dx)r/2 ≤(C|BR|q/3∫B2R\BR|f |2 dx)r/2 ≤ C∥fχR3\BR∥F qr ≤ Cε.(3.4.8)Here we used (3.4.6). Therefore g satisfies all the conditions of the lemma.Next we will prove the main theorem:Proof. Take n ∈ N, from lemma (3.4.4) there exists un0 ∈ L2(R3) such that∥u0 − un0∥F qr ≤ 1n and ∥u0 − un0∥F q,mr ≤ 1n for any m ∈ N. Let (un, p¯n) be aglobal Leray solution for initial data un0 . By Lemma (3.4.3) there exists asequence Nn such that Nn > n and un is bounded uniformly on BNn× [0, n].Hence, there exists a sub-sequence u1,k that converges on BN1× (0, 1) in thefollowing sense:u1,k∗⇀ u1 in L∞(0, 1;L2(BN1)),u1,k ⇀ u1 in L∞(0, 1;H1(BN1)),u1,k → u1 in L3(0, 1;L3(BN1)).(3.4.9)By Lemma (3.4.3) the sequence u1,k is also uniformly bounded on BNn×[0, n]for any n ∈ N. Therefore by induction we construct a subsequence {un,k}k∈Nfrom {un−1,k}k∈N which converges to a vector field un on BNn × (0, n) ask →∞ in the following sense:un,k∗⇀ un in L∞(0, n;L2(BNn)),un,k ⇀ un in L∞(0, n;H1(BNn)),un,k → un in L3(0, n;L3(BNn)).(3.4.10)Denote u˜n as a 0 extension of un to R3 × (0,∞). From our constructionu˜n = ˜un−1 on BNn−1 × (0, n − 1). Let u = limn→∞ u˜n. Then, u = un onBNn × (0, n) for every n ∈ N.703.4. Global ExistenceLet u(k) = uk,k on BNk × (0, k) and 0 elsewhere. Then for any n ∈ N wehave the following as k →∞:u(k)∗⇀ u in L∞(0, n;L2(BNn)),u(k) ⇀ u in L∞(0, n;H1(BNn)),u(k) → u in L3(0, n;L3(BNn)).(3.4.11)Using Lemma (3.4.3) we have the following bound for u:sup0<t<n∥u(·, t)∥rF q,Nr+∞∑m=1( 1|Qm|q3∫ n0∫Qm|∇u|2 dx dt) r2 ≤ c0∥u0∥rF q,Nr(3.4.12)The pressure is dealt similarly to [3], indeed we know that there is weak limitp(k) ⇀ p in L3/2((0, n) × QNn), we will check the pressure decompositionformula and strong convergence. Take Q ∈ C, T > 0, then for any k(u(k), p(k)) are global Leray energy solutions and there exists p(k)Q such thatfor all (x, t) ∈ Q× [0, T ]:p(k)(x, t)− p(k)Q = GQij(u(k)i u(k)j ) = −13δiju(k)i u(k)j +p.v.∫y∈Q∗∗Kij(x− y)(u(k)i u(k)j )(y) dy++p.v.∫y/∈Q∗∗(Kij(x− y)−Kij(xQ − y))(u(k)i u(k)j )(y) dy(3.4.13)Choose m > T so that Q∗∗ ⊂ Qm = Q(0, 2m) and separate GQij(uiuj) intotwo termsp(k)1 (x, t) = −13δiju(k)i u(k)j + p.v.∫y∈Q∗∗Kij(x− y)(u(k)i u(k)j )(y) dy++p.v.∫Qm\Q∗∗(Kij(x− y)−Kij(xQ − y))(u(k)i u(k)j )(y) dyp(k)2 (x, t) = p.v.∫y/∈Qm(Kij(x− y)−Kij(xQ − y))(u(k)i u(k)j )(y) dy(3.4.14)We denote p1, p2 as similar terms but where u(k) is substituted by u. To713.4. Global Existenceprove convergence, choose ε > 0 arbitrary small, then|p2 − p(k)2 | ≤∫y/∈Qm(Kij(x− y)−Kij(xQ − y))|u(k)i u(k)j − uiuj |(y) dy ≤∑Q˜∈C |Q˜|1/3≥2m−1∫Q˜(Kij(x− y)−Kij(xQ − y))|u(k)i u(k)j − uiuj |(y) dy ≤∑Q˜∈C |Q˜|1/3≥2m−1∫Q˜|Q|1/3|Q˜|4/3 |u(k)i u(k)j − uiuj |(y) dy.(3.4.15)Here we used that Q∗∗ ⊂ Qm and therefore |Kij(x − y) − Kij(xQ − y)| ≤C |Q|1/3|Q˜|4/3 . Next we will apply Holder inequality∑Q˜∈C |Q˜|1/3≥2m−1∫Q˜|Q|1/3|Q˜|4/3 |u(k)i u(k)j − uiuj |(y) dy ≤|Q|1/3( ∑Q˜∈C |Q˜|1/3≥2m−1|Q˜|(q−4)r3(r−2)) r−2r( ∑Q˜∈C |Q˜|1/3≥2m−1( 1|Q˜|q/3∫Q˜|u(k)i u(k)j − uiuj |(y) dy) r2)2/r ≤|Q|1/3( ∑Q˜∈C |Q˜|1/3≥2m−1|Q˜|(q−4)r3(r−2)) r−2r(∥u∥2F q,mr + ∥u(k)∥2F q,mr ) ≤C|Q|1/32m(q−4)(∥u0∥2F q,mr + ∥u(k)0 ∥2F q,mr ).(3.4.16)Here we used (3.4.3), since we have uniform bound for u, u(k) up to time Twith our choice of m. Notice that q < 4 and u(k)0 converges to u0, thereforewe choose m, k large enough so that ∥u(k)0 ∥2F q,mr ≤ ε+ ∥u0∥2F q,mrandC|Q|1/32m(q−4)(∥u0∥2F q,mr + ∥u(k)0 ∥2F q,mr ) ≤ 2ε (3.4.17)Next we apply (3.4.11) to get large enough k so that∥p(k)1 − p1∥L3/2(Q×[0,T ] ≤ ε (3.4.18)This proves that GQij(u(k)i u(k)j ) → GQij(uiuj) in L3/2(Q × [0, T ]). Lastly, inany fixed domain Q × [0, T ] pair (u,GQij(uiuj) solves (3.1.1). Therefore,723.5. Eventual regularity∇p = ∇GQij(uiuj) in D′(R3) at every time t and so there exists a function oftime pQ(t) such thatp(x, t) = GQij(uiuj)(x) + pQ(t), for (x, t) ∈ Q× [0, T ]Also from this identity we get pQ(t) ∈ L 32 (0, T ). The rest of the proof followsfrom the similar argument to [3], because we have the same convergence ofuk, pk on any Bn × T0 for any fixed T0 > 0.3.5 Eventual regularityIn this Section we will prove Theorem (3.1.3) , we will use one of Caffarelli-Kohn-Nirenberg epsilon-regularity criteria variations, see [10]Lemma 3.5.1. For any σ ∈ (0, 1), there exists a universal constant ε∗(σ) >0 such that, if a pair (u, p) is a suitable weak solution of (3.1.1) in Qr =Br(x0)× (t0 − r2, t0), andε3 =1r2∫Qr(|u|3 + |p| 32 ) dx dt < ε∗, (3.5.1)Then u ∈ L∞(Qσr). Moreover,∥∇ku∥L∞(Qσr) ≤ Ckεr−k−1, ∀k ∈ Z+ (3.5.2)for universal constants Ck = Ck(σ).Next we will prove Theorem (3.1.3)Proof. Choose sufficiently large T,N1 such that ∥u0∥F q,mr < 1 for allm > N1and Lemma 3.4.3 holds for N > N1 and t < T . We take some N > N1 anddenote the following:Q = (−2N , 2N )3. (3.5.3)Note that Q ∈ CN . By lemma (3.2.1),(3.4.3) we have thatJ =1|Q| 23∫ t0∫Q|u|3 + |p− (p)Q(s)| 32 dx dx ≤≤ C|Q| 1−q2∫ t0( ∑Q′∩(Q∗∗ ̸=∅)c( 1|Q′| q3∫Q′|u|2 dy) r2) 3rds+C|Q| 23∫ t0∫Q∗∗|u|3 dx ds,(3.5.4)733.5. Eventual regularityhere cubes Q′ are part of family CN . Next we apply (3.3.5):1|Q| 23∫ t0∫Q|u|3 dx dt ≤ C(ε)|Q|q− 53∫ t0( 1|Q| q3∫Q|u|2 dx)3dt+ε|Q| 13∫ t0∫Q|∇u|2 dx ds+ C|Q| q2− 76∫ t0( 1|Q| q3∫Q|u|2 dx) 32ds(3.5.5)Since q ≤ 1 we can apply Lemma 3.4.3 to get the following:J =1|Q| 23∫ t0∫Q|u|3 + |p− (p)Q(s)| 32 dx dx ≤ ∥u0∥F 1,rN , ∀t < T. (3.5.6)Take σ < 1 from Lemma (3.5.1) arbitrary small. From Lemma (3.1.2) wecan choose m large enough so that ∥u0∥F 1,rm ≤ ε(σ). Then by Remark (3.4.2)we can take T = c∗2m with c∗ < 1 and get that1c∗22m∫ c∗22m0∫B2m (0)(|u|3 + |p− (p)Q| 32 ) dx dt ≤ ε(σ) (3.5.7)By epsilon-regularity Lemma (3.5.1) we get that u is regular inZm = Bσ√c∗2m × [(1− σ2)c∗22m, c∗22m], (3.5.8)and ∥u∥L∞(Zm) ≤ C2−m. Take δ < 1 and choose c∗, σ so that Zm containsPm = {(x, t) ∈ R4+ : δ|x|2 ≤ t, (1− σ2)c∗22m ≤ t ≤ 4(1− σ2)c∗22m}.(3.5.9)Taking the union by m ≥ N1 we prove the statement of (3.1.3).74Chapter 4Scalar elliptic equations witha singular driftIn this chapter we investigate the weak solvability and properties of weaksolutions to the Dirichlet problem for a scalar elliptic equation −∆u+ b(α) ·∇u = f in a bounded domain Ω ⊂ R2 containing the origin, where f ∈W−1q (Ω) with q > 2 and b(α) := b− α x|x|2 , b is a divergence–free vector fieldand α ∈ R is a parameter. Work is done with Professor Timofei Shilkin4.1 Introduction and Main ResultsAssume Ω ⊂ R2 is a bounded simply connected domain with a C1–smoothboundary ∂Ω and 0 ∈ Ω. We consider the following boundary value problem: −∆u+(b− α x|x|2)· ∇u = −div f in Ω,u|∂Ω = 0.(4.1.1)Here u : Ω → R, b : Ω → R2 and f : Ω → R2 are given functions andα ∈ R is a parameter. We always assume that the vector field b satisfies theassumptionsb ∈ L2,w(Ω), div b = 0 in D′(Ω), (4.1.2)where L2,w(Ω) is a weak Lebesgue space equipped with the norm∥b∥L2,w(Ω) := supλ>0λ |{ x ∈ Ω : |b(x)| > λ }| 12 . (4.1.3)Note thatdivx|x|2 = 2πδ0 in D′(Ω),where δ0 is the delta-function concentrated at x = 0. We also define thetotal driftb(α) := b− α x|x|2 . (4.1.4)754.1. Introduction and Main ResultsThen from (4.1.2) we obtainb(α) ∈ L2,w(Ω), ∥b(α)∥L2,w(Ω) ≤ C(|α|+ ∥b∥L2,w(Ω)) ,and the divergence of b(α) is sign-definite: div b(α) ≤ 0 in the sense of dis-tributions if α ≥ 0 and div b(α) > 0 if α < 0. The negative sign of div b(α)means that the drift term generates the positive increase to the quadraticform of our differential operator while the negative sign of div b(α) means op-posite. The goal of this chapter is to investigate the existence and propertiesof weak solutions to the problem (4.1.1).Our research is partly motivated by axially symmetric problems for the3D Navier-Stokes equations. In the study of such problems the equation∂tu−∆u+(v − α x′|x′|2)· ∇u = 0 in R3 × (0, T ) (4.1.5)plays an important role. Here v = v(x, t) is the divergence-free velocityfield, x′ = (x1, x2, 0)T , and u = u(x, t) is some auxiliary scalar function.For example, for axially symmetric solutions without swirl (i.e. if v(x, t) =vr(r, z, t)er + vz(r, z, t)ez where er, eφ, ez is the standard cylindrical basis)the equation (4.1.5) is satisfied for α = 2 and u =ωφr , where ωφ := vr,z−vz,rand r = |x′|. In the case of general axially symmetric solutions v(x, t) =vr(r, z, t)er+vφ(r, z, t)eφ+vz(r, z, t)ez the equation (4.1.5) holds for α = −2and u = rvφ.It is well-known in the Navier-Stokes theory (see, for example, [41], [49],[50], [52], [56]) that while in the case α > 0 some results like Liouville-typetheorems assume no special conditions on the solutions u to the equation(4.1.5) besides a proper decay of the drift v, the analogous results in thecase α < 0 require the additional condition u|Γ = 0 where Γ := {x ∈ R3 :x1 = x2 = 0}. Our equation (4.1.1) can be considered as a 2D elliptic modelfor the general equation (4.1.5) and one of the goals of the present thesis isto investigate from the general point of view the role the condition u(0) = 0(which is modeling the condition u|Γ = 0 in the 3D situation) plays in thetheory.During the last few decades the problem (4.1.1) was intensively studiedin the case of the divergence-free drift (i.e. for α = 0), see, for example, thepapers [32], [33], [37], [44], [45], [47], [51], [53], [55], [57], [58] and referencesthere. Papers devoted to the non-divergence free drifts are not so numerous(see [38], [39], [40] for the references). In the general situation the drift canbe decomposed into divergence-free and potential parts. So, our problem(4.1.1) can also be interpreted as a model problem containing a singularpotential part of the drift whose divergence is sign-defined.764.1. Introduction and Main ResultsOur work was motivated to a certain extent by the recent paper [40]where the problem (4.1.1) was studied in the case of Ω ⊂ Rn with n ≥ 3 andb ∈ Ln,w(Ω) such that div b ∈ Ln2,w(Ω), div b ≤ 0. Our results concerningthe case α ≥ 0 can be viewed as a 2D version of results in [40]. The maintechnical difference between [40] and our work is that in the 2D situationthe divergence of the drift is not an integrable function but only a measure.We define the bilinear form Bα[u, η] byBα[u, η] :=∫Ωη b(α) · ∇u dx. (4.1.6)This form is well-defined at least for u ∈ W 1p (Ω) with p > 2 and η ∈ Lq(Ω)with q > 2pp−2 . Note that as b(α) is singular the drift term in (4.1.6) generallyspeaking is not integrable for u ∈W 12 (Ω). So, instead of the standard notionof weak solutions from the energy class W 12 (Ω) following [40] we introducethe definition of p-weak solutions:Definition 4.1.1. Assume p > 2, f ∈ Lp(Ω) and b satisfies (4.1.2). We sayu is a p-weak solution to the problem (4.1.1) if u ∈◦W 1p(Ω) and u satisfiesthe relation∫Ω∇u · ∇η dx + Bα[u, η] =∫Ωf · ∇η dx, ∀ η ∈◦W 12(Ω). (4.1.7)From the imbedding theorem we obtain that for p > 2 every p-weaksolution to the problem (4.1.1) is Ho¨lder continuous:u ∈ C1− 2p (Ω¯).Moreover, the form Bα[u, u] in (4.1.7) possesses the following property.Proposition 4.1.1. Assume b satisfies (4.1.2). Then for any α ∈ R andany v ∈◦W 1p(Ω) with p > 2 the quadratic form Bα[v, v] satisfies the identityBα[v, v] = πα |v(0)|2. (4.1.8)Proof. Taking into account (4.1.2) and div x|x|2 = 0 in Ω \ {0} from (4.1.6)integrating by parts we obtainBα[v, v] = − limϵ→0∫Ω\Bϵαx|x|2 · ∇|v|22dx = limϵ→0α2ϵ∫∂Bϵ|v|2 ds = πα |v(0)|2.Here we denote Bϵ := { x ∈ R2 : |x| < ϵ }.774.1. Introduction and Main ResultsTo demonstrate effects concerning weak solvability of the problem (4.1.1)we take f ≡ 0, b ≡ 0, and consider the equation −∆u− αx|x|2 · ∇u = 0 in B,u|∂B = 0,(4.1.9)in the unit ball B := { x ∈ R2 : |x| < 1 }. For radial solutions u(x) = v(|x|)the equation (4.1.9) reduces to the ODE for the Darboux-type operatorv′′(r) +α+ 1rv′(r) = 0, r ∈ (0, 1).The family of solutions to this ODE is v(r) = c1 r−α + c2 if α ̸= 0 andc1 ln r+c2 if α = 0. If α ≥ 0 then the only weak radial solution u(x) = v(|x|)to the problem (4.1.9) such that u ∈W 12 (B) is identically zero. In contrast,if α < 0 then we have a one-parameter family v(x) = c(r|α| − 1) of non-trivial p-weak solutions to the problem (4.1.9) with some p = p(α), p > 2.In particular, for α < 0 p-weak solutions of the problem (4.1.9) are non-unique, see also [46] for a related result. Nevertheless, if for α < 0 we add theadditional condition u(0) = 0 to the problem (4.1.9) then any radial p-weaksolution to (4.1.9) satisfying this requirement is identically zero. This hintsthat in the case of α < 0 the condition u(0) = 0 can serve as an additionalrequirement that provides uniqueness for the problem (4.1.1). Our goal isto investigate existence of p-weak solutions satisfying this requirement.The main results of this chapter are the following two theorems:Theorem 4.1.2. Assume α ≥ 0, b satisfies (4.1.2) and q > 2. Then thereexists p > 2 depending only on q, Ω, α and ∥b∥L2,w(Ω) such that for anyf ∈ Lq(Ω) there exists a unique p-weak solution u to the problem (4.1.1).Moreover, this solution satisfies the estimate∥u∥W 1p (Ω) ≤ C ∥f∥Lq(Ω), (4.1.10)with the constant C depending only on q, p, α, Ω and ∥b∥L2,w(Ω).Theorem 4.1.3. Assume α < 0, b satisfies (4.1.2) and q > 2. Then thereexists p > 2 depending only on q, Ω, α and ∥b∥L2,w(Ω) such that for anyf ∈ Lq(Ω) there exists a unique p-weak solution u to the problem (4.1.1) sat-isfying the condition u(0) = 0. Moreover, this solution satisfies the estimate(4.1.10) with the constant C depending only on q, p, α, Ω and ∥b∥L2,w(Ω).784.2. Higher Integrability ResultsNote that the assumption ∂Ω is of class C1 in Theorems 4.1.2 and 4.1.3actually is made only for brevity and our proofs can be extended for moregeneral domains.This chapter is organized as follows: in Section 4.2 we prove some apriori estimates of p-weak solutions including higher integrability results, inSection 4.3 we prove the estimate of the Ho¨lder norm of p-weak solutionsin the case of α < 0 which turns out to be a crucial step in our proof ofthe higher integrability of ∇u if α < 0, in Section 4.4 we present the proofsof Theorems 4.1.2 and 4.1.3, in Appendix we recall some known facts andprove an approximation result for divergence-free functions from L2,w(Ω).In the chapter we use the following notation. For any a, b ∈ Rn wedenote by a · b their scalar product in Rn. We denote by Lp(Ω) and W kp (Ω)the usual Lebesgue and Sobolev spaces. C∞0 (Ω) is the space of smoothfunctions compactly supported in Ω. The space◦W 1p(Ω) is the closure ofC∞0 (Ω) in W 1p (Ω) norm. The space of distributions on Ω is denoted byD′(Ω). By C(Ω¯) and Cα(Ω¯), α ∈ (0, 1) we denote the spaces of continuousand Ho¨lder continuous functions on Ω¯. For q ∈ [1,+∞) let Lq,w(Ω) be aweak Lebesgue space.The symbols ⇀ and → stand for the weak and strong convergence re-spectively. We denote by BR(x0) the ball in Rn of radius R centered at x0and write BR if x0 = 0. We write also B instead of B1. For ω ⊂ R2 wedenote the average of f over ω by(f)ω := −∫ωf dx =1|ω|∫ωf dx.For f , g : R2 → R we denote by f ∗ g the convolution(f ∗ g)(x) :=∫R2f(x− y)g(y) dy.4.2 Higher Integrability ResultsIn this Section we recall a priori estimates for weak solutions to the problem(4.1.1). Our first theorem is the energy estimate:794.2. Higher Integrability ResultsLemma 4.2.1. Assume α ∈ R and b satisfies (4.1.2). Let u be a p-weaksolution to the problem (4.1.1), corresponding to some f ∈ Lp(Ω) with p > 2.In the case of α < 0 assume additionally that u(0) = 0. Then the estimate∥u∥W 12 (Ω) ≤ C ∥f∥L2(Ω) (4.2.1)holds with some constant C > 0 depending only on Ω.Proof. We take η = u in (4.1.7) and use (4.1.8).The next result is the global estimate of L∞-norms of p-weak solutions.Lemma 4.2.2. Assume α ∈ R and b satisfies (4.1.2). Let u ∈◦W 1p(Ω) bea p-weak solution to the problem (4.1.1) corresponding to some f ∈ Lq(Ω)with q > 2. In the case of α < 0 assume additionally that u(0) = 0. Thenu ∈ L∞(Ω) and the estimate∥u∥L∞(Ω) ≤ C ∥f∥Lq(Ω) (4.2.2)holds with some constant C depending only on Ω and q.Proof. Assume k ≥ 0 and take η = (u − k)+ := max{0, u − k} in (4.1.7).Note that if u(0) = 0 and k ≥ 0 then (u − k)+(0) = 0. From (4.1.8) weobtainBα[u, (u− k)+] = Bα[(u− k)+, (u− k)+] = πα |(u− k)+|2(0).The last term in non-negative for α ≥ 0 and vanishes if α < 0, u(0) = 0,k ≥ 0. Hence from (4.1.7) we obtain∫Ak|∇u|2 dx ≤∫Akf · ∇u dx, ∀ k ≥ 0.where we denote Ak := { x ∈ Ω : u(x) > k }. The rest of the proof goesas in the usual elliptic theory. Applying the Ho¨lder inequality we obtain∫Ak|∇u|2 dx ≤ ∥f∥2Lq(Ω) |Ak|δ, ∀ k ≥ 0,where δ := 2(12 − 1q)> 0. This inequality yields the following estimate, see[43, Chapter II, Lemma 5.3],esssupΩ u+ ≤ C(q,Ω) ∥f∥Lq(Ω).where u+ := max{0, u}. The estimate of esssupΩ u− where u− := max{0,−u}can be obtained in a similar way if we replace u by −u.804.2. Higher Integrability ResultsNow we prove a higher integrability result for ∇u in the case of α ≥ 0.Our proof is based on the reverse Ho¨lder inequality for ∇u, see, for example,[35, Chapter V].Theorem 4.2.3. Assume α ≥ 0, b satisfies (4.1.2) and q > 2. Then thereexists p > 2 depending only on q, Ω, α and ∥b∥L2,w(Ω) such that for any f ∈Lq(Ω) and any p-weak solution u to the problem (4.1.1) the estimate (4.1.10)holds with some constant C depending only on Ω, q, α and ∥b∥L2,w(Ω).Proof. First we consider an internal point x0 ∈ Ω. Take R > 0 such thatB2R(x0) ⊂ Ω and choose a cut-off function ζ ∈ C∞0 (B2R(x0)) so that ζ ≡ 1on BR(x0) and ∥∇ζ∥L∞(R2) ≤ C/R. Denote u¯ = u − (u)B2R(x0). Takingη = ζ2u¯ in (4.1.7) and using the relationBα[u, ζ2u¯] = Bα[ζu¯, ζu¯] −∫B2R(x0)b(α) · ζ|u¯|2∇ζ dxafter routine computations we arrive at∫BR(x0)|∇u|2 dx + Bα[ζu¯, ζu¯] ≤ 14∫B2R(x0)|∇u|2 dx −∫B2R(x0)b(α) · ζ|u¯|2∇ζ dx + C∫B2R(x0)(|f |2 + |u¯|2) dx.(4.2.3)Taking into account Proposition 4.1.1 and the identity ζ(0) = 1 we obtainBα[ζu¯, ζu¯] = πα |u¯(0)|2 (4.2.4)As α ≥ 0 we haveBα[ζu¯, ζu¯] ≥ 0 (4.2.5)and hence we can drop this term from (4.2.3). For the second term on theright-hand side of (4.2.3) we get∫B2R(x0)b(α) · ζ|u¯|2∇ζ dx ≤ CR∥b(α)∥L2,∞(Ω)∥|u¯|2∥L2,1(B2R(x0)) ≤≤ CR∥b(α)∥L2,∞(Ω)∥u¯∥2L4,2(B2R(x0)).Using the Ho¨lder inequality for Lorentz spaces (see [42, Proposition 2.1],also cited in [48], [34, Section 4.1].) we obtain:∥u¯∥L4,2(B2R(x0)) ≤ C ∥1∥L20,10/3(B2R(x0)) ∥u¯∥L5,5(B2R(x0)).814.2. Higher Integrability ResultsTaking into account the property of the indicator function ∥χE∥Lp,s(Ω) =(p/s)1/s|E ∩ Ω|1/p we derive∥u¯∥L4,2(B2R(x0)) ≤ C |B2R|120 ∥u¯∥L5(B2R(x0)).Hence we arrive at∫B2R(x0)b(α) · ζ|u¯|2∇ζ dx ≤ CR4/5∥b(α)∥L2,w(Ω)∥u¯∥2L5(B2R(x0)).Using the imbedding theorem we estimate∫B2R(x0)b(α) · ζ|u¯|2∇ζ dx ≤ CR4/5∥b(α)∥L2,w(Ω)∥∇u∥2L 107(B2R(x0)). (4.2.6)Using (4.2.5) and (4.2.6) from (4.2.3) we obtain −∫BR(x0)|∇u|2 dx12≤ 12 −∫B2R(x0)|∇u|2 dx12+ C −∫B2R(x0)|f |2 dx12++ C(1 + ∥b(α)∥1/2L2,w(Ω)) −∫B2R(x0)|∇u| 107 dx710.(4.2.7)Now we derive similar estimates near the boundary. Extend u and f byzero outside Ω and denote this extension by u˜ and f˜ . Note that as ∂Ω isC1–smooth there exists d > 0 depending on Ω such that for any R < d andany x ∈ ∂Ω |BR(x)\Ω| > 14 |BR|. Without loss of generality one can assumed < dist{0, ∂Ω}. Take x0 ∈ ∂Ω and R < d/2. Choose a cut-off functionζ ∈ C∞0 (B2R(x0)) so that ζ ≡ 1 on BR(x0). Then ζ2u ∈◦W 12(Ω2R(x0)) wherewe denote ΩR(x0) := BR(x0) ∩ Ω. Taking η = ζ2u in (4.1.7) after routinecomputations we arrive at∫BR(x0)|∇u˜|2 dx + Bα[u, ζ2u] ≤ 14∫B2R(x0)|∇u˜|2 dx +C∫B2R(x0)(|f˜ |2 + |u˜|2) dx.(4.2.8)824.2. Higher Integrability ResultsAs 0 ̸∈ Ω2R(x0) we conclude div b(α) = 0 in D′(Ω2R(x0)) and henceBα[u, ζ2u] = −∫B2R(x0)b(α) · ζ|u˜|2∇ζ dx.Repeating the arguments in the internal case we obtain∫B2R(x0)b(α) · ζ|u˜|2∇ζ dx ≤ CR4/5∥b(α)∥L2,w(Ω)∥u˜∥2L5(B2R(x0)).As |{x ∈ B2R(x0) : u˜(x) = 0 }| ≥ 14 |B2R| we can apply Lemma A.0.1 andobtain∥u˜∥L5(B2R(x0)) ≤ C ∥∇u˜∥L 107(B2R(x0)) = C ∥∇u∥L 107(Ω2R(x0))with some absolute constant C > 0. Hence from (4.2.8) we obtain −∫ΩR(x0)|∇u|2 dx12≤ 12 −∫Ω2R(x0)|∇u|2 dx12+ C −∫Ω2R(x0)|f |2 dx12++ C(1 + ∥b(α)∥1/2L2,w(Ω)) −∫Ω2R(x0)|∇u| 107 dx710(4.2.9)which holds for any x0 ∈ ∂Ω and any R < d. Combining the interior andboundary estimates in the standard way we obtain (4.2.9) for any x0 ∈ Ω¯and any R < d/2. This estimate is the reverse Ho¨lder inequality for ∇u, see[35, Chapter V]. Hence there exists p ∈ (2, q] such that ∇u ∈ Lp(Ω) and thefollowing estimate holds:∥∇u∥Lp(Ω) ≤ C (∥∇u∥L2(Ω) + ∥f∥Lq(Ω))with some constant C > 0 depending only on q, Ω, α and ∥b∥L2,w(Ω). Com-bining this estimate with (4.2.1) we obtain (4.1.10).The estimate of the Ho¨lder norm of solutions to the problem (4.1.1) isthe crucial step in our proof of the higher integrability of ∇u for α < 0. Thisestimate was originally obtained in [28], [29], see also [47]. As the mentionedpapers formally deal with a bit stronger assumption on the divergence-freepart of the drift b (which in 2D-case corresponds to b ∈ L2(Ω) instead ofb ∈ L2,w(Ω) in our case) we outline the proof of this result.834.2. Higher Integrability ResultsTheorem 4.2.4. Assume α < 0, b satisfies (4.1.2) and q > 2. Then thereexists µ ∈ (0, 1) depending only on q, Ω, α and ∥b∥L2,w(Ω) such that for anyf ∈ Lq(Ω), any p > 2 and any p-weak solution u to the problem (4.1.1) suchthat u(0) = 0 the estimate∥u∥Cµ(Ω¯) ≤ C(∥f∥Lq(Ω) + ∥u∥L∞(Ω)), (4.2.10)holds with some constant C depending only on α, Ω, q and ∥b∥L2,w(Ω).Proof. For any α ∈ R the drift b(α) is divergence free in Ω \ {0}:div b(α) = 0 in D′(Ω \ {0}).It is well-known that solutions of elliptic equations with the divergence freedrifts b(α) ∈ L2,w(Ω\BR) are Ho¨lder continuous, see, for example, [47], [31],[37]. Hence for any R > 0 such that B2R ⋐ Ω we have u ∈ Cµ(Ω \BR) andfor any x0 ∈ Ω¯ \BR and any 0 < ρ < R the following estimate holds:oscΩ∩Bρ(x0) u ≤ C( ρR)µ (oscΩ∩BR(x0) u + ∥f∥Lq(Ω∩BR(x0)))(4.2.11)with some constant µ ∈ (0, 1) and C > 0 depending only on q, Ω, α and∥b∥L2,w(Ω).On the other hand, for any 0 < ρ < R we have the estimateoscBρ u ≤ C( ρR)µ (oscBR u + ∥f∥Lq(BR)). (4.2.12)Though this estimate is not new (see, for example [56, Theorem 10.7]),for the reader’s convenience we present its proof in the next Section, seeProposition 4.3.1 below. Now take arbitrary x0 ∈ Ω and denote r := |x0|.For r < R we obtain Br(x0) ⊂ B2r ⊂ B2R andoscBρ(x0) u ≤ C(ρr)µ (oscBr(x0) u + ∥f∥Lq(Br(x0)))≤≤ C(ρr)µ (oscB2r u + ∥f∥Lq(B2r))≤C(ρr)µ ( rR)µ(oscB2R u+ ∥f∥Lq(B2R))≤ C( ρR)µ (∥u∥L∞(Ω) + ∥f∥Lq(Ω)).The last estimate together with (4.2.11), (4.2.12) implies (4.2.10).The last theorem of the present Section is the analogue of Theorem 4.2.3in the case of α < 0.844.3. Ho¨lder Continuity of p–Weak SolutionsTheorem 4.2.5. Assume α < 0, b satisfies (4.1.2) and q > 2. Then thereexists p > 2 depending only on q, Ω, α and ∥b∥L2,w(Ω) such that for anyf ∈ Lq(Ω) and any p-weak solution u to the problem (4.1.1) satisfying the ad-ditional assumption u(0) = 0 the estimate (4.1.10) holds with some constantC > 0 depending only on Ω, q, α and ∥b∥L2,w(Ω). Moreover, without lossof generality one can assume the constant C in (4.1.10) is a non-decreasingfunction with respect to ∥b∥L2,w(Ω).Proof. Assume α < 0 and u is a p-weak solution to the problem (4.1.1).From Theorem 4.2.4 we obtain u ∈ Cµ(Ω¯) with some µ ∈ (0, 1) and henceintegrating by parts and taking into account u(0) = 0 we obtain∫Ωx|x|2 · ∇u η dx = −∫Ωux|x|2 · ∇η dx, ∀ η ∈ C∞0 (Ω). (4.2.13)The last relation means that u is a p-weak solution to the problem{−∆u+ b · ∇u = −div g in Ω,u|∂Ω = 0, u(0) = 0,(4.2.14)whereg = f − α x|x|2 u.Note that g ∈ Lq0(Ω) for any q0 ∈(2,min{q, 21−µ})and∥g∥Lq0 (Ω) ≤ C(∥f∥Lq(Ω) + |α| ∥u∥Cµ(Ω¯))(4.2.15)with a constant C > 0 depending only on µ and Ω. Applying Theorem 4.2.3to the problem (4.2.14) we conclude there exist constants p > 2 and C > 0depending only on q0, Ω and ∥b∥L2,w(Ω) such that∥u∥W 1p (Ω) ≤ C ∥g∥Lq0 (Ω).Taking into account (4.2.15) and (4.2.10) we obtain the estimate (4.1.10).4.3 Ho¨lder Continuity of p–Weak SolutionsThe main result of this Section is estimate (4.2.12):854.3. Ho¨lder Continuity of p–Weak SolutionsProposition 4.3.1. Assume α < 0, b satisfies (4.1.2), b(α) is defined in(4.1.4). Assume q > 2, f ∈ Lq(Ω) and B2R ⋐ Ω. Assume p > 2 and u is ap-weak solution to−∆u+ b(α) · ∇u = −div f in Ω (4.3.1)such that u(0) = 0. Then there exist µ ∈ (0, 1) and C > 0 depending only onq, α, ∥b∥L2,w(Ω) such that for any 0 < ρ < R the following estimate holds:oscBρ u ≤ C( ρR)µ (oscBR u + ∥f∥Lq(BR)). (4.3.2)Our proof is based on the ideas of [56, Section 10.3], see also [47] wherea similar result is obtained. We split the proof of Proposition 4.3.1 ontoseveral steps. We start from the Caccioppolli-type inequality:Lemma 4.3.2. Assume α < 0, b satisfies (4.1.2) in Ω = B2 and p > 2.Then for any f ∈ Lp(B2), any p-weak solution u to the equation (4.3.1) inB2 such that u(0) ≤ 0, and any ζ ∈ C∞0 (B2) the following estimate holds:∫B2ζ2|∇u+|2dx ≤ C∫B2|u+|2(|∇ζ|2 + |b(α)| |∇ζ|)dx + C∫B2ζ2|f |2 dx.