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Existence and ill-posedness for fluid PDEs with rough data Kwon, Hyunju 2019

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Existence and ill-posedness for fluid PDEs with rough databyHyunju KwonA THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mathematics)The University of British Columbia(Vancouver)April 2019c© Hyunju Kwon, 2019The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:Existence and ill-posedness for fluid PDEs with rough datasubmitted by Hyunju Kwon in partial fulfillment of the requirements for the de-gree of Doctor of Philosophy in Mathematics.Examining Committee:Tai-Peng Tsai, MathematicsSupervisorDong Li, MathematicsSupervisorStephen Gustafson, MathematicsSupervisory Committee MemberYoung-Heon Kim, MathematicsUniversity ExaminerMatthew Choptuik, Physics and AstronomyUniversity ExamineriiAbstractIt has been of great interest in recent decades to know whether the incompress-ible Euler equations are well-posed in the borderline spaces. In order to under-stand the behavior of solutions in these spaces, the logarithmically regularized 2DEuler equations were introduced. In the borderline Sobolev space, the local well-posedness was proved by Chae-Wu when the regularization is sufficiently strong,while strong ill-posedness of the unregularized case was established by Bourgain-Li. The first part of the dissertation closes the gap between the two results, byestablishing the strong ill-posedness in the remaining intermediate regime of theregularization.The second part of the thesis considers the Cauchy problem of incompressible3D Navier-Stokes equations with uniformly locally square integrable initial data. Ifthe square integral of the initial datum on a ball vanishes as the ball goes to infinity,the existence of a time-global weak solution has been known. However, such datado not include constants, and the only known global solutions for non-decayingdata are either for perturbations of constants or when the velocity gradients are inLp with finite p. This work presents how to construct global weak solutions fornon-decaying initial data whose local oscillations decay, no matter how slowly.iiiLay SummaryFor any given evolutionary partial differential equations, one of the fundamentalquestions is the existence and uniqueness of a solution to an equation. Here, theanswer depends on the solution space. Once the existence of a solution is guaran-teed, the follow-up questions are its lifespan and stability under a small change ininitial data. The second one has a significance, especially in the physical applica-tion.In this dissertation, two fluid models are considered to have a deeper under-standing of the motion of fluid flows: the logarithmically regularized 2D Eulerequations and Naiver-Stokes equations. For the former equations, we explain howto find initial data in some borderline space such that a corresponding unique solu-tion exists in some other space but leaves the borderline space instantaneously. Forthe latter ones, a construction scheme is introduced for a globally existing solutionto the Naiver-Stokes equations with non-decaying initial data.ivPrefaceThis dissertation is based on two different previous works. One is submitted forpublication in an academic journal, and the other will be submitted soon.The material in Chapter 2 is original, independent work by the author, H.Kwon. I was responsible for developing the methodology, the solution, as wellas the manuscript composition.The material in Chapter 3 is based on the paper “Global Navier-Stokes flowsfor non-decaying initial data with slowly decaying oscillation”. This work is donein collaboration with T. Tsai, and its preprint can be found in [30]. I provedthe local existence of a local energy solution, defined as in [26], to the Navier-Stokes equations, under a revised scheme of the one in [31]. For the main the-orem (see Theorem 3.1.1), based on the scheme suggested by T. Tsai, I laid thegroundwork for the global existence under stronger assumptions on initial data,lim|x0|→∞´B(x0,1)|∇v0|dx → 0 and supx0∈R3´B(x0,1)|v0|3dx ≤ C for some positiveconstant C. Then, T. Tsai weakened these assumptions to the ones in the theorem.I wrote most part of the manuscript, but subchapter 3.2.4 is written by T. Tsai.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Strong ill-posedness of logarithmically regularized 2D Euler equa-tions in the borderline Sobolev spaces . . . . . . . . . . . . . . . . . 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Large Lagrangian deformation . . . . . . . . . . . . . . . . . . . 132.4 Local critical Sobolev norm inflation . . . . . . . . . . . . . . . . 212.5 Patching argument . . . . . . . . . . . . . . . . . . . . . . . . . 292.6 Proof of Theorem 2.1.1 . . . . . . . . . . . . . . . . . . . . . . . 432.7 The compact case . . . . . . . . . . . . . . . . . . . . . . . . . . 442.8 Analysis of the velocity . . . . . . . . . . . . . . . . . . . . . . . 642.8.1 Kernel for the velocity . . . . . . . . . . . . . . . . . . . 642.8.2 Operator norm of Tγ on Lp . . . . . . . . . . . . . . . . . 662.8.3 Estimate for ∆−1∂iiTγW . . . . . . . . . . . . . . . . . . . 66vi3 Global Navier-Stokes flows for non-decaying initial data with slowlydecaying oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.2 Notations and preliminaries . . . . . . . . . . . . . . . . . . . . . 753.2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 753.2.2 Uniformly locally integrable spaces . . . . . . . . . . . . 763.2.3 Heat and Oseen kernels on Lquloc . . . . . . . . . . . . . . 783.2.4 Heat kernel on L1uloc with decaying oscillation . . . . . . . 823.3 Local existence . . . . . . . . . . . . . . . . . . . . . . . . . . . 873.4 Spatial decay estimates . . . . . . . . . . . . . . . . . . . . . . . 1063.5 Global existence . . . . . . . . . . . . . . . . . . . . . . . . . . . 1264 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138viiAcknowledgmentsMy deepest gratitude goes to my advisors Dong Li and Tai-Peng Tsai. Withouttheir guidance, tolerance, and endless support, this thesis would not have beenpossible.I would like to thank Prof. Tsai for suggesting suitable problems at differentstages of my doctoral study. In addition, I very much value his advice on differentaspects of leading a life in the academia. His role has been invaluable in helpingme network with other researchers. I want to extend my thanks to Prof. Li. Allthe discussions with him have not only stimulated my enthusiasm for research, butalso deepened my understanding of the subject. His constant encouragement andfeedback have been pivotal in helping me grow as a researcher.Furthermore, I wish to thank Professor Dongho Chae for introducing me thefield of fluid PDE, and Professor Kyungkeun Kang for nudging me in the rightdirection during my days as an undergraduate.Lastly, but definitely not the least, I thank all my friends and family for theconstant love and support through thick and thin.viiiChapter 1IntroductionIn the study of partial differential equations (PDEs), one of the first questions thatone would ask is the existence of a solution to the equation. In mathematics, theexistence is proved by indicating the space where the solution lies, so the answerfor the existence depends on the solution space. Once we have the existence, thequestion on the uniqueness of the solution follows. If the existence and uniquenessof a solution of a time-evolutionary PDE are guaranteed in some space, an initialdata in the solution space identifies the solution associated with the data. It is theusual way of indicating an object of discussion in PDEs.On the other hand, it is also imperative to ask whether a small change in thedata leads to a small change in the corresponding solutions. Here, the smallnessis measured by the norm defined in the solution space. In the derivation of theequations describing a certain phenomenon, we often assume an ideal situation inwhich all the minor factors are ignored. In this way, we can simplify the complexityof the setting. However, it could lead a small error, for example in experiments,between the data measured in the