(4.3.3)Here u+ := max{u, 0} and C > 0 is some absolute constant.Proof. We take η := ζ2u+ in (4.1.7). The convective term satisfies theidentityBα[u, ζ2u+] = Bα[ζu+, ζu+] −∫B2ζ |u+|2 b(α) · ∇ζ dx.As u(0) ≤ 0 we obtain u+(0) = 0 and hence from (4.1.8) we concludeBα[ζu+, ζu+] = πα ζ2(0)|u+(0)|2 = 0.Now the result follows by the Ho¨lder and Young inequalities.Now we proceed with the following maximum estimate:864.3. Ho¨lder Continuity of p–Weak SolutionsLemma 4.3.3. Assume α < 0, b satisfies (4.1.2) in Ω = B, q > 2 andf ∈ Lq(B). Then there is a constant C > 0 depending only on q, α and∥b∥L2,w(B) such that for any p > 2 and any p-weak solution u of the equation(4.3.1) in B satisfying u(0) ≤ 0 the following estimate holds:supB1/2u+ ≤ C(−∫B|u+|q1 dx)1/q1+ C ∥f∥Lq(B), (4.3.4)where q1 := max{5, 2qq−2}.Proof. The proof of (4.3.4) follows by application of the standard Moseriteration technique. Let β ≥ 0 be arbitrary and denoteu¯ := u+ + k, z := u¯β+22 , w := u¯β+22 − k β+22 ,where we take k := ∥f∥Lq(B) if f ̸≡ 0 and k = 0 otherwise. Assume ζ ∈C∞0 (B) is an arbitrary cut-off function and take η = ζ2u¯β2w in (4.1.7). Wehave∫B∇u · ∇η dx ≥ 2β + 2∫Bζ2|∇w|2dx − 4β + 2∫B|w∇ζ · ζ∇w| dx,Bα[u, η] = 2β + 2Bα[w, ζ2w],∫Bf · ∇η dx ≤ 2∫Bζ |f | u¯β2(ζ |∇w|+ |∇ζ|w)dx.As u(0) ≤ 0 we obtain w(0) = 0. Combining all above inequalities from(4.1.7) similar to (4.3.3) we obtain∫B|ζ∇w|2dx ≤ C∫Bw2(|∇ζ|2 + |b(α)| |∇ζ|)dx + C (β + 2)2∫Bζ2|f |2u¯β dx.Note that k ≤ u¯, w ≤ z and ∇z = ∇w. So, in the case f ̸≡ 0 from the lastinequality we obtain∫B|ζ∇z|2dx ≤ C∫Bz2(|∇ζ|2 + |b(α)| |∇ζ|)dx + C (β + 2)2∫Bζ2∣∣∣fk∣∣∣2z2 dx.874.3. Ho¨lder Continuity of p–Weak SolutionsRecall that∥∥fk∥∥Lq(B)= 1. Hence the last term can be estimated by theHo¨lder inequality:∫Bζ2∣∣∣fk∣∣∣2z2 dx ≤ ∥∥∥fk∥∥∥2Lq(B)∥ζz∥2L 2qq−2(B) ≤ C ∥ζz∥2Lq1 (B),and we arrive at∫B|∇(ζz)|2dx ≤ C∫Bz2(|∇ζ|2 + |b(α)| |∇ζ|)dx + C (β + 2)2∥ζz∥2Lq1 (B).Applying the imbedding theorem∥ζz∥L2q1 (B) ≤ C ∥∇(ζz)∥L2(B),taking arbitrary 12 ≤ r < R ≤ 1, and choosing ζ ∈ C∞0 (BR) so that ζ ≡ 1on Br, |∇ζ| ≤ C/(R− r), we get∫Bz2|b(α)| |∇ζ| dx ≤ CR− r ∥b(α)∥L2,w(BR)∥z∥2L4,2(BR)and∫Bz2|∇ζ|2 dx ≤ ∥|∇ζ|2∥L2,w(BR\Br)∥z2∥L2,1(BR) ≤C(R− r)3/2 ∥z∥2L4,2(BR).Combining these estimates we obtain∥z∥L2q1 (Br) ≤C√R− r(∥b(α)∥12L2,w(B)+ (R− r)− 14)∥z∥L4,2(BR)+ C (β + 2) ∥z∥Lq1 (BR).Using the Ho¨lder inequality for Lorentz norms and taking into account R ∈(12 , 1] we conclude∥z∥L4,2(BR) ≤ C R2(1/4−1/q1) ∥z∥Lq1 (BR) ≤ C ∥z∥Lq1 (BR).Applying the Cauchy inequality we derive the estimate∥z∥L2q1 (Br) ≤ C(β + 2 +K(R− r)−1) ∥z∥Lq1 (BR), K := ∥b(α)∥ 12L2,w(B)+1,884.3. Ho¨lder Continuity of p–Weak Solutionswhich is valid for any 12 ≤ r < R ≤ 1. Hence we have∥u¯∥L2γq1 (Br) ≤ C1γ(2γ +K(R− r)−1) 1γ ∥u¯∥Lγq1 (BR) (4.3.5)with an arbitrary γ ≥ 1, γ := β+22 . Denote s0 = q1, sm := 2sm−1, anddenote also Rm =12+12m+1. Taking r = Rm, R = Rm−1, γ =sm−1q1in (4.3.5)we obtain∥u¯∥Lsm (BRm ) ≤ exp(c0m2−m)(1 +K)21−m ∥u¯∥Lsm−1 (BRm−1 ).Iterating this inequality we get∥u¯∥Lsm (B1/2) ≤ C0 (1 +K)2 ∥u¯∥Lq1 (B).with some positive constant C0 independent onm. Takingm→∞ we arriveat (4.3.4).Lemma 4.3.4. Assume α < 0, b satisfies (4.1.2) in Ω = B, q > 2 andf ∈ Lq(B). Then there are constants C > 0 and ϵ0 ∈ (0, 1) depending onlyon q, α and ∥b∥L2,w(B) such that for any p > 2 and any p-weak solution uof the equation (4.3.1) in B such that u(0) ≤ 0 if|{x ∈ B : u(x) > 0 }| ≤ ϵ0 |B|thensupB1/2u+ ≤ 12supBu+ + C ∥f∥Lq(B).Proof. From Lemma 4.3.3 we obtain (4.3.4) with q1 := max{5, 2qq−2}. Thenwe haveC(−∫B|u+|q1dx)1/q1 ≤ C supBu+|{x ∈ B : u(x) > 0 }|1/q1|B|1/q1 ≤ C ϵ1/q10 supBu+.Choosing Cϵ1/q10 =12 we obtain the required statement.Next we prove that a small value subset has density less then 1.894.3. Ho¨lder Continuity of p–Weak SolutionsLemma 4.3.5. Assume α < 0, b satisfies (4.1.2) in Ω = B. Then for anyp > 2, any g ∈ Lp(B) and any p-weak solution v to the problem−∆v + b(α) · ∇v = −div g in B (4.3.6)satisfying the assumptions0 ≤ v ≤ 2 in B, v(0) ≥ 1, (4.3.7)if∥g∥L2(B) ≤ c⋆ |α|, where c⋆ :=√π4, (4.3.8)then the following estimate holds:|{x ∈ B : v(x) > λ }| > γ. (4.3.9)Here λ, γ ∈ (0, 1) are some constants depending on α and ∥b∥L3/2(B) in theexplicit way described in the proof below.Proof. Assume there exist g and v satisfying (4.3.6), (4.3.7), (4.3.8) suchthat|{x ∈ B : v(x) > λ }| ≤ γ. (4.3.10)Integrating by parts in (4.1.7) for any η ∈ C∞0 (B) we obtain2π|α|v(0)η(0) = −∫Bv(∆η + b(α) · ∇η)dx −∫Bg · ∇η dx. (4.3.11)Note that v(0) ≥ 1. Choose η so that η = 1 on B1/2, ∥∇η∥L2(B) ≤ 4√π and∥η∥C2(B¯) ≤ c⋆⋆ where c⋆⋆ > 0 is some sufficiently large absolute constant.Then from (4.3.8) and the Ho¨lder inequality we obtain∣∣∣ ∫Bg · ∇η dx∣∣∣ ≤ π|α|.Hence from (4.3.11) we obtainπ|α| ≤∣∣∣ ∫Bv(∆η + b(α) · ∇η)dx∣∣∣. (4.3.12)DenoteB[v > λ] := { x ∈ B : v(x) > λ }, B[v ≤ λ] := { x ∈ B : v(x) ≤ λ }.904.3. Ho¨lder Continuity of p–Weak SolutionsFrom the Ho¨lder inequality we obtain∫B[v>λ]v(∆η+b(α)·∇η)dx ≤ ∥v∥L∞(B) ∥η∥C2(B¯)(1+∥b(α)∥L 32(B))|B[v > λ]| 13 .Taking into account (4.3.7), (4.3.10) and ∥η∥C2(B) ≤ c⋆⋆ we conclude∣∣∣ ∫B[v>λ]v(∆η + b(α) · ∇η)dx∣∣∣ ≤ 2c⋆⋆ (1 + ∥b(α)∥L 32(B))γ13 .On the other hand∣∣∣ ∫B[v≤λ]v(∆η + b(α) · ∇η)dx∣∣∣ ≤ c⋆⋆ (1 + ∥b(α)∥L1(B))λ.Finally, we obtain|α| ≤ 2c⋆⋆(1 + ∥b(α)∥L 32(B))γ13 + c⋆⋆(1 + ∥b(α)∥L1(B))λ.This inequality leads to the contradiction if we fix values of λ, γ ∈ (0, 1) sothat2c⋆⋆(1 + ∥b(α)∥L 32(B))γ13 + c⋆⋆(1 + ∥b(α)∥L1(B))λ < π|α|.Lemma 4.3.6. Assume α < 0, b satisfies (4.1.2) in Ω = B2 and let λ,γ ∈ (0, 1) be the constants defined in Lemma 4.3.5. Then for any ϵ > 0there exists s0 ∈ N depending only on ϵ, λ, α, ∥b∥L2,w(B) such that for anyp > 2, any g ∈ Lp(B) and any p-weak solution v to the problem (4.3.6) inB2 satisfying (4.3.7) in B2 either|{x ∈ B : v(x) ≤ 2−s0λ }| ≤ ϵ |B| (4.3.13)or2−s0λ ≤ 1c⋆|α| ∥g∥L2(B2). (4.3.14)Proof. Assume s0 ∈ N is arbitrary and g and v satisfy (4.3.6), (4.3.7). If∥g∥L2(B2) ≥ c⋆|α|914.3. Ho¨lder Continuity of p–Weak Solutionsthen (4.3.14) holds as λ ∈ (0, 1). Assume now (4.3.8) is valid. Then (4.3.9)is true.Let k ∈ (0, 1) be arbitrary. Denote u := k − v. Then u satisfies (4.3.1)and u(0) = k − v(0) ≤ 0. Then for any ζ ∈ C∞0 (B2) from (4.3.3) we obtain∫B2ζ2|∇(v− k)−|2 dx ≤ C∫B2(v− k)2−(|∇ζ|2+ |b(α)| |∇ζ|)dx+C∫B2|g|2 dx.Choosing ζ so that ζ ≡ 1 in B, |∇ζ| ≤ 4, for any k ∈ (0, 1) we obtain∫B|∇(v − k)−|2 dx ≤ C∫B2(v − k)2−(1 + |b(α)|)dx+ C∫B2|g|2 dx. (4.3.15)For s = 0, 1, 2, . . . we denote ks := 2−sλ andAs := {x ∈ B : v(x) < ks } = {x ∈ B : (v − ks)−(x) > 0 }.Applying the De Giorgi inequality (see [43, Chapter II §3, Lemma 3.9]) weobtain(ks − ks+1) |As+1|1/2 ≤ C|B \As|∫As\As+1|∇v(x)| dx.As B \ As ⊃ B \ A0 and |B \ A0| ≥ γ|B| using the Ho¨lder inequality weobtain2−s−1λ |As+1|1/2 ≤ Cγ|As \As+1|1/2( ∫As\As+1|∇v|2 dx)1/2.Hence|As+1| ≤ Cλ2γ222s+2 |As \As+1|∫B|∇(v − ks)−|2 dx.Using (4.3.15) with k = ks we arrive at|As+1| ≤ C 22sλ2γ2|As \As+1|∫B2[(v − ks)2−(1 + |b(α)|)+ |g|2] dx.As v ≥ 0 in B2 we have (v − ks)− ≤ ks = 2−sλ in B2 and|As+1| ≤ Cγ2|As \As+1|∫B2(1 + |b(α)|+ 22sλ2|g|2)dx.924.3. Ho¨lder Continuity of p–Weak SolutionsTaking the sum from s = 0 to s = s0 − 1 and using inclusions B ⊃ A0 ⊃A1 ⊃ . . . ⊃ As0 we obtains0|As0 | ≤C0γ2(1 + ∥b(α)∥L1(B2) +22s0λ2∥g∥2L2(B2))(4.3.16)with some absolute positive constant C0. Now let us fix the value s0 ∈ N sothatC0γ2s0(1 + c2⋆|α|2 + ∥b(α)∥L1(B2))≤ ϵ |B| (4.3.17)and consider two cases. If2s0λ∥g∥L2(B2) ≥ c⋆|α|then (4.3.14) follows. Otherwise2s0λ∥g∥L2(B2) ≤ c⋆|α|and from (4.3.16), (4.3.17) we obtain (4.3.13).Lemma 4.3.7. Assume α < 0, b satisfies (4.1.2) in Ω = B2, q > 2 andg ∈ Lq(B2). Let λ, γ ∈ (0, 1) be the constants defined in Lemma 4.3.5,ϵ0 ∈ (0, 1) be defined in Lemma 4.3.4 and s0 ∈ N be the number fromLemma 4.3.6 defined in (4.3.17) for ϵ = ϵ0. Denote δ := 2−s0λ, δ ∈ (0, 1).Then for any p > 2 and any p-weak solution v to the problem (4.3.6) in B2satisfying (4.3.7) in B2 eitherc⋆ |α| δ ≤ ∥g∥L2(B2) (4.3.18)orinfB1/2v + c ∥g∥Lq(B) ≥δ2. (4.3.19)Here c > 0 is some constant depending only on q, α and ∥b∥L2,w(B2).Proof. If g satisfies (4.3.14) then (4.3.18) follows. Assume now∥g∥L2(B) < c⋆|α| δ.Then applying Lemma 4.3.6 we obtain|{x ∈ B : v(x) ≤ δ }| ≤ ϵ0|B|.934.3. Ho¨lder Continuity of p–Weak SolutionsDenote u = δ − v. Then u(0) = δ − v(0) < 0 and u satisfies all assumptionsof Lemma 4.3.4 with f = −g. Hence we obtainsupB1/2(δ − v)+ ≤ 12supB(δ − v)+ + c ∥g∥Lq(B) ≤δ2+ c ∥g∥Lq(B)which givesδ − infB1/2v ≤ δ2+ c ∥g∥Lq(B).Lemma 4.3.7 is proved.Lemma 4.3.8. Assume α < 0, b satisfies (4.1.2) in Ω = B2, q > 2 andf ∈ Lq(B). There exist constants δ ∈ (0, 1) and c > 0 depending only on q,α and ∥b∥L2,w(B2) such that for any p > 2 and any a p-weak solution u tothe equation (4.3.1) in B2 satisfying u(0) = 0 the following inequality holds:oscB1/2 u ≤(1− δ2)oscB2 u + c ∥f∥Lq(B2). (4.3.20)Proof. For any r ∈ (0, 2) we denoteM(r) := supBru, m(r) := infBru, ω(r) :=M(r)−m(r),Note that m(2) ≤ 0 ≤M(2). Denote k0 := m(2)+M(2)2 , and define v and gso thatv(x) := 2u(x)−m(2)ω(2), g(x) :=2f(x)ω(2), if k0 < 0,v(x) := 2M(2)− u(x)ω(2), g(x) := −2f(x)ω(2), if k0 ≥ 0.Then v is a p-weak solution to the equation (4.3.6) in B2 satisfying theadditional conditions (4.3.7) in B2. From Lemma 4.3.7 we concludeeither δ ≤ c ∥g∥L2(B2) or infB1/2 v + c ∥g∥Lq(B) ≥δ2.In the first case from the Ho¨lder inequality we obtainω(2) ≤ cδ∥f∥L2(B2) ≤ c(δ, q) ∥f∥Lq(B2)944.3. Ho¨lder Continuity of p–Weak Solutionsand hence (4.3.20) follows. In the second case if k0 < 0 we havem(1/2) − m(2) + 2c ∥f∥Lq(B) ≥δ2ω(2)and hence (1− δ2)ω(2) + 2c ∥f∥Lq(B) ≥ ω(1/2)and we obtain (4.3.20). If k0 ≥ 0 thenM(2) − M(1/2) + 2c ∥f∥Lq(B) ≥δ2ω(2)and we again obtain (4.3.20).Lemma 4.3.9. Assume α < 0, b satisfies (4.1.2) in Ω. Assume q > 2,f ∈ Lq(Ω) and B2R ⋐ Ω. Then there exist constants δ ∈ (0, 1) and c > 0depending only on q, α and ∥b∥L2,w(B2R) such that for any p > 2 and any p-weak solution u to the equation (4.3.1) in Ω satisfying u(0) = 0 the followinginequality holds:oscBR/2 u ≤(1− δ2)oscB2R u + cR1− 2q ∥f∥Lq(B2R). (4.3.21)Proof. For x ∈ B2 we denoteuR(x) = u(Rx), b(α)R (x) = Rb(α)(Rx), fR(x) = Rf(Rx).Then uR is a solution to−∆uR + b(α)R · ∇uR = −div fR in B2and, moreover,∥b(α)R ∥L2,w(B2) = ∥b(α)∥L2,w(B2R), ∥fR∥Lq(B2) = R1−2q ∥f∥Lq(B2R).From Lemma 4.3.8 we obtainoscB1/2 uR ≤(1− δ2)oscB2 uR + c ∥fR∥Lq(B2)which implies (4.3.21).Now the inequality (4.3.2) follows from (4.3.21) by the standard iterationtechnique. Proposition 4.3.1 is proved.954.4. Proof of Main Results4.4 Proof of Main ResultsWe start with the proof of Theorem 4.1.2. To construct a p-weak solutionto the problem (4.1.1) we approximate our drift b(α) by smooth functionsusing Lemma 4.4.2. In the case α ≥ 0 our construction shows that p-weak solutions are always approximative solutions in the sense they can beobtained as limits of smooth solutions of the equations with smooth drifts.Note that this is not true if α < 0 as the approximative solution must satisfythe maximum principle (locally), but the radial solutions constructed in thecase of α < 0 in Section 4.1 do not possess this property.Proposition 4.4.1. Assume α ≥ 0, b ∈ C∞(Ω¯) satisfies div b = 0 in Ω andq > 2. Assume f ∈ Lq(Ω) and ϵ > 0. Defineb(α)ϵ := b− αx|x|2 + ϵ2 .Then there exists a unique q–weak solution u to the problem{−∆u+ b(α)ϵ · ∇u = −div f in Ω,u|∂Ω = 0.(4.4.1)Moreover, there exists p ∈ (2, q] depending only on q, Ω, α and ∥b∥L2,w(Ω)such that the following estimate holds:∥u∥W 1p (Ω) ≤ C ∥f∥Lq(Ω). (4.4.2)Here C > 0 is a constant which can be chosen to be a non-decreasing functionof q, Ω, α and ∥b∥L2,w(Ω).Proof. The existence and uniqueness of the weak solution u ∈ W 12 (Ω) forequation (4.4.1) with the smooth drift b(α)ϵ is well-known, see, for example,[43]. Moreover, from Lq–theory for equations with smooth coefficients weobtain uϵ ∈W 1q (Ω). Note that as α ≥ 0 we have div b(α)ϵ ≤ 0 in Ω. Hence theestimate (4.4.2) follows by the arguments similar to the proof of Theorem4.2.3.Now we state an approximation result in weak Lebesgue space:Proposition 4.4.2. Let Ω ⊂ R2 be a bounded simply connected domainof class C1. Assume b ∈ L2,w(Ω), div b = 0 in D′(Ω). Then there existbϵ ∈ C∞(Ω¯), div bϵ = 0 in D′(Ω), such that964.4. Proof of Main Results1) ∥bϵ∥L2,w(Ω) ≤ C ∥b∥L2,w(Ω),2) bϵ → b a.e. in Ω as ϵ→ 0.Here the constant C > 0 depends only on Ω.Proof. Assume p ∈ (1,+∞) and denoteJp(Ω) := { u ∈ Lp(Ω;R2) : div u = 0 in D′(Ω) }.Consider any bounded simply connected domain Ω0 ⊂ R2 such that Ω ⋐ Ω0.Then there exists an extension operator T : Jp(Ω) → Jp(R2) such that forall b ∈ Jp(Ω) the function b˜ := Tb has the following propertiesb˜ ≡ 0 in R2 \ Ω0, b˜|Ω = b,div b˜ = 0 in D′(R2), ∥b˜∥Lp(R2) ≤ cp∥b∥Lp(Ω).The existence of the extension operator T is shown, for example, in [36,§3.1, Theorem 3.1], and Lp-estimates for this operator follow from the cor-responding estimates for weak solutions to the Neumann problem involvedinto the construction of the operator T . Indeed, denote by E :W 1p′(Ω0\Ω)→W 1p′(Ω0) any bounded linear extension operator, i.e. (Eη)∣∣Ω0\Ω = η. Thenthe formulal(η) := −∫Ωb · ∇η˜ dx, η˜ := Eη, η ∈W 1p′(Ω0 \ Ω)determines a bounded linear functional on the Banach space W 1p′(Ω0 \ Ω).Moreover, the condition div b = 0 in D′(Ω) implies that the functional lis independent of the extension operator E. Hence there exists a uniquesolution φ ∈W 1p (Ω0 \ Ω) of the Neumann problem∫Ω0\Ω∇φ · ∇η dx = l(η), ∀ η ∈W 1p′(Ω0 \ Ω),∫Ω0\Ωφ dx = 0, ∥∇φ∥Lp(Ω0\Ω) ≤ c ∥b∥Lp(Ω),and one can take b˜ := b in Ω and b˜ := ∇φ in Ω0 \ Ω and then extend b˜ byzero to the whole R2. It is easy to see that b˜ ∈ Jp(R2) and Tb := b˜ satisfiesall necessary properties.974.4. Proof of Main ResultsFrom the Marcinkiewicz interpolation theorem (see [34, Theorem 1.4.19])we conclude that T is also bounded as an operator from L2,w(Ω) into L2,w(R2):∥b˜∥L2,w(R2) ≤ c2,w∥b∥L2,w(Ω), ∀ b ∈ L2,w(Ω) : div b = 0 in D′(Ω).Let ωϵ be the standard Sobolev kernel ωϵ(x) = ϵ−2ω(x/ϵ), ω ∈ C∞0 (B),∫R2 ω(x) dx = 1, and denoteb˜ϵ := ωϵ ∗ b˜, bϵ := b˜ϵ|Ω.Then for b ∈ L2,w(Ω) we haveb˜ϵ ∈ C∞0 (R2), div b˜ϵ = 0 in D′(R2),b˜ϵ → b˜ in Lp(R2), ∀ p ∈ [1, 2).(4.4.3)Moreover, as the convolution operator Tϵb˜ := ωϵ ∗ b˜ ≡ b˜ϵ is bounded inLp(R2) for any p ∈ (1,+∞) with ∥Tϵ∥Lp→Lp = 1, i.e.∥b˜ϵ∥Lp(R2) ≤ ∥b˜∥Lp(R2), ∀ p ∈ (1,+∞),from the Marcinkiewicz interpolation theorem (see [34, Theorem 1.4.19]) weconclude that Tϵ is also bounded as an operator from L2,w(R2) into L2,w(R2)and the norm ∥Tϵ∥L2,w→L2,w is independent of ϵ:∃ M > 0 : ∥b˜ϵ∥L2,w(R2) ≤ M ∥b˜∥L2,w(R2).Hence bϵ possesses all required properties.Now we can prove Theorem 4.1.2:Proof. Let p > 2 be the exponent defined in Proposition 4.4.1. Note that pdepends only on q, Ω, α and ∥b∥L2,w(Ω). From Proposition 4.4.2 we obtainthe existence of a sequence bϵ ∈ C∞(Ω¯) such thatdiv bϵ = 0 in Ω, bϵ → b a.e. in Ω, ∥bϵ∥L2,w(Ω) ≤ C ∥b∥L2,w(Ω).(4.4.4)Denoteb(α)ϵ := bϵ − αx|x|2 + ϵ2 (4.4.5)and note that∥b(α)ϵ ∥L2,w(Ω) ≤ c(|α|+ ∥b∥L2,w(Ω)). (4.4.6)984.4. Proof of Main ResultsFrom Proposition 4.4.1 we conclude that for any ϵ > 0 there exists a q–weaksolution uϵ to the problem (4.4.1) (with b(α)ϵ defined in (4.4.5)). Moreover,as the constant C in (4.4.2) is a non-decreasing function of ∥b∥L2,w(Ω) from(4.4.6) we conclude that the estimate∥uϵ∥W 1p (Ω) ≤ C ∥f∥Lq(Ω) (4.4.7)holds with some constant C > 0 independent on ϵ. From (4.4.7) we obtainexistence of u ∈W 1p (Ω) such that for some subsequence uϵ we haveuϵ ⇀ u in W 1p (Ω). (4.4.8)On the other hand, from (4.4.4) we conclude there exists a subsequence suchthatb(α)ϵ → b(α) in Lp′(Ω), p′ =pp− 1 . (4.4.9)Hence for any η ∈ C∞0 (Ω) from (4.4.8) and (4.4.9) we obtain∫Ωb(α)ϵ · ∇uϵη dx →∫Ωb(α) · ∇u η dxas ϵ → 0. Passing to the limit in the equations for uϵ (in a weak form) weobtain (4.1.7). Hence u is a p-weak solution to the problem (4.1.1). Theuniqueness of p-weak solutions follows from (4.2.1).Now we present a proof of Theorem 4.1.3. We split it into several steps.Lemma 4.4.3. Assume α < 0. Then there exists p1 > 2 depending only onα and Ω such that for any Bϵ ⋐ Ω and any f ϵ ∈ C∞0 (Ω \ Bϵ) there exists aunique p1–weak solution wϵ ∈◦W 1p1(Ω) to the problem −∆wϵ − |α| x|x|2 · ∇wϵ = −|x|α div f ϵ in D′(Ω),wϵ|∂Ω = 0.(4.4.10)Proof. As the right-hand side of the equation (4.4.10) is smooth it can berepresented in the form div gϵ for some gϵ ∈ Lq1(Ω) with some q1 > 2. Hencethe result follows from Theorem 4.1.2.Lemma 4.4.4. Assume α < 0. Then there exists p2 > 2 depending only onα and Ω such that for any Bϵ ⋐ Ω and any f ϵ ∈ C∞0 (Ω \ Bϵ) there exists aunique p2–weak solution uϵ to the problem −∆uϵ − α x|x|2 · ∇uϵ = −div f ϵ in D′(Ω),uϵ|∂Ω = 0, uϵ(0) = 0.(4.4.11)994.4. Proof of Main ResultsProof. Let wϵ be a unique p1-weak solution to the problem (4.4.10) withsome p1 > 2 depending only on α and Ω. Define the functionuϵ(x) := |x||α|wϵ(x).It is easy to see that uϵ ∈ W 1p2(Ω) for some p2 > 2 depending only on p1and |α|. The direct computation shows that uϵ is a p2-weak solution to theproblem (4.4.11).Lemma 4.4.5. Assume α < 0 and q > 2. There exists p > 2 dependingonly on q, Ω and α such that for any f ∈ Lq(Ω) there exists a unique p-weaksolution u to the problem −∆u− αx|x|2 · ∇u = −div f in Ω,u|∂Ω = 0, u(0) = 0.(4.4.12)Moreover, u satisfies the estimate∥u∥W 1p (Ω) ≤ C ∥f∥Lq(Ω)where C > 0 is some constant depending only on q and Ω.Proof. Assume f ∈ Lq(Ω) and take f ϵ ∈ C∞0 (Ω\Bϵ) so that f ϵ → f in Lq(Ω)as ϵ → +0. Denote by p > 2 the exponent determined in Theorem 4.2.5(without loss of generality we can assume p ≤ p2 where p2 is determinedin Lemma 4.4.4). Denote by uϵ a unique p-weak solution to the problem(4.4.11). Then from Theorem 4.2.5 we obtain∥uϵ∥Lp(Ω) ≤ C ∥f ϵ∥Lq(Ω)and hence there exists u ∈◦W 1p(Ω) such that for some subsequence the weakconvergence uϵ ⇀ u in W 1p (Ω) takes place. It is easy to see that u satisfiesthe integral identity (4.1.7) with b = 0. Moreover, as uϵ(0) = 0 and theimbedding W 1p (Ω) ↪→ C(Ω¯) is compact we obtain u(0) = 0. So, u is a p-weak solution to the problem (4.4.12). The uniqueness of u follows from(4.2.1).Lemma 4.4.6. Assume α < 0, b ∈ C∞(Ω¯), div b = 0 in Ω, q > 2 and p > 2is the exponent depending only on Ω, q, α, ∥b∥L2,w(Ω) defined in Lemma1004.4. Proof of Main Results4.4.5. Then for any f ∈ Lq(Ω), v ∈ Lq(Ω) there exists a unique p-weaksolution u ∈◦W 1p(Ω) to the problem −∆u− αx|x|2 · ∇u = −div(f + bv) in Ω,u|∂Ω = 0, u(0) = 0.Moreover, the operator A : Lq(Ω) → Lq(Ω), A(v) := u, is continuous andcompact.Proof. For given f , v ∈ Lq(Ω) the existence of a unique p-weak solutionu satisfying u(0) = 0 follows from Lemma 4.4.5. Moreover, u satisfies theestimate∥u∥W 1p (Ω) ≤ C (∥f∥Lq(Ω) + ∥b∥L∞(Ω)∥v∥Lq(Ω)).For a given f ∈ Lq(Ω) the last estimate implies∥A(v1)−A(v2)∥W 1p (Ω) ≤ C ∥b∥L∞(Ω)∥v1 − v2∥Lq(Ω)and hence the operator A is continuous as an operator from Lq(Ω) intoLq(Ω). The compactness of this operator in Lq(Ω) follows from the com-pactness of the imbedding of W 1p (Ω) into Lq(Ω).Lemma 4.4.7. Assume α < 0, b ∈ C∞(Ω¯), div b = 0 in Ω, q > 2 and p > 2is the exponent depending only on Ω, q, α, ∥b∥L2,w(Ω) defined in Lemma4.4.5. Then for any f ∈ Lq(Ω) there exists a unique p-weak solution u tothe problem −∆u+(b− α x|x|2)· ∇u = −div f in Ω,u|∂Ω = 0, u(0) = 0.Proof. We apply the Leray-Schauder fixed point theorem for the operatorA : Lq(Ω) → Lq(Ω) defined in Lemma 4.4.6. Assume λ ∈ [0, 1] and v ∈Lq(Ω) satisfies v = λA(v). Denote u := A(v). Then u is a unique p-weaksolution to the problem −∆u+(λb− α x|x|2)· ∇u = −div f in Ω,u|∂Ω = 0, u(0) = 0.1014.4. Proof of Main ResultsFrom Theorem 4.2.5 we obtain the estimate∥u∥W 1p (Ω) ≤ C ∥f∥Lq(Ω)with some constant depending only on q, Ω, α and ∥b∥L2,w(Ω) and indepen-dent of λ ∈ [0, 1]. Hence there exists u ∈◦W 1p(Ω) satisfying u = A(u).Finally, we can relax the smoothness conditions on the divergence-freepart of the drift and prove Theorem 4.1.3:Proof. Assume b satisfies (4.1.2) and let p > 2 be the exponent defined inLemma 4.4.7. From Lemma 4.4.2 we obtain existence of bϵ ∈ C∞(Ω¯) suchthat div bϵ = 0 in Ω, ∥bϵ∥L2,w(Ω) ≤ c ∥b∥L2,w(Ω) ≤ and bϵ → b in Lp′(Ω) wherep′ = pp−1 . From Lemma 4.4.7 we conclude that for any ϵ > 0 there exists aunique p-weak solution uϵ to the problem{−∆uϵ + b(α)ϵ · ∇uϵ = −div f in Ω,uϵ|∂Ω = 0, uϵ(0) = 0,where b(α)ϵ = bϵ − α x|x|2 . From Theorem 4.2.5 we obtain the estimate∥uϵ∥W 1p (Ω) ≤ C ∥f∥Lq(Ω)where the constant C > 0 depends only on a, Ω, α and ∥b∥L2,w(Ω). Thenthere exists u ∈◦W 1p(Ω) such that for some subsequence uϵ the convergence(4.4.8) holds. The rest of the proof repeats the proof of Theorem 4.1.2 inthe beginning of this Section. The identity u(0) = 0 follows from uϵ(0) = 0and compactness of the imbedding of W 1p (Ω) into C(Ω¯). The uniqueness ofp-weak solutions follows from (4.2.1).102Chapter 5Elliptic equations with asingular drift from a weakMorrey spaceIn this chapter we prove the existence, uniqueness and regularity of weaksolutions to the Dirichlet problem for an elliptic equation with a drift bsatisfying div b ≤ 0 in Ω. We assume b belongs to some weak Morrey classwhich includes in the 3D case, in particular, drifts having a singularity alongthe axis x3 with the asymptotic b(x) ∼ c/r, where r =√x21 + x22. Work isdone with Professor Timofie Shilkin.5.1 Introduction and Main ResultsAssume Ω ⊂ Rn is a bounded Lipschitz domain, n ≥ 3. We consider thefollowing boundary value problem:{−∆u+ b · ∇u = −div f in Ω,u|∂Ω = 0.(5.1.1)Here u : Ω→ R is unknown, b : Ω→ Rn and f : Ω→ Rn are given functions.In particular, we are interested in the drift of typen = 3, b(x) = −α x′|x′|2 , x′ = (x1, x2, 0), α ∈ R. (5.1.2)Motivated by this example, we assume that the drift b satisfies the conditionb ∈ L2,n−2w (Ω), (5.1.3)where Lp,λw (Ω) is the weak Morrey space equipped with the quasi-norm∥b∥Lp,λw (Ω):= supx0∈ΩsupR<diamΩR−λp ∥b∥Lp,w(BR(x0)∩Ω),1035.1. Introduction and Main Resultsand Lp,w(Ω) is the weak Lebesgue space equipped with the quasi-norm∥b∥Lp,w(Ω) := sups>0s |{ x ∈ Ω : |b(x)| > s }| 1p , (5.1.4)where |Ω| denotes the n-dimensional Lebesgue measure of Ω. Note that thecondition (5.1.3) implies in particular that b belongs to the usual Morreyspace Lp,n−p(Ω) for any p ∈ [1, 2) (see Proposition 5.2.1 below and thedefinition of Morrey spaces at the end of this section). Note also that bothscales of spaces Lp,n−p(Ω) and Lp,n−pw (Ω) are critical ones, i.e. these spacesare invariant under the scalingbλ(x) = λ b(λx), λ > 0 =⇒ ∥bλ∥Lp,n−pw (Ωλ) = ∥b∥Lp,n−pw (Ω), (5.1.5)where we assume Ω is star-shaped and denote Ωλ := {x ∈ Rn : x/λ ∈ Ω }.Concerning further properties of weak Morrey spaces we refer to [69], [84].Here we emphasise only that the scale of weak Morrey spaces Lp,n−pw (Ω) isconvenient for the description of drifts b which have a certain asymptoticnear some singular submanifold Σ ⊂ Ω. For example, for n = 3 we haveb(x) ∼ 1x3 , Σ = a plane =⇒ b ∈ L1,2w (Ω),b(x) ∼ 1√x21+x22, Σ = a line =⇒ b ∈ L2,1w (Ω),b(x) ∼ 1√x21+x22+x23, Σ = a point =⇒ b ∈ L3,0w (Ω) = L3,w(Ω).We define the bilinear form B[u, η] byB[u, η] :=∫Ωη b · ∇u dx. (5.1.6)Note that for b satisfying (5.1.3) the bilinear form (5.1.6) generally speakingis not well-defined for u ∈W 12 (Ω) and η ∈ C∞0 (Ω) whereW 1p (Ω) := {u ∈ Lp(Ω) : ∇u ∈ Lp(Ω) }, p ∈ [1,+∞],is the usual Sobolev space and C∞0 (Ω) is the linear space of C∞–smoothfunctions compactly supported in Ω.Nevertheless, the bilinear form