real world and the one in the ideal setting. Also,data plugged into a computer program, which usually discretize the continuum,can carry small error with respect to the true one. In order to see that the solutionobtained either from experiments or by the assistance of a computer program isclose to the mathematical solution, the continuity of the solution in the initial datamust be guaranteed.When we have positive answers to all of the three fundamental questions, the1existence, uniqueness, and continuous dependence of solutions in the initial datafor an evolutionary PDE, we say that the PDE is well-posed. Then, we analyze thebehavior of the solution further by examining the lifespan of the solution, its rateof growth, or its asymptotic behavior.In this dissertation, we narrow our focus down to the incompressible fluid mod-els, the Navier-Stokes equations, Euler, and equations derived from them.The Navier-Stokes equation describes the flow of incompressible, homoge-neous, viscous fluids, which is given as follows∂tu+(u ·∇)u+∇p = ν∆udivu = 0u|t=0 = u0.Here, the unknown functions u and p represent the flow velocity and a pressure,respectively, and a positive number ν is the kinematic viscosity. The second term(u ·∇)u in the first equation, called the non-linear term, explains convective phe-nomena of the fluids, while the last term ν∆u describes diffusive phenomena. Also,the divergence-free condition corresponds to the incompressibility of the fluids.Even though the equations were introduced in the 19th Century, we only have alimited understanding of the solutions. One of the difficulties arises from turbu-lence, which remains as one of the unsolved problems in physics. Mathematicianshave made many efforts to understand the motion of the fluids, and the global-in-time existence of a unique smooth solution is one of the seven Millennium PrizeProblems stated by the Clay Mathematics Institute.The Euler equation, on the other hand, describes the motion of incompressible,homogeneous, ideal fluids as follows,∂tu+(u ·∇)u+∇p = 0divu = 0u|t=0 = u0.In other words, we examine the fluid under a zero viscosity hypothesis. However,the Euler equation is often considered to have the same level of the difficulty as the2Navier-Stokes.In the past decades, the local-in-time well-posedness theories of both Navier-Stokes and Euler equations have been well established for solutions with suitableregularity. However, for solutions starting from rough initial data, the question ofthe well-posedness has great mathematical challenges. Even in a solution spacehaving some threshold regularity, the solution behavior is unpredictable. There-fore, analysis of such solutions has attracted massive interest among mathemati-cians. In this direction, two different problems are studied in this dissertation.One is about strong ill-posedness of Euler-type equations in the borderline Sobolevspace, and the other is the existence of time-global Navier-Stokes solutions withnon-decaying initial data.The well-posedness of the Euler equations in the borderline spaces is a long-standing open problem. Many efforts have been made in order to solve this prob-lem. One direction is finding a borderline solution space on which the standardenergy method works. In particular, in such space, the L∞(Rn)-norm of the gradi-ent of the velocity is under control. For example, the local well-posedness of theEuler equations in Rn, n ≥ 2, is known on the critical Besov spaces Bnp+1p,1 (Rn) for1 < p ≤ ∞ (See [9, 39–41]). However, the borderline Sobolev space H n2+1(Rn) isnot included in these critical Besov spaces; because the Sobolev embedding barelyfails there and the gradient of velocity is out of control in L∞ space. To tackle thewell-posedness in the critical Sobolev space, regularized Euler equations are in-troduced [10, 11]. For example, in [11], the logarithmically regularized 2D Eulerequations ∂tω+(u ·∇)ω = 0u = ∇⊥ψ, ∆ψ = Tγωω|t=0 = ω0are studied, where Tγ , γ > 0, is a Fourier multiplier defined by Tγ = log−γ(10−∆).In the absence of the operator Tγ , these equations correspond to the 2D