B[u, η] is well-defined at least for u ∈W 1p (Ω) with p > 2 and η ∈ Lq(Ω) with q > 2pp−2 . So, instead of the standardnotion of weak solutions from the energy class W 12 (Ω) we introduce thedefinition of p-weak solutions to the problem (5.1.1), see also [38], [40], [81]:1045.1. Introduction and Main ResultsDefinition 5.1.1. Assume p > 2, b ∈ Lp′(Ω), p′ = pp−1 , and f ∈ L1(Ω).We say u is a p-weak solution to the problem (5.1.1) if u ∈◦W 1p(Ω) and usatisfies the identity∫Ω∇u · ∇η dx + B[u, η] =∫Ωf · ∇η dx, ∀ η ∈ C∞0 (Ω). (5.1.7)Note that if u is a weak or a p-weak solution to (5.1.1) and f ∈ L2(Ω)then by density arguments we can extend the class of test functions in (5.1.7)from η ∈ C∞0 (Ω) to all functions η ∈◦W 12(Ω) ∩ L∞(Ω). In this chapter wealways assume that the vector field b has a sign-defined divergence:div b ≤ 0 in D′(Ω), (5.1.8)where D′(Ω) is the space of distributions on Ω. In the case (5.1.2) the con-dition (5.1.8) corresponds to α ≥ 0. The important impact of the condition(5.1.8) is due to the fact that in this case the quadratic form B[u, u] providesa positive support to the quadratic form of the elliptic operator in (5.1.1)(see Proposition 5.2.2 below), while in the opposite case div b ≥ 0 in Ω thequadratic form B[u, u] is (formally) non-positive and hence it “shifts” theoperator to the “spectral area”. It is well-known that in the case of a non-regular drift b the violation of the condition (5.1.8) potentially can ruin theuniqueness for the problem (5.1.1) even in the class of smooth solutions. Forexample, in the case Ω = B where B := {x ∈ Rn : |x| < 1 } the functionsu(x) = c(|x|2 − 1), b(x) = n x|x|2 , b ∈ Ln,w(B), div b ≥ 0 in D′(B)(5.1.9)satisfy to (5.1.1) with f ≡ 0. Note also that in the regular case b ∈ Ln(Ω)the uniqueness for the problem (5.1.1) holds regardless of the sign of div bdue to the maximum principle, see, for example, [33] and reference there.Together with (5.1.3) and (5.1.8) we will assume that b also possess thefollowing approximation property:∃ {bk}∞k=1 ⊂ C∞(Ω¯) : div bk ≤ 0 in Ω,∥bk∥L2,n−2w (Ω) ≤ c ∥b∥L2,n−2w (Ω), ∥bk − b∥L1(Ω) → 0.(5.1.10)To construct a smooth approximation (5.1.10) one can use, for example, theconvolution of b with a standard smooth Sobolev kernel. It is well-knownthat such convolution preserves Morrey-type norms. Nevertheless, to pre-serve the sign of the divergence in the whole domain Ω we need to construct1055.1. Introduction and Main Resultsan “extension–approximation” of the superharmonic function representingthe potential part of the drift b. Though an extension of a superharmonicfunction is a non-trivial problem (see, for example, [72]), in the case of astar-shaped domain the corresponding “extension–approximation” can beobtained easily by the dilatation. So, for the reader convenience we presenta construction of the approximation (5.1.10) for an arbitrary b satisfying(5.1.3), (5.1.8) in the case of a star-shaped Ω, see Proposition B.0.1 in Ap-pendix.The main results of this chapter are the following two theorems:Theorem 5.1.1. Assume b satisfies (5.1.3), (5.1.8), (5.1.10). Then thereexist p > 2 depending only on n, Ω and ∥b∥L2,n−2w (Ω)such that for anyf ∈ Lp(Ω) there exists a unique p-weak solution u ∈ W 1p (Ω) to the problem(5.1.1). Moreover, this solution satisfies the estimate∥u∥W 1p (Ω) ≤ c ∥f∥Lp(Ω), (5.1.11)with a constant c > 0 depending only on Ω, n and ∥b∥L2,n−2w (Ω).Theorem 5.1.2. Assume b satisfies (5.1.3), (5.1.8) and p > 2. Then for anyq > n there exists µ ∈ (0, 1) depending only on n, p, q, Ω and ∥b∥L2,n−2w (Ω)such that if u is a p–weak solution to the problem (5.1.1) corresponding tothe right-hand side f ∈ Lq(Ω) then u is Ho¨lder continuous on Ω¯ with theexponent µ and the estimate∥u∥Cµ(Ω¯) ≤ c ∥f∥Lq(Ω), (5.1.12)holds with the constant c > 0 depending only on q, p, n, Ω and ∥b∥L2,n−2w (Ω).The study of the weak solvability and properties of weak solutions todrift-diffusion equations has a long history. Below we focus on the briefoverview of results concerning the existence of weak solutions, their unique-ness, boundedness and Ho¨lder continuity.Very often the study of the problem (5.1.1) is coupled with the study ofthe dual problem {−∆v − div(bv) = div g in Ω,v|∂Ω = 0.(5.1.13)In the case of the divergence–free driftdiv b = 0 in D′(Ω) (5.1.14)1065.1. Introduction and Main Resultsproblems (5.1.1) and (5.1.13) are equivalent, but for the general drifts inis not the case. The reason is that the dual problem (5.1.13) is presentedin the divergent form. Hence weak solutions, depending on the regularityof b, in principle are well-defined in a wider class than the energy spaceW 12 (Ω) (one can consider classes of very weak solutions, renormalized so-lutions etc.) For example, in a series of papers [61], [62], [63] (see also[64] and reference there) the existence of weak (distributional) solutions forthe problem (5.1.13) in various functional classes under various critical andsupercritical assumptions on b was studied.Another approach to the solvability of the problem (5.1.1) with the roughcoefficients was developed in a series of papers [76], [77], [78], [79] (see alsoreferences there). This approach is applicable to elliptic operators in thenon-divergent form (with the main part −ajk(x) ∂2u∂xj∂xk with a non-smoothuniformly elliptic matrix a(x) = (ajk(x)) instead of the Laplacian) and thedrift b belonging to the critical Morrey spaces. In this approach the existenceof solutions in Sobolev spaces W 2p (Ω) with some p > 1 is investigated andthe equation (5.1.1) is understood a.e. in Ω.For the problem (5.1.1) weak solutions from W 12 (Ω) formally are well-defined only for b ∈ L2(Ω) which does not cover the case (5.1.3). Neverthe-less, weak solutions to elliptic and parabolic equations with rough coefficientscommonly known to possess the higher integrability property. This meansthat the gradient of a weak solution is integrable with some exponent whichis slightly greater than 2. One can ask what are the optimal conditions onthe drift b which provide the higher integrability of weak solutions to theproblem (5.1.1)? The first results in this direction were obtained earlierin [39], [40], [81] where the critical drift b ∈ Ln,w(Ω) satisfying (5.1.8) wasstudied. So, our Theorem 5.1.1 can be viewed as a far away extension ofthese results. Note also that the 2D case was investigated separately in [65]and [82].All results concerning weak solvability and properties of weak solutionsto the problems (5.1.1) and (5.1.13) formally can be split onto three groups:the results related to the subcrtitical drifts, the results for critical driftsand supercritical results. We remind that we call some functional classX supercritical if for b ∈ X and bλ defined in (5.1.5) the norm ∥bλ∥X(B)computed over a unit ball B blows up as λ→ 0.The results concerning the drift b belonging to subcritical Lesbeguespaces are classical one, see [43]. The same is true for drifts belongingto the critical Lebesgue spaceb ∈ Ln(Ω). (5.1.15)1075.1. Introduction and Main ResultsWe call drifts satisfying (5.1.15) regular. In the regular case the drift termprovides a compact perturbation of the Laplace operator and hence the prob-lem (5.1.1) possesses the Fredholm property (i.e. the existence for (5.1.1)follows from the uniqueness). Moreover, in the case (5.1.15) weak solutionsare Ho¨lder continuous and the maximum principle holds (which provides theuniqueness of weak solutions), see the survey of the results, for example, in[33]. So, we can consider the regular case as a starting point of our investiga-tion. Note that the necessary and sufficient conditions for the boundednessof the bilinear form (5.1.6) on the Sololev space W 12 (Rn) were investigatedin [45].The counterexample (5.1.9) shows that even the minimal relaxation ofthe condition (5.1.15) (still within the critical scale) can ruin the maximumprinciple and the uniqueness for the problem (5.1.1). So, to recover theuniqueness we need to impose some additional assumptions on b. For ex-ample, one can consider the condition (5.1.8) or its particular case (5.1.14).(Motivated by the properties of the quadratic form B[u, u] we call the con-dition (5.1.8) “non-spectral”.) In this result we focus only on the critical“non-spectral” case and we are going to investigate the critical “spectral”case div b ≥ 0 in a separate forthcoming paper, see [65] and [82] for therelated 2D results.For b ∈ L2(Ω) satisfying (5.1.14) the uniqueness of weak solutions inthe energy class W 12 (Ω) can be found in [58] (see also [57] for the paraboliccase). In [58] a counterexample to the uniqueness for the problem (5.1.1)with a divergence-free drift was also constructed if the condition b ∈ L2(Ω)is violated (see [67] and references there for the further developments in thisdirection). For b ∈ L2(Ω) satisfying the “non-spectral” condition (5.1.8)the corresponding uniqueness result in the energy class W 12 (Ω) can be easilyobtained by a straightforward adaptation of the methods of [57], [58]. For b ∈L2,w(Ω) satisfying (5.1.8) our Proposition 5.3.1 below provides a uniquenessresult for the problem (5.1.1) in a class of weak solutions which possess thehigher integrability property.The next issue is the boundedness of weak solutions. It is well-knownthat even in the divergence–free case (5.1.14) the local boundedness of weaksolutions requires some extra regularity of the drift, see [47], [33], [71], [59],see also a recent survey [60] and reference there. Nevertheless, our Propo-sition 5.3.1 shows that for a drift b ∈ L2,w(Ω) satisfying (5.1.8) p–weaksolutions to the boundary value problem (5.1.1) are always bounded with-out any extra assumptions on the regularity of b. We emphasize that ourresult is global (i.e. it requires smooth boundary conditions) and can not belocalized. The phenomena of the dependence of local boundedness of weak1085.1. Introduction and Main Resultssolutions on their behaviour on the boundary of the domain is discussed in[33].The last issue we are going to discuss is Ho¨lder continuity of weak so-lutions. Note that under the critical condition (5.1.15) Ho¨lder continuityis the optimal regularity. For the problems (5.1.1), (5.1.13) with a drift bbelonging to the subcritical Morrey space b ∈ L2,λ(Ω) with λ ∈ (n − 2, n)Ho¨lder continuity of weak solutions is known for a long time, see [68] and[83]. On the other hand, the counterexample in [32] shows that even in thedivergence–free case (5.1.14) the smallest violation of the critical scale canlead to the loss of continuity by weak solutions. So, in this work we focuson results concerning drifts belonging to critical spaces.For the divergence-free drifts from critical spaces the best known for to-day results were obtained in [31], [51] (see also [53]) where Ho¨lder continuityof weak solutions was proved for the heat equation with a drift term sat-isfying (5.1.14) (see also the related result in [44]). In its elliptic version,these results assume that the drift b satisfying (5.1.14) belongs to the spaceBMO−1(Ω), i.e. there exists a skew-symmetric matrix A ∈ BMO(Ω) suchthat b = divA (see the definition of BMO space at the end of this section).If we assume ∇A ∈ L1,n−1(Ω) (which is in a sense the optimal regularityfor the critical Morrey scale Lp,n−p(Ω)) then from Trudinger’s imbedding[85] we obtain A ∈ BMO(Ω) and hence Ho¨lder continuity of weak solutionsfollows from [31], [51].For the non–divergence free critical drifts satisfying the condition (5.1.8)local a priori estimates of the Ho¨lder norms for Lipschitz continuous solutionsto the problem (5.1.1) were obtained earlier in [47] under the additionalrestriction b ∈ Lr,n−r(Ω) with r ∈ (n2 , n]. So, in our Theorem 5.1.2 weremove this restriction and require only b ∈ L2,n−2w (Ω). Note also that ourpresent contribution can be viewed as a multidimensional analogue of theresults obtained earlier in [65] in the 2D case.Here are some comments and possible extensions: With minor changes in the proofs our technique