Euler equa-tions for the vorticity ω = ∂1u2− ∂2u1. Hence, the operator Tγ regularizes thevelocity in the Euler vorticity equations at the level of logarithm of the Laplacian.In particular, a large positive number γ , in the index of the operator Tγ , indicates3more regular velocity. As a result of the regularization, for γ > 12 , the local well-posedness in the borderline Sobolev space has been proved, [11]. In the end, Bour-gain and Li recently have established that the Euler equation in Rn, n = 2,3, isstrongly ill-posed in the borderline Sobolev space Hn2+1(Rn) in [4]. Furthermore,they showed the strong ill-posedness in Cm(Rn) and Cm−1,1(Rn), for m≥ 1 in [5].Here, the strong ill-posedness holds in the sense that for any given smooth initialdata, we can always find a perturbation such that it is arbitrarily small in the border-line space, but the perturbed solution leaves the borderline space instantaneously.Indeed, it breaks the existence of the solution in the borderline space and its contin-uous dependence at any initial data in a dense subset of the borderline spaces. Eventhough this long-standing problem is solved in two and three-dimensional spaces,it is unknown whether the strong ill-posedness of the logarithmically regularized2D Euler equations still holds in the intermediate region 0< γ ≤ 12 . This problem isdiscussed in Chapter 2 of this thesis. Indeed, the following theorem is established.Theorem 1.0.1. The logarithmically regularized 2D Euler equation for 0 < γ ≤ 12is strongly ill-posed in the borderline Sobolev space H1(R2)∩ H˙−1(R2).More precisely, we construct two types of perturbations; one is compactly sup-ported, and the other is not. The proof is based on the scheme for the strong ill-posedness invented by Bourgain and Li in [4]: the creation of the large Lagrangiandeformation, critical norm inflation, and the gluing procedure. However, in orderto deal with the regularization, new technical ingredients and a new approach inthe gluing procedure in the compact case are required. Combined with the localwell-posedness result for γ > 12 , Theorem 1.0.1 gives a complete solution to thewell-posedness problem of the logarithmically regularized 2D Euler equations inthe borderline Sobolev space.The work on the Navier-Stokes equation is about classifying non-decaying ini-tial data for which the existence of a time-global weak solution is guaranteed. Non-decaying Navier-Stokes flows at spatial infinity are widely known in practice suchas constant flows or periodic flows. However, most research has been toward de-caying flows, because of difficulties arising from the pressure. For initial data withfinite kinetic energy (i.e., square integrable), the global existence of a weak solu-tion, called Leray-Hopf solution, is well-known. Unfortunately, such initial data4decays at spatial infinity, in the sense that the square integral of the data in a unitball vanishes as the center of the ball goes to infinity. In order to work on non-decaying data, we consider a local version of the Leray-Hopf solution, called alocal energy solution, whose square integrals on unit balls are uniformly bounded.Such a solution is physically reasonable because the kinetic energy is rarely con-centrated in a small region all of a sudden. The local existence is the only availableresult for a local energy solution with non-decaying initial data. In Chapter 3 of thedissertation, we consider non-decaying initial data with arbitrarily slow oscillationdecay at spatial infinity and find a time-global weak solution with uniformly localkinetic energy. The more precise statement is as follows.Theorem 1.0.2. For any divergence-free vector field v0 ∈ E2σ (R3) + L3uloc,σ (R3)satisfyinglim|x0|→∞ˆB(x0,1)|v0− (v0)B(x0,1)|dx = 0, (1.0.1)we can find a time-global local energy solution (v, p) to the Navier-Stokes equa-tions (NS) in R3× (0,∞), in the sense of Definition 3.3.1.Here, Lquloc,σ (R3), 1 ≤ q < ∞, is the space of divergence-free vector fieldswhose Lq(R3)-norms on unit balls are uniformly bounded, and E2σ is the subspaceof L2uloc,σ (R3) with additional