allows us to obtain re-sults similar to our Theorems 5.1.1–5.1.2 if we replace the Laplace op-erator in (5.1.1) by an elliptic operator in the divergence form−div(a(x)∇u)with a uniformly elliptic matrix a(x) = (ajk(x)), ajk ∈ L∞(Ω). Weconsider this extension to be obvious. In the case of the Laplace operator in (5.1.1) (or, more generally, in thecase of the operator −div(a(x)∇u) with a sufficiently smooth matrixa(x)) our p-weak solutions in Theorem 5.1.1 possess locally integrable1095.1. Introduction and Main Resultssecond derivatives u ∈ W 2s,loc(Ω) with some s > 1 (if f is sufficientlyregular). Hence for p-weak solutions the equation (5.1.1) is valid a.e.in Ω. See the related results in [76], [77], [78], [79] for the case of ellipticoperators in the non-divergent form with non-smooth coefficients. In our investigation of Ho¨lder continuity of weak solutions in Theorem5.1.2 we focus on the “optimal” conditions for b but not for f . It is well-known (see, for example, [73, Theorem 5.17]) that Ho¨lder continuityof weak solutions to the problem (5.1.1) is valid if the right hand sidef belongs to some appropriate Morrey space. So, we believe that thecondition f ∈ Lq(Ω) with q > n in Theorem 5.1.2 can be relaxed, seerelated results, for example, in [68], [69].This chapter is organized as follows. In Section 5.2 we introduce someauxiliary results which are basic tools of our work. In Section 5.3 we obtainthe higher integrability of weak solutions to the problem (5.1.1) and proveTheorems 5.1.1. In Section 5.4 we introduce some De Giorgi–type classeswhich are convenient for the study of the equations with the coefficients fromMorrey spaces. These De Giorgi classes are very similar to the classical ones(see [73], [74], [43]) and readers familiar with the De Giorgi technique canomit this Section. In Section 5.5 we show that p–weak solutions to theproblem (5.1.1) belong to the above De Giorgi classes and hence they areHo¨lder continuous. So, in Section 5.5 we prove Theorem 5.1.2. Finally, inAppendix we show that in the case of a star-shaped domain Ω any functionb satisfying (5.1.3), (5.1.8) possesses the approximation property (5.1.10)which allows us to prove existence of weak solutions via a limiting procedure.In the chapter we use the following notation. For any a, b ∈ Rn we denoteby a · b = akbk their scalar product in Rn. Repeated indexes assume thesummation from 1 to n. An index after comma means partial derivative withrespect to xk, i.e. f,k :=∂f∂xk. We denote by Lp(Ω) and Wkp (Ω) the usualLebesgue and Sobolev spaces. We do not distinguish between functionalspaces of scalar and vector functions and omit the target space in notation.C∞0 (Ω) is the space of smooth functions compactly supported in Ω. Thespace◦W 1p(Ω) is the closure of C∞0 (Ω) in W1p (Ω) norm. For p > 1 we denotep′ = pp−1 . The space of distributions on Ω is denoted by D′(Ω). By Cµ(Ω¯),µ ∈ (0, 1) we denote the spaces of Ho¨lder continuous functions on Ω¯ equippedwith the norm∥u∥Cµ(Ω¯) := supx∈Ω|u(x)| + supx,y∈Ω: x̸=y|u(x)− u(y)||x− y|µ .1105.2. Auxiliary resultsThe symbols ⇀ and → stand for the weak and strong convergence respec-tively. We denote by BR(x0) the ball in Rn of radius R centered at x0 andwrite BR if x0 = 0. We write B instead of B1. For a domain Ω ⊂ Rn we alsodenote ΩR(x0) := Ω∩BR(x0). For an open set Ω0 ⊂ Rn we write Ω ⋐ Ω0 ifΩ¯ is compact and Ω¯ ⊂ Ω0. For u ∈ L∞(Ω) we denoteoscΩ u := ess supΩu− essinfΩ u.We denote by Lp,λ(Ω) the Morrey space equipped with the norm∥u∥Lp,λ(Ω) := supx0∈ΩsupR<diamΩR−λp ∥u∥Lp(ΩR(x0)).We denote by χω the indicator function of the set ω ⊂ Rn. For u : Ω → Rand k ∈ R we denote (u − k)+ := max{u − k, 0}. We denote by BMO(Ω)the space of functions with a bounded mean oscillation equipped with thesemi-norm[u]BMO(Ω) := supx0∈ΩsupR<diamΩR−n∫ΩR(x0)|u(x)− (u)ΩR(x0)| dx,where (f)ω stands for the average of f over the domain ω ⊂ Rn. Also wedenote by f ∗ g the convolution of functions f , g : Rn → R:(f)ω := −∫ωf dx =1|ω|∫ωf dx, (f ∗ g)(x) :=∫Rnf(x− y)g(y) dy.5.2 Auxiliary resultsIn this Section we present several auxiliary results. For reader’s conveniencewe formulate these results not in their full generality but only in the formthey are used in our work.The first result shows that by relaxing an exponent of the integrabilityof a function we can always switch from a weak Morrey norm to a regularone.1115.2. Auxiliary resultsProposition 5.2.1. For any p ∈ (1, n) and q ∈ [1, p) there is a constantc > 0 depending only on n, p, q and Ω such that the following inequalityholds:∥b∥Lq,n−q(Ω) ≤ c ∥b∥Lp,n−pw (Ω)Proof. The result follows from the Ho¨lder inequality for Lorentz spaces, see[34, Section 1.4].The next result shows that under the condition (5.1.8) the quadraticform corresponding the drift term is non-negative.Proposition 5.2.2. Assume p ≥ 2, p′ = pp−1 , and b ∈ Lp′(Ω) satisfies(5.1.8). Let B[u, η] be the bilinear form defined in (5.1.6). Then∀u ∈ L∞(Ω) ∩◦W 1p(Ω) B[u, u] ≥ 0. (5.2.1)Proof. The result follows from the inclusions |u|2 ∈◦W 1p(Ω), b ∈ Lp′(Ω),p′ = pp−1 , and the inequalityB[u, u] = 12∫Ωb · ∇|u|2 dx ≥ 0.The next result is the general Marcinkiewicz interpolation theorem inLorentz spaces, see [2, Theorem 5.3.2].Proposition 5.2.3. Assume Ω, Ω′ ⊂ Rn are bounded domains and assumefor some 1 < p0 < p1 < ∞ a linear operator T : Lpj (Ω) → Lpj (Ω′) isbounded for j = 0, 1 and∥T∥Lp0 (Ω)→Lp0 (Ω′) ≤ M0, ∥T∥Lp1 (Ω)→Lp1 (Ω′) ≤ M1Then for any p ∈ (p0, p1) the operator T is bounded from Lp,w(Ω) to Lp,w(Ω′)and∥T∥Lp,w(Ω)→Lp,w(Ω′) ≤ M1−θ0 M θ1where θ ∈ (0, 1) satisfies 1p = 1−θp0 + θp1 .The next proposition proved by Chiarenza and Frasca in [66] is the well-known extension of the result of C. Fefferman [70] for p = 2. This theoremis one of basic tools in our proofs of both Theorems 5.1.1 and 5.1.2.1125.3. Higher integrability of weak solutionsProposition 5.2.4. Assume p ∈ (1, n), r ∈ (1, np ] and V ∈ Lr,n−pr(Ω).Then∫Ω|V | |u|p dx ≤ cn,r,p ∥V ∥Lr,n−rp(Ω)∥∇u∥pLp(Ω), ∀u ∈ C∞0 (Ω).with the constant cn,r,p > 0 depending only on n, r and p.5.3 Higher integrability of weak solutionsIn this Section we present the proof of Theorem 5.1.1. First we show thatin the case of (5.1.8) any p–weak solution corresponding to a sufficientlyregular right-hand side is bounded.Proposition 5.3.1. Assume p ≥ 2, p′ = pp−1 , b ∈ Lp′(Ω) satisfies (5.1.8),q > n and f ∈ Lq(Ω). Then any weak solution u ∈ W 1p (Ω) to the problem(5.1.1) (in the sense of the integral identity (5.1.7)) is essentially boundedand satisfies the estimate∥u∥L∞(Ω) ≤ c ∥f∥Lq(Ω), (5.3.1)with a constant c > 0 depending only on Ω, n and q.Proof. For m > 0 we define a truncation Tm : R→ R by Tm(s) := m for s ≥m, Tm(s) = s for s < m. For any s ∈ R we also denote (s)+ := max{s, 0}.Now we fix some m > 0 and denote u¯ := Tm(u). Then for any k ≥ 0 wehave(u¯− k)+ ∈ L∞(Ω) ∩◦W 1p(Ω), ∇(u¯− k)+ = χΩ[k<u<m]∇uwhere Ω[k < u < m] = {x ∈ Ω : k < u(x) < m }. Approximatingη := (u¯ − k)+ by smooth functions we can take η as a test function in(5.1.7). For k ≥ m we have η ≡ 0 and hence B[u, η] = 0. For k < m thecondition (5.1.8) impliesB[u, η] = 12∫Ω[k<u<m]b · ∇|η|2 dx + (m− k)∫Ω[u≥m]b · ∇u dx= B[η, η] + (m− k)∫Ωb · ∇(u−m)+ dx ≥ 01135.3. Higher integrability of weak solutionsHence for Ak := {x ∈ Ω : u¯(x) > k } from (5.1.7) we obtain∫Ω|∇(u¯− k)+|2 dx ≤ ∥f∥2Lq(Ω)|Ak|1− 2q , ∀ k ≥ 0,which implies (see [43, Chapter II, Lemma 5.3])ess supΩu¯ ≤ c |Ω|δ ∥f∥Lq(Ω), δ := 1n − 1q ,with some constant c > 0 depending only on n and q. As this estimate isuniform with respect to m > 0 we conclude u is essentially bounded fromabove andess supΩu ≤ c |Ω|δ ∥f∥Lq(Ω).Applying the same procedure to u¯ := Tm(−u) instead of u¯ := Tm(u) weobtain u ∈ L∞(Ω) as well as (5.3.1).Proposition 5.3.1 implies the uniqueness theorem for the problem (5.1.1)in the class of p-weak solutions.Proposition 5.3.2. Assume p ≥ 2, p′ = pp−1 , and b ∈ Lp′(Ω) satisfies(5.1.8). Then for any f ∈ Lp(Ω) the problem (5.1.1) can have at most oneweak solution in the class W 1p (Ω).Proof. Assume u1 ∈ W 1p (Ω) and u2 ∈ W 1p (Ω) are two weak solutions of(5.1.1) corresponding to the same f ∈ Lp(Ω). Then u := u1 − u2 is a weaksolution of (5.1.1) corresponding to f ≡ 0. From Proposition 5.3.1 we obtainu ≡ 0 in Ω.The next proposition is the energy estimate. Note that the constantc > 0 in this estimate does not depend on the drift b satisfying the condition(5.1.8).Proposition 5.3.3. Assume p ≥ 2, p′ = pp−1 , b ∈ Lp′(Ω) satisfies (5.1.8),and f ∈ Lq(Ω) with q > n. Then for any weak solution u ∈ W 1p (Ω) to theproblem (5.1.1) the following estimate hold:∥u∥W 12 (Ω) ≤ c ∥f∥L2(Ω)with some constant c > 0 depending only on n and Ω.1145.3. Higher integrability of weak solutionsProof. From Proposition 5.3.1 we obtain u ∈ L∞(Ω) and hence the testfunction η := u ∈ L∞(Ω)∩◦W 1p(Ω) is admissible for (5.1.7). From Proposition5.2.2 we obtain B[u, u] ≥ 0 and hence from (5.1.7) we obtain the inequality∥∇u∥2L2(Ω) ≤∫Ωf · ∇u dxwhich the result follows from.Our next result is the estimate of the bilinear form corresponding to thedrift term. This result is a direct consequence of the Feffermann–Chiarenza–Frasca estimate, see Proposition 5.2.4.Proposition 5.3.4. Assume r ∈(2nn+2 , 2)and b ∈ Lr,n−r(Ω). Then thereexists c > 0 depending only on n and r such that for any u ∈ W 12 (Ω) andζ ∈ C∞0 (Ω) the following estimate holds:∫Ωζ3 |b| |u|2 dx ≤ c ∥b∥Lr,n−r(Ω) ∥∇(ζ2u)∥L2(Ω) ∥∇(ζu)∥L 2nn+2(Ω). (5.3.2)Moreover, for any θ ∈ (nr − n2 , 1) there exists c > 0 depending only on n, rand θ such that for any u ∈W 12 (Ω) and any ζ ∈ C∞0 (Ω) satisfying 0 ≤ ζ ≤ 1we have∫Ωζ1+θ |b| |u|2 dx ≤ c ∥b∥Lr,n−r(Ω)∥u∥1−θL2(Ω)∥∇(ζu)∥1+θL2(Ω)|Ω|θ/n. (5.3.3)If we additionally assume u ∈◦W 12(Ω) then the estimates (5.3.2) and (5.3.3)remains true for an arbitrary ζ ∈ C∞0 (Rn).Proof. To prove (5.3.2) we apply the Ho¨lder inequality∫Ωζ3 |b| |u|2 dx ≤ c ∥ζ2u∥L 2nn−2(Ω)∥ζbu∥L 2nn+2(Ω).Denote q := 2nn+2 , V := |b|q, r1 := rq , r1 > 1. ThenV ∈ Lr1,n−qr1(Ω), ∥V ∥Lr1,n−qr1 (Ω) = ∥b∥qLr,n−r(Ω)and from Proposition 5.2.4 we obtain∥ζbu∥Lq(Ω) ≤ c ∥V ∥1/qLr1,n−qr1 (Ω)∥∇(ζu)∥Lq(Ω) ≤ c ∥b∥Lr,n−r(Ω) ∥∇(ζu)∥Lq(Ω).1155.3. Higher integrability of weak solutionsOn the other hand, by the Gagliardo-Nirenberg inequality we obtain∥ζ2u∥L 2nn−2(Ω) ≤ c ∥∇(ζ2u)∥L2(Ω)and we arrive at (5.3.2).Now we turn to the proof of (5.3.3). Take q ∈(2nn+2 , r)such that1q′ =1−θ2 + θn−22n , q′ = qq−1 .Then ∫Ωζ1+θ |b| |u|2 dx ≤ c ∥ζθu∥Lq′ (Ω)∥ζbu∥Lq(Ω).Denote V := |b|q, r1 := rq , r1 > 1. Then V ∈ Lr1,n−qr1(Ω) and∥ζbu∥Lq(Ω) ≤ c ∥V ∥1/qLr1,n−qr1 (Ω)∥∇(ζu)∥Lq(Ω) ≤ c ∥b∥Lr,n−r(Ω) ∥∇(ζu)∥Lq(Ω)On the other hand, we can interpolate∥ζθu∥Lq′ (Ω) ≤ cθ ∥u∥1−θL2(Ω)∥∇(ζu)∥θL2(Ω).So, we obtain∫Ωζ1+θ |b| |u|2 dx ≤ c ∥u∥1−θL2(Ω)∥∇(ζu)∥θL2(Ω)∥b∥Lr,n−r(Ω) ∥∇(ζu)∥Lq(Ω).Using the Ho¨lder inequality we obtain∥∇(ζu)∥Lq(Ω) ≤ ∥∇(ζu)∥L2(Ω)|Ω|1q− 12 = ∥∇(ζu)∥L2(Ω)|Ω|θnwhich completes the proof.The next proposition is the higher integrability estimate which is a mainresult of the present Section. In this proposition we assume that b is smoothand we derive (5.1.11) as an a priori estimate for a sufficiently regular drift.Proposition 5.3.5. Assume b ∈ C∞(Ω¯) satisfies (5.1.8) and assume f ∈Lq(Ω) with q > n. Let u ∈W 12 (Ω)∩L∞(Ω) be a weak solution to the problem(5.1.1) (i.e. the identity (5.1.7) holds). Then there exists p > 2 dependingonly on n, q , Ω and ∥b∥L2,n−2w (Ω)such that u ∈ W 1p (Ω) and the estimate(5.1.11) holds with some constant c > 0 depending only on n, q, Ω and∥b∥L2,n−2w (Ω).1165.3. Higher integrability of weak solutionsProof. Let us fix some r ∈ ( 2nn+2 , 2). First we consider an internal pointx0 ∈ Ω. Take R > 0 such that B2R(x0) ⊂ Ω and choose a cut-off functionζ ∈ C∞0 (B2R(x0)) so that ζ ≡ 1 on BR(x0) and |∇ζ| ≤ c/R. Denoteu¯ = u − (u)B2R(x0). Then the function η = ζ4u¯ is admissible for (5.1.7). Itis easy to see thatB[u, ζ4u¯] = B[ζ2u¯, ζ2u¯] − 2∫B2R(x0)ζ3|u¯|2 b · ∇ζ dx.Taking into account (5.2.1) from (5.1.7) we obtain∥ζ2∇u∥2L2(B2R(x0)) ≤ c(∥u¯∇ζ∥2L2(B2R(x0)) + ∥f∥2L2(B2R(x0)))+c∫B2R(x0)ζ3|u¯|2 b · ∇ζ dx.We estimate the last term with the help of (5.3.2):∫B2R(x0)ζ3|u¯|2 b · ∇ζ dx ≤≤ ∥∇ζ∥L∞(B2R(x0)) ∥b∥Lr,n−r(Ω) ∥∇(ζ2u¯)∥L2(B2R(x0)) ∥∇(ζu¯)∥L 2nn+2(B2R(x0)).Using the Young inequality we get∥ζ2∇u∥2L2(B2R(x0)) ≤ c(∥u¯∇ζ∥2L2(B2R(x0)) + ∥f∥2L2(B2R(x0)))++ c ∥∇ζ∥2L∞(B2R(x0)) ∥b∥2Lr,n−r(Ω) ∥∇(ζu¯)∥2L 2nn+2(B2R(x0)).Taking into account ζ ≡ 1 on BR(x0) and ∥∇ζ∥L∞(B2R(x0)) ≤ cR and theinequalities∥u¯∥L2(B2R(x0)) ≤ c ∥∇u∥L 2nn+2(B2R(x0)),∥∇(ζu¯)∥L 2nn+2(B2R(x0)) ≤ c ∥∇u∥L 2nn+2(B2R(x0)),we arrive at∥∇u∥L2(BR(x0)) ≤ c ∥f∥L2(B2R(x0)) +cR(1+∥b∥Lr,n−r(Ω))∥∇u∥L 2nn+2(B2R(x0)).Now we derive similar estimates near the boundary. Extend u and f by zerooutside Ω and denote this extension by u˜ and f˜ . Note that as ∂Ω is Lipschitz1175.3. Higher integrability of weak solutionsthere exist R0 > 0 and δ0 > 0 depending on Ω such that for any R < R0and any x0 ∈ ∂Ω |BR(x0) \ Ω| ≥ δ0|BR|. Take x0 ∈ ∂Ω, R < R0/2 anddenote ΩR(x0) := BR(x0) ∩ Ω. Choose a cut-off function ζ ∈ C∞0 (B2R(x0))so that ζ ≡ 1 on BR(x0). Then the functionη := ζ4u ∈ L∞(Ω2R(x0)) ∩◦W 12(Ω2R(x0))is admissible for the identity (5.1.7). After routine computations similar toabove we obtain for any x0 ∈ ∂Ω and R < R0∥∇u∥L2(ΩR(x0)) ≤ c ∥f∥L2(Ω2R(x0)) +cR(1 + ∥b∥Lr,n−r(Ω))∥∇u∥L 2nn+2(Ω2R(x0)).(5.3.4)Combining internal and boundary estimates in the standard way and divid-ing the result by Rn/2 for any x0 ∈ Ω¯ and R < R0 we obtain the reverseHo¨lder inequality for ∇u: −∫ΩR(x0)|∇u|2 dx12≤ c1 −∫Ω2R(x0)|f |2 dx12++ c2(1 + ∥b∥Lr,n−r(Ω)) −∫Ω2R(x0)|∇u| 2nn+2 dxn+22n.From Proposition 5.2.1 we conclude∥b∥Lr,n−r(Ω) ≤ c ∥b∥L2,n−2w (Ω).Now the higher integrability of u follows via Gehring’s lemma, see, for exam-ple, [35, Chapter V]. Namely, there exists p > 2 and c∗ > 0 depending onlyon c1, c2 and ∥b∥L2,n−2w (Ω) such that ∇u ∈ Lp(Ω) and the following estimateholds:∥∇u∥Lp(Ω) ≤ c∗ (∥∇u∥L2(Ω) + ∥f∥Lp(Ω)).Combining this estimate with Proposition 5.3.3 we obtain (5.1.11).Finally we can prove Theorem 5.1.1.Proof. Let us fix some q > n. Let p > 2 be the exponent defined in Propo-sition 5.3.5 and denote p′ = pp−1 . For f ∈ Lp(Ω) we can find fk ∈ C∞0 (Ω)1185.4. Modified De Giorgi classessuch that ∥fk − f∥Lp(Ω) → 0. As b possesses the approximation property(5.1.10) there exists bk ∈ C∞(Ω¯) such that div bk ≤ 0 in Ω and∥bk∥L2,n−2w (Ω) ≤ c ∥b∥L2,n−2w (Ω), ∥bk − b∥Lp′ (Ω) → 0 as k → 0.Let uk ∈W 12 (Ω) ∩ L∞(Ω) be weak solutions to the regularized problem{−∆uk + bk · ∇uk = −div fk in Ω,uk|∂Ω = 0.From Proposition 5.3.5 we obtain the estimate∥uk∥W 1p (Ω) ≤ c ∥fk∥Lp(Ω)with a constant c > 0 depending only on n, q, Ω and ∥b∥L2,n−2w (Ω). Hence wecan take a subsequence uk such thatuk ⇀ u in W1p (Ω).As bk → b in Lp′(Ω) we can pass to the limit in the identity∫Ω∇uk ·(∇η + bkη)dx =∫Ωfk · ∇η dx, ∀ η ∈ C∞0 (Ω),and obtain (5.1.7). Hence u ∈◦W 1p(Ω) is a p-weak solution to the problem(5.1.1). Its uniqueness follows from Proposition 5.3.2.5.4 Modified De Giorgi classesIn this Section we introduce the modified De Giorgi classes which are con-venient for the study of solutions to the elliptic equations with coefficientsfrom Morrey spaces. The only difference of our classes from the standard DeGiorgi classes (see, for example [73], [74], [43]) is the additional multiplier1 + Rα(R−ρ)α in the inequalities (5.4.1) and (5.4.2).Definition 5.4.1. We say u ∈ DG(Ω, k0) if u ∈ W 12 (Ω) and there existconstants γ > 0, F > 0, α ≥ 0, q > n such that for any BR(x0) ⊂ Ω, any1195.4. Modified De Giorgi classes0 < ρ < R and any k ≥ k0 the following inequality holds∫Ak∩Bρ(x0)|∇u|2 dx ≤ γ2(R− ρ)2(1 +Rα(R− ρ)α) ∫Ak∩BR(x0)|u− k|2 dx ++ F 2 |Ak ∩BR(x0)|1−2q(5.4.1)where we denote Ak := {x ∈ Ω : u(x) > k }. We write u ∈ DG(Ω) ifu ∈ DG(Ω, k0) for any k0 ∈ R.The second modified De Giorgi class characterizes the behavior of Sobolevfunctions near the boundary of the domain:Definition 5.4.2. We say u ∈ DG(∂Ω) if u ∈◦W 12(Ω) and there existconstants γ > 0, F > 0, α ≥ 0, q > n, R0 > 0 such that for any x0 ∈ ∂Ω,any 0 < ρ < R ≤ R0 and any k ≥ 0 the following inequality holds∫Ak∩Ωρ(x0)|∇u|2 dx ≤ γ2(R− ρ)2(1 +Rα(R− ρ)α) ∫Ak∩ΩR(x0)|u− k|2 dx ++ F 2 |Ak ∩ ΩR(x0)|1−2q(5.4.2)where we denote ΩR(x0) := Ω ∩BR(x0).We emphasize that while u ∈ DG(Ω) implies that (5.4.1) holds for anyk ∈ R the inclusion u ∈ DG(∂Ω) implies that (5.4.2) holds only for non-negative k ∈ R. Such choice of the boundary De Giorgi class is motivatedby the homogeneous Dirichlet boundary conditions in (5.1.1).The main result of this Section is the following theorem:Theorem 5.4.1. Assume Ω ⊂ Rn is a bounded Lipschitz domain and denoteby DG(Ω) and DG(∂Ω) the modified De Giorgi classes corresponding tosome positive constants F , γ, α and q > n.(i) Assume ±u ∈ DG(Ω). Then there exists µ ∈ (0, 1) depending only onn, q, γ such that u ∈ Cµloc(Ω) and for any Ω′ ⋐ Ω∥u∥Cµ(Ω¯′) ≤ c(Ω,Ω′)(∥u∥L2(Ω) + F)with some constant c(Ω,Ω′) > 0 depending only on Ω, Ω′, n, q, γ andα.1205.4. Modified De Giorgi classes(ii) Assume ±u ∈ DG(Ω)∩DG(∂Ω). Then there exists µ ∈ (0, 1) depend-ing only on n, q, γ such that u ∈ Cµ(Ω¯) and∥u∥Cµ(Ω¯) ≤ c(Ω)(∥u∥L2(Ω) + F)with some constant c(Ω) > 0 depending only on Ω, R0, n, q, γ and α.Our modified De Giorgi classes in Definitions 5.4.1 and 5.4.2 are verysimilar to ones introduced earlier in [43]. Our proof of Theorem 5.4.1 isstandard and follows to the general line in [43]. But due to the minordifference in the proof and for the reader convenience we present the proofof Theorem 5.4.1 in this Section.We start with the local maximum estimate:Lemma 5.4.2. Assume u ∈ DG(BR, k0). Then u is essentially boundedfrom above in BR/2 andsupBR/2(u− k0)+ ≤ c∗( −∫BR|(u− k0)+|2 dx)1/2+ F R1−nq (5.4.3)where (u− k0)+ := max{u− k0, 0} and c∗ > 0 depends only on n, γ and α.Proof. Without loss of generality we can assume R = 1 and k0 = 0. DenoteH := c∗( −∫Bu2+ dx)1/2+ F ,where the precise value of the constant c∗ = c∗(n, γ, α) > 0 will be fixedlater. Defineρm :=12 +12m+1, km := H − H2m , Jm :=∫Bρm(u− km)2+ dx,µm := |{x ∈ Bρm : u(x) > km+1 }|.From Jm ≥ (km+1 − km)2µm we obtainµm ≤ 22m+2H2Jm. (5.4.4)1215.4. Modified De Giorgi classesDefine ρ′m :=12(ρm+1 + ρm) and define cut-off functionsζm ∈ C∞0 (Bρ′m), ζm ≡ 1 on Bρm+1 , |∇ζm| ≤cρm − ρm+1 .From the Ho¨lder inequality and the imbedding W 12 (Bρ′m) ↪→ L 2nn−2 (Bρ′m ) weobtainJm+1 ≤ c(n) µ2nm(∥∇(u−km+1)+∥2L2(Bρ′m ) +c(n)(ρm − ρm+1)2 Jm). (5.4.5)Using u ∈ DG(B) we estimate∥∇(u−km+1)+∥2L2(B′ρm ) ≤γ2(ρm − ρm+1)2(1 +ραm(ρm − ρm+1)α)Jm + F2µ1− 2qm(5.4.6)and henceJm+1 ≤ c(n, γ)(ρm − ρm+1)2 µ2nm(1 +ραm(ρm − ρm+1)α)Jm + F2µ1+ 2n− 2qm .DenoteYm :=JmH2.Taking into account ρm−ρm+1 = 12m+2 , ρm ≤ 1 from (5.4.4), (5.4.5), (5.4.6)we obtainYm+1 ≤ c 22m(1+ 2n ) (1 + 2αm) Y 1+2nm + cF 2H22m(1+ 2n− 2q)Y1+ 2n− 2qm .Using and FH ≤ 1 we arrive atYm+1 ≤ c0 bm Y 1+ϵm (1 + Y ϵ1m ), (5.4.7)where b := 22+α+4n , ϵ = 2n − 2q , ϵ1 = 2q and c0 depends only on n, γ, α. Nowwe can use the following lemma:Lemma 5.4.3. Assume {Ym}∞m=1 ⊂ R are nonnegative and there exist ϵ >0, ϵ1 ≥ 0, b ≥ 1 and c0 ≥ 1 such that (5.4.7) holds. Assume Y0 ≤ θ whereθ := (2c0)−1/ϵ b−1/ϵ2. ThenYm ≤ θ b−m/ϵ, ∀ m ∈ N.1225.4. Modified De Giorgi classesProof. We use the induction with respect to m. For m = 0 the statementholds by assumption. Now assuming Ym ≤ θ b−m/ϵ from (5.4.7) we estimateYm+1 ≤ c0 bm Y 1+ϵm (1 + Y ϵ1m ) ≤ c0 bm (θ b−m/ϵ)1+ϵ(1 + (θ b−m/ϵ)ϵ1)≤≤ 2 c0 θ1+ϵ b−mϵ = 2 c0 θ1+ϵ b1/ϵ b−m+1ϵTo finish the proof we note that θ = (2c0)−1/ϵ b−1/ϵ2 implies 2c0 θ1+ϵ b1/ϵ =θ.Now fix c∗ > θ−1/2 where θ is defined in Lemma 5.4.3. Then Y0 < θholds and hence by Lemma 5.4.3 we obtain Ym → 0 which gives (5.4.3).Now we prove so-called “Density lemma” (or “Thin set lemma”):Lemma 5.4.4. Assume u ∈ DG(Ω, k0). Then there exists θ0 ∈ (0, 1),θ0 = θ0(n, γ, α) such that for any B2ρ := B2ρ(x0) ⋐ Ω if|B2ρ ∩Ak0 | ≤ θ0 |B2ρ|then eithersupBρ(u− k0)+ ≤ 12supB4ρ(u− k0)+ (5.4.8)orsupB4ρ(u− k0)+ ≤ 4c∗ F (2ρ)1−nq (5.4.9)where c∗ > 0 is a constant from (5.4.3).Proof. Take θ0 ∈ (0, 1) such that c∗ θ1/20 = 14 . Assume (5.4.9) does not hold.Then from (5.4.3) we obtain (5.4.8).Now we prove a result called “How to find a thin set”:Lemma 5.4.5. Assume u ∈ DG(Ω, k0). Then for any δ ∈ (0, 1) andany θ ∈ (0, 1) there exists s ∈ N, s = s(δ, θ, n, γ) such that if for someB4ρ := B4ρ(x0) ⋐ Ω the following estimate is valid|B2ρ \Ak0 | ≥ δ |B2ρ|then either|B2ρ ∩Ak¯| ≤ θ |B2ρ|, (5.4.10)orM(4ρ)− k0 ≤ 2s F ρ1−nq . (5.4.11)Here we denote k¯ =M(4ρ)− 12s (M(4ρ)− k0) and M(4ρ) := supB4ρu.1235.4. Modified De Giorgi classesProof. Denote M :=M(4ρ) and assume s ∈ N is some number whose valuewill be fixed later. From u ∈ DG(B4ρ) for any k0 ≤ k ≤ k¯ we obtain∫B2ρ∩Ak|∇u|2 dxdt ≤ c(n, γ, α) ρn−2((M − k)2 + F 2ρ2(1−nq )). (5.4.12)Assume (5.4.11) is not valid. Then for any k0 ≤ k ≤ k¯F ρ1−nq ≤ 12s(M − k0) = M − k¯ ≤ M − kand from (5.4.12) we obtain∫B2ρ∩Ak|∇u|2 dxdt ≤ 2 c(n, γ, α) ρn−2 (M − k)2. (5.4.13)Define km :=M − 12m (M − k0) and denoteAm := { x ∈ B2ρ | u(x) > km },Dm := { x ∈ B2ρ | km ≤ u(x) < km+1 }.From the De Giorgi inequality (see [43, Lemma 3.9]) we derive(km+1 − km)|Am+1| ≤ C(n) |B2ρ |1+ 1n|B2ρ \Am |∫Dm|∇u(x)| dx.Note that|B2ρ \Am | ≥ δ |B2ρ|and hence(km+1 − km) |Am+1| ≤ C(n)δρ∫Dm|∇u| dx.Using the Ho¨lder inequality and (5.4.13) we obtain(km+1 − km)|Am+1| ≤ C(n, γ)δρn2 |Dm |1/2(M − km).Note thatkm+1 − km = 12m+1(M − k0), M − km = 12m(M − k0).1245.4. Modified De Giorgi classesHence|Am+1|2 ≤ C(n, γ, δ) ρn |Dm |, m = 1, 2, 3, . . .Taking a summation over m from 1 to s− 1 we gats−1∑m=1|Am+1|2 ≤ C(n, γ, δ) ρns−1∑m=1|Dm |.Note thats−1∑m=1|Dm | = |A1 \A2|+ . . .+ |As−1 \As| = |A1 \As| ≤ |B2ρ|,As ⊂ Am, m = 1, . . . , s− 1 =⇒s−1∑m=1|Am+1|2 ≥ (s− 1)|As|2.So, for any s ∈ N we obtain|As| ≤ C(n, γ, δ)√s− 1 ρn2 |B2ρ| 12 .Let θ ∈ (0, 1) be arbitrary. Find s ∈ N, s = s(δ, θ, n, γ), so thatC(n, γ, δ)√s− 1 ρn2 |B2ρ| 12 ≤ θ |B2ρ|.Then|As| ≤ θ |B2ρ|,which implies (5.4.10).Our next result is a local estimate of oscillations:Lemma 5.4.6. Assume θ0 ∈ (0, 1) is a constant from Lemma 5.4.4 andassume ±u ∈ DG(Ω). Then for any B4ρ ≡ B4ρ(x0) such that B4ρ ⋐ ΩeitheroscBρ u ≤(1− 12s0+2)oscB4ρ u, (5.4.14)oroscB4ρ u ≤ c∗∗ F ρ1−nq , (5.4.15)where s0 = s(12 , θ0, n, γ), and s ∈ N, s = s(δ, θ, n, γ) is from Lemma 5.4.5and c∗∗ > 0 is some constant depending only on c∗ and s0.1255.4. Modified De Giorgi classesProof. Take c∗∗ = max{4c∗, 2s0} where c∗ is from (5.4.9) and 2s0 is from(5.4.11). Assume (5.4.15) does not hold. Then both (5.4.9) and (5.4.11) arenot valid. Denotem(ρ) = infBρu, M(ρ) = supBρu, k0 :=m(4ρ) +M(4ρ)2.and consider two cases:(1)∣∣ { u ≤ k0 } ∩B2ρ ∣∣ ≥ 12 |B2ρ|(2)∣∣ { u ≥ k0 } ∩B2ρ ∣∣ ≥ 12 |B2ρ|Case (1). Using Lemma 5.4.5 with δ = 12 and θ = θ0 we find s0 such thatfork¯ :=M(4ρ)− 12s0(M(4ρ)− k0)we have|B2ρ ∩Ak¯| ≤ θ0 |B2ρ|.Then applying Lemma 5.4.4 we obtainsupBρ(u− k¯)+ ≤ 12supB2ρ(u− k¯)+,and henceM(ρ) ≤ 12(M(4ρ) + k¯),which meansM(ρ) ≤ M(4ρ)− 12s0+2(M(4ρ)−m(4ρ)).Adding this with −m(ρ) ≤ −m(4ρ) we arrive atoscBρ u ≤(1− 12s0+2)oscB4ρ u.Case (2). We denotev := −u, l0 := −k0, Mv(ρ) := supBρv, mv(ρ) := infBρv.Hence we have v ∈ DG(Ω). From (2) we get| {v ≤ l0} ∩B2ρ} | ≥ 12 |B2ρ|.1265.4. Modified De Giorgi classesHence similar to aboveMv(ρ) ≤ Mv(4ρ)− 12s0+2(Mv(4ρ)−mv(4ρ)),which givesoscBρ v ≤(1− 12s0+2)oscB4ρ v.AsoscBρ v = oscBρ u, oscB4ρ v = oscB4ρ u,the result follows.The next result follows easily by iterations of the estimate of oscillations:Lemma 5.4.7. Assume u ∈ L∞(Ω) satisfies the oscillation estimate∀ B4R(x0) ⊂ Ω, oscBR(x0) u ≤ σ oscB4R(x0) u+c1 F R1−nq , σ ∈ (0, 1).Then∀ BR(x0) ⊂ Ω, ∀ ρ < R oscBρ(x0) u ≤ c2(( ρR)µ ∥u∥L∞(Ω) + F ρµ)where µ ∈ (0, 1) depends only on σ, n and q and c2 depends on σ, n, q andc1.Now part (i) of Theorem 5.4.1 follows from Lemma 5.4.7.Now we turn to the investigation of the boundary regularity. Take ar-bitrary Ω0 ⋑ Ω and denote by u˜ the zero extension of u from Ω onto Ω0.Obviously, if ±u ∈ DG(Ω) ∩ DG(∂Ω) then ±u˜ ∈ DG(Ω0, 0) and hence, inparticular, u ∈ L∞(Ω).As ∂Ω is Lipschitz there exists δ0 ∈ (0, 1) and 0 < R0 < dist{Ω, ∂Ω0}such that for any x0 ∈ ∂Ω and any R ∈ (0, R0)|{x ∈ Ω0 : u˜(x) = 0 } ∩BR(x0)| ≥ δ0 |BR|.Lemma 5.4.8. Assume θ0 ∈ (0, 1) is a constant from Lemma 5.4.4 andassume ±u ∈ DG(Ω) ∩DG(∂Ω). Then for any ρ ≤ R04 eitheroscΩρ(x0) u ≤(1− 12s0+2)oscΩ4ρ(x0) u, (5.4.16)oroscΩ4ρ(x0) u ≤ c∗∗ F ρ1−nq (5.4.17)where s0 = s(12 , θ0, n, γ), and c∗∗ > 0 are defined in Lemma 5.4.6.1275.4. Modified De Giorgi classesProof. Denote v˜ := −u˜. Then u˜, v˜ ∈ DG(Ω0, 0). Denotem(ρ) = infBρ(x0)u˜, M(ρ) = supBρ(x0)u˜,mv˜(ρ) = infBρ(x0)v˜, Mv˜(ρ) = supBρ(x0)v˜,Note thatm(ρ) ≤ 0 ≤M(ρ)and|{x ∈ Ω0 : u˜(x) ≤ 0 } ∩B2ρ(x0)| ≥ δ0 |B2ρ|,|{x ∈ Ω0 : v˜(x) ≤ 0 } ∩B2ρ(x0)| ≥ δ0 |B2ρ|.Assume (5.4.17) does not hold. Then (5.4.15) does not hold for both u˜ andv˜. Using Lemma 5.4.5 with δ = δ0, θ = θ0 and k0 = 0 we obtainM(ρ) ≤(1− 12s0+2)M(4ρ),Mv˜(ρ) ≤(1− 12s0+2)Mv˜(4ρ).Taking the sum of these two inequalities and taking into accountMv˜(ρ) = −m(ρ), Mv˜(4ρ) = −m(4ρ)we obtainoscΩρ(x0) u ≤(1− 12s0+2)oscΩ4ρ(x0) u.The next result is the boundary analogue of Lemma 5.4.7.Lemma 5.4.9. Assume u ∈ L∞(Ω) satisfies the oscillation estimate∀ x0 ∈ ∂Ω, ∀ ρ < 4R ≤ R0 oscΩR(x0) u ≤ σ oscΩ4R(x0) u+c1 F R1−nq .Where σ ∈ (0, 1), then∀ x0 ∈ ∂Ω, ∀ ρ < R ≤ R0 oscΩρ(x0) u ≤ c2(( ρR)µ ∥u∥L∞(Ω) + F ρµ)where µ ∈ (0, 1) depends only on σ, n and q and c2 depends on σ, n, q andc1.Now part (ii) of Theorem 5.4.1 follows by the standard combination ofestimates in Lemmas 5.4.7 and 5.4.9.1285.5. Ho¨lder continuity of weak solutions5.5 Ho¨lder continuity of weak solutionsIn this Section we give the proof of Theorem 5.1.2. Assume p > 2 andu ∈◦W 1p(Ω) is a p-weak solution to the problem (5.1.1) corresponding to theright-hand side f ∈ Lq(Ω) with q > n. From Proposition 5.3.1 we concludeu ∈ L∞(Ω).Let us fix some r ∈(2nn+2 , 2)and θ ∈ (nr − n2 , 1). Then from (5.1.3) andProposition 5.2.1 we obtainb ∈ Lr,n−r(Ω), ∥b∥Lr,n−r(Ω) ≤ c ∥b∥L2,n−2w (Ω).First we consider the case BR(x0) ⊂ Ω. Take some radius ρ < R and acut-off function ζ ∈ C∞0 (BR(x0)) such that0 ≤ ζ ≤ 1, ζ ≡ 1 on Bρ(x0), |∇ζ| ≤ cR− ρ. (5.5.1)Assume k ∈ R is arbitrary and denoteu˜ := (u− k)+ ≡ max{u− k, 0}, u˜ ∈ L∞(Ω) ∩W 1p (Ω).Fix m := 11−θ and note that 2m− 1 = m(1 + θ). Take η = ζ2mu˜ in (5.1.7).It is easy to see thatB[u, η] = B[ζmu˜, ζmu˜] − m∫Ωζ2m−1 b · ∇ζ |u˜|2 dx.Taking into account (5.2.1) and 2m− 1 = m(1 + θ) we obtain∥ζm∇u˜∥2L2(BR(x0)) ≤ c ∥u˜∇ζ∥2L2(BR(x0)) +m∫Ωζm(1+θ) b · ∇ζ |u˜|2 dx ++ ∥f∥2Lq(Ω) |Ak ∩BR(x0)|1− 2q(5.5.2)where Ak := {x ∈ Ω : u(x) > k }. Taking into account (5.5.1) and applyingthe estimate (5.3.3) we obtain∫Ωζm(1+θ) b·∇ζ |u˜|2 dx ≤ cRθR− ρ ∥b∥Lr,n−r(Ω) ∥∇(ζmu˜)∥1+θL2(BR(x0)) ∥u˜∥1−θL2(BR(x0)).1295.5. Ho¨lder continuity of weak solutionsTaking arbitrary ϵ > 0 and applying the Young inequality we obtain∫Ωζmb · ∇ζ |u˜|2 dx ≤ ϵ ∥∇ (ζmu˜)∥2L2(BR(x0)) ++cϵ(R− ρ)2( RR− ρ) 2θ1−θ ∥b∥21−θLr,n−r(BR(x0))∥u˜∥2L2(BR(x0)).So, if we fix sufficiently small ϵ > 0 from (5.5.2) for any k ∈ R and 0 < ρ < Rwe obtain12 ∥∇(u− k)+∥2L2(Bρ(x0)) ≤≤ c(R− ρ)2(1 +( RR− ρ) 2θ1−θ ∥b∥21−θLr,n−r(Ω))∥(u− k)+∥2L2(BR(x0))++ ∥f∥2Lq(Ω) |Ak ∩BR(x0)|1− 2q .Hence we obtain that u ∈ DG(Ω). Applying the same arguments to −uinstead of u we also obtain −u ∈ DG(Ω).Now we turn to the estimates near the boundary. Assume x0 ∈ ∂Ωand denote ΩR(x0) := Ω∩BR(x0). Take a cut-off function ζ ∈ C∞0 (BR(x0))satisfying (5.5.1). Then for any k ≥ 0 the function η = ζ2m(u−k)+ vanisheson ∂Ω and hence it is admissible for the identify (5.1.7). Repeating allprevious arguments we arrive at12 ∥∇(u− k)+∥2L2(Ωρ(x0)) ≤≤ c(R− ρ)2(1 +( RR− ρ) 2θ1−θ ∥b∥21−θLr,n−r(Ω))∥(u− k)+∥2L2(ΩR(x0))++ ∥f∥2Lq(Ω) |Ak ∩ ΩR(x0)|1− 2q ,which holds for any k ≥ 0. Hence u ∈ DG(∂Ω). Applying the same argu-ments to −u instead of u we also obtain −u ∈ DG(∂Ω). Now Theorem 5.1.2follows directly from Theorem 5.4.1.130Chapter 6Concluding ChapterThis thesis contains two parts, but both of them are related to Navier-Stokesequations. {∂tv −∆v + (v · ∇)v +∇p = 0div v = 0(6.0.1)Our first research topic is directly related to Navier-Stokes equations andour goal was to construct global-in-time local energy solutions of (6.0.1) forinitial data that is allowed to grow at spatial infinity. Previously on thistopic there were mainly results in either uniform local spaces L2uloc or inHerz spaces, so the goal of this research is to identify a continuous familyof spaces between uniform local and Herz spaces where we can constructglobal-in-time local energy solutions. Additionally, we considered a separatefamily of weighted spaces F qr (see Definition 1.1.2) which are correspondingto different Herz spaces and have a norm that does not include supremumof weighted integrals, unlike L2uloc or intermediate spaces Fpα,λ. Next, let usrecall the definition of intermediate spaces F pα,λ.Definition 6.0.1. Let 1 ≤ p <∞, 0 ≤ α ≤ 1 and 0 ≤ λ <∞. Define F pα,λto be the space of f ∈ Lploc(R3) with finite norm ∥f∥F pα,λ(R) where for anyR ≥ 1,∥f∥F pα,λ(R) = supx∈R3(1rλx∫B(x,rx)|f |p)1/p, rx =18max{R, |x|}α. (6.0.2)We denote by F pα,λ,σ the space of divergence-free vector fields in R3 with allcomponents in F pα,λ. Furthermore, we denote by F˚pα,λ and F˚pα,λ,σ the closureof C∞c (R3) and C∞c,σ(R3) = {ζ ∈ C∞c (R3;R3),div ζ = 0} in F pα,λ-norm,respectively.In our definition of intermediate spaces we have two parameters α, λ andwe proved global existence of the solutions only with additional assumptionin the case (3− α)λ < 1.limρ→∞ ρα(λ−2)∥u0∥2F 2α,λ(ρ) = 0 (6.0.3)131Chapter 6. Concluding ChapterIn the future we might try to improve it or consider alternative spaces toachieve the result with less restrictive additional assumptions. To summarizewe found a continuous class of spaces, connecting L2uloc and Herz spaces, inwhich we proved global in time existence and eventual regularity of localenergy solutions.The second topic is elliptic equations with drift from critical spaces, themotivation for this project came from Navier-Stokes equations. Let (v, p) besolutions to the Navier Stokes system (6.0.1), in axi-symmetric coordinatesv = vr(r, z, t)er + vφ(r, z, t)eφ + vz(r, z, t)ez and we encounter the followingequation∂tψ −∆ψ +(v + α x′|x′|2)· ∇ψ = 0, where x′ = (x1, x2, 0). (6.0.4)Here we have two separate cases, depending if the angle component of thesolution is zero or not.If vφ ≡ 0, α = −2 then, ψ = ωφr, ωφ := vr,z − vz,r,If vφ ̸≡ 0, α = 2 then, ψ = rvφ.Equation (6.0.4) is a parabolic equation with critical drift b =(v + α x′|x′|2),notice that v is divergence free function as a solution to Navier Stokes equa-tions, while the second term in the drift is responsible for the sign of div b.This kind of drift was the main motivation for our work, and the reason whywe chose weak Morrey spaces as the class for our drift. Notice that the axisymmetric case of Navier Stokes includes both cases of spectral (α = 2) andnon-spectral (α = −2) drift. In our work we studied elliptic two and threedimensional equation with critical drift, in two dimensional case we provedexistence, uniqueness, Ho¨lder continuity and higher integrability of solutionsfor both cases of positive and negative div b. For three dimensional caseswe studied only non-spectral case, but we continued the work and have adraft for three dimensional spectral case. In the future we are planning toprove similar results for parabolic equation in the form of (6.0.4). Anotherinteresting topic related to equations with critical drift are Liouville-typetheorems, it is reasonable to expect those results since the technique weused to prove Holder continuity relies on oscillation estimatesoscBR u ≤ θ oscB2R u.These kind of estimates can both be used to prove Holder continuity ofsolutions and Liouville-type theorems, depending on how we iterate them,and is another direction of research that we are considering.132Bibliography[1] A. 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J. 17 No. 5 (1968), 473-483.140Appendix AImbedding TheoremHere we present a variant of the imbedding theorem that we used in Section4.2:Proposition A.0.1. Assume BR ⊂ Rn, n ≥ 2, p ∈ [1, n) and p∗ := pnn−p .Then for any λ ∈ (0, 1) there exists C = C(n, p) > 0 such that if u ∈W 1p (BR) satisfies|{x ∈ BR : u(x) = 0 }| ≥ λ|BR|then∥u∥Lp∗ (BR) ≤ C(n, p)λ−1 ∥∇u∥Lp(BR).Proof. From the Sobolev imbedding theorem we conclude∥u∥Lp∗ (BR) ≤cR∥u∥Lp(BR) + c ∥∇u∥Lp(BR).Denote E := {x ∈ BR : u(x) = 0 } and uΩ :=∫−Ωu dx. Then we have∥u∥Lp(BR) = ∥u− uE∥Lp(BR) ≤ ∥u− uBR∥Lp(BR) + |BR|1/p|uE − uBR | ≤≤ ∥u− uBR∥Lp(BR) +|BR|1/p|E| ∥u− uBR∥L1(BR).Now the result follows from the usual Poincare and Ho¨lder inequalities:∥u− uBR∥Lp(BR) ≤ cR ∥∇u∥Lp(BR)∥u− uBR∥L1(BR) ≤ |BR|1/p′∥u− uBR∥Lp(BR) ≤ c |BR|1/n+1/p′∥∇u∥Lp(BR).141Appendix BApproximation PropositionIn this Section we show that in the case of a bounded star-shaped domainΩ ⊂ Rn the approximation property (5.1.10) is valid for an arbitrary bsatisfying (5.1.3), (5.1.8).Proposition B.0.1. Assume Ω ⊂ Rn is a bounded star-shaped domain andlet b satisfy (5.1.3), (5.1.8). Then there exists a sequence bk ∈ C∞(Ω¯) suchthat (5.1.10) holds.Proof. Take λ > 1 and consider the scaling (5.1.5). Obviously, for λ > 1 wehaveΩ ⋐ Ωλ where Ωλ := {x ∈ Rn : x/λ ∈ Ω }.Denote dλ := dist{Ω, ∂Ωλ}. Thanks to (5.1.5) we obtain∥bλ∥L2,n−2w (Ωλ)= ∥b∥L2,n−2w (Ω), div bλ ≤ 0 in D′(Ωλ),and we have a convergence∥bλ − b∥L1(Ω) → 0 as λ→ 1.Now let us fix λ > 1 and consider ϵ ∈ (0, dλ). Denotebλϵ (x) :=∫Ωλωϵ(x− y)bλ(y) dy, x ∈ Ωλ,where ωϵ(x) := ϵ−nω(x/ϵ) and ω ∈ C∞0 (Rn) is the standard Sobolev kernel,i.e.ω ≥ 0, suppω ∈ B¯,∫Rnω(x) dx = 1, ω(x) = ω0(|x|).Then bλϵ ∈ C∞(Ω¯) and, moreover, from (5.1.8) for ϵ < dλ we obtaindiv bλϵ ≤ 0 in Ω.142Appendix B. Approximation PropositionWe also have a convergence∥bλϵ − bλ∥L1(Ω) → 0 as ϵ→ 0.Take arbitrary x0 ∈ Ω. Note that Bϵ(x) ⊂ Ωλ for any x ∈ Ω. First weconsider the case R ∈ (0, ϵ). From the definition of the convolution we havea pointwise estimate|bλϵ (x)| ≤1ϵn∫Bϵ(x)|bλ(y)| dy, ∀x ∈ BR(x0) ∩ Ωand hence from the definition of the Morrey norm L1,n−1(Ωλ) we obtain|bλϵ (x)| ≤1ϵ∥bλ∥L1,n−1(Ωλ) ∀x ∈ BR(x0) ∩ Ω.Taking into account the estimate∥bλ∥L1,n−1(Ωλ) = ∥b∥L1,n−1(Ω) ≤ c ∥b∥L2,n−2w (Ω)with a constant c > 0 depending only on Ω we arrive at|bλϵ (x)| ≤1ϵ∥bλ∥L1,n−1(Ωλ) ≤cϵ∥b∥L2,n−2w (Ω), ∀x ∈ BR(x0) ∩ Ω.Hence we obtain∥bλϵ ∥2L2(BR(x0)∩Ω) ≤ c(Rϵ)2∥b∥2L2,n−2w (Ω)Rn−2.Applying the trivial estimate∥bλϵ ∥L2,w(BR(x0)∩Ω) ≤ ∥bλϵ ∥L2(BR(x0)∩Ω)for R ∈ (0, ϵ) we arrive at∥bλϵ ∥2L2,w(BR(x0)∩Ω) ≤ c ∥b∥2L2,n−2w (Ω)Rn−2. (B.0.1)Now consider the case R ∈ (ϵ, diamΩ). For any f ∈ L1(B2R(x0) ∩ Ωλ) wedefine a convolution operator(Tϵf)(x) :=∫Bϵ(x)ωϵ(x− y)f(y) dy, x ∈ BR(x0) ∩ Ω.143Appendix B. Approximation PropositionFrom the Ho¨lder inequality for p ∈ [1,+∞) we obtain|(Tϵf)(x)|p ≤∫Bϵ(x)ωϵ(x− y)|f(y)|p dy, x ∈ BR(x0) ∩ Ω.As for ϵ < min{R, dλ} we have the inclusion∀x ∈ BR(x0) ∩ Ω Bϵ(x) ⊂ B2R(x0) ∩ Ωλ,we conclude∫BR(x0)∩Ω|(Tϵf)(x)|p dx ≤∫B2R(x0)∩Ωλ|f(y)|p dy, x ∈ BR(x0) ∩ Ω.The last inequality implies that for ϵ < min{R, dλ} the linear operator Tϵis bounded from Lp(B2R(x0) ∩ Ωλ) to Lp(BR(x0) ∩ Ω) for any p ∈ [1,+∞)and the following estimate holds:Mp := ∥Tϵ∥Lp(B2R(x0)∩Ωλ)→Lp(BR(x0)∩Ω) ≤ 1.Using Proposition 5.2.3 we conclude that the following interpolation esti-mate holds:∥Tϵ∥L2,w(B2R(x0)∩Ωλ)→L2,w(BR(x0)∩Ω) ≤ cM1/23 M1/23/2 ≤ cwith some constant c > 0 depending only on n. Hence we obtain∥bλϵ ∥L2,w(BR(x0)∩Ω) ≤ c ∥bλ∥L2,w(B2R(x0)∩Ωλ).From the definition of the Morrey quasinorm L2,n−2w (Ωλ) we obtain∥bλϵ ∥2L2,w(BR(x0)∩Ω) ≤ c ∥bλ∥2L2,n−2w (Ωλ) (2R)n−2.Taking into account (5.1.5) for any R ∈ (ϵ, diamΩ) we arrive at the estimate(B.0.1) for any R ∈ (ϵ,diamΩ). So, the estimate (B.0.1) is valid for allR ∈ (0, diamΩ). Hence for any λ > 1 and any ϵ ∈ (0, dλ) we obtain∥bλϵ ∥L2,n−2w (Ω) ≤ c ∥b∥L2,n−2w (Ω).Now taking some λk → 1+ and choosing ϵk → +0 so thatϵk ∈ (0, dλk), ∥bλkϵk − bλk∥L1(Ω) <1kwe obtain the sequence bk := bλkϵk∈ C∞(Ω¯) such that (5.1.10) hold.144
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Tsai, Tai-Peng, 1969-
On problems of regularity and existence for critical drift elliptic equations and Navier-Stokes equations
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