spatial decay assumption. (v0)B denotes the averageof v0 on a set B. The oscillation decay assumption is motivated by a recent result[38] for global Navier-Stokes flows with initial data in L∞(R3) whose gradients arein Lq(R3) for some q > 3. However, applying the idea of Caldero´n in [7] to thelocal energy solution, the result in Theorem 1.0.2 made a great improvement fromthe most recent result [38].We close the chapter by giving a brief presentation of subsequent chapters.In Chapter 2, we discuss the strong ill-posedness of the logarithmically regularized2D Euler equations for 0< γ ≤ 12 in the borderline Sobolev space. The constructionof a non-compactly supported perturbation is considered in Section 2.3-2.6, whilethe compactly supported one is studied in Section 2.7. In Chapter 3, we considerthe global existence of local energy solutions to the Navier-Stokes equation in R3for non-decaying initial data with slowly decaying oscillations. In Section 3.3, we5reprove the local existence of local energy solutions for initial data in L2uloc,σ (R3).Then, based on the key estimate in Section 3.4, the global existence is establishedin Section 3.5.6Chapter 2Strong ill-posedness oflogarithmically regularized 2DEuler equations in the borderlineSobolev spaces2.1 IntroductionThe incompressible Euler equation describes the behaviour of inviscid and volume-preserving fluids. It has two unknown functions u and p which present the fluidvelocity and pressure, respectively. The Euler equation for the vorticity ω =∇⊥ ·uin the domain R2 is∂tω+(u ·∇)ω = 0, (x, t) ∈ R2×Ru = ∇⊥ψ, ∆ψ = ω,ω|t=0 = ω0,(E)where ∇⊥ = (−∂2,∂1). It is well-known that the 2D Euler vorticity equation (E)is well-posed globally in time in the Sobolev spaces W s,p(R2)∩ H˙−1(R2), p > 2s ,s ≥ 1 in the literature. (For example, see [12, 37]). In the effort of understanding7the behaviour of the solutions in the borderline Sobolev space H1(R2)∩ H˙−1(R2),Chae and Wu [11] introduce logarithmically regularized 2D Euler equations∂tω+(u ·∇)ω = 0, (x, t) ∈ R2×Ru = ∇⊥ψ, ∆ψ = T (|∇|)ω,ω|t=0 = ω0,where the Fourier multiplier T (|∇|) satisfiesˆ ∞1T 2(r)rdr <+∞. (2.1.1)Such operator T regularizes the velocity in the Euler vorticity equation (E) at thelevel of logarithm of the Laplacian. The particular integrability assumption (2.1.1)on T is imposed to guarantee the local well-posedness of the regularized model inthe critical Sobolev space. As typical examples of T satisfying (2.1.1), we haveT̂γω(k) = ln−γ(e+ |k|2)ωˆ(k), T̂γω(k) = ln−γ(e+ |k|)ω̂(k), ∀k ∈ R2. (2.1.2)for γ > 12 . In this chapter, we narrow our focus on these examples.The logarithmically regularized 2D Euler equations∂tω+(u ·∇)ω = 0, (x, t) ∈ R2×Ru = ∇⊥ψ, ∆ψ = Tγω,ω|t=0 = ω0.(LE)can be seen as a family of transport equations for a scalar function ω :R2×R→Rwith the non-local velocity u : R2×R→ R2. The Fourier multiplier Tγ = Tγ(|∇|)is defined as in (2.1.2) but we extend the range of γ to γ > 0. By its definition, theoperator Tγ in the extended range still plays a role of a logarithmic regularization.In the case of γ = 0, this operator is considered as the identity, so that (LE) withγ = 0 corresponds to the 2D Euler vorticity equation.Under suitable regularity assumptions on the solution, the transport phenomena8of the equation (LE) can be better described by the equivalent formω(φ(x, t), t) = ω0(x),with the help of the characteristic φ : R2×R→ R2 which solves∂tφ(x, t) = u(φ(x, t), t)φ(x,0) = x.The global well-posedness result of the 2D Euler vorticity equation (the caseγ = 0 in (LE)) in the subcritical spaces W s,p(R2)∩H˙−1(R2), p> 2s can be extendedto that of (LE) for γ ≥ 0. The local well-posedness follows from the usual energymethod which requires two key estimates: commutator estimates and Sobolev in-equalities. The critical space is determined by the Sobolev inequality‖∇u‖∞ =∥∥∥D∇⊥∆−1Tγω∥∥∥∞. ‖ω‖W s,p(R2) , p >2s.Then, we can extend the local solution to the global one by the Beale-Kato-Majdacriterion in [3].In [11], the regularized velocity u=∇⊥∆−1Tγω leads to the local well-posednessof (LE) even in the critical space H1(R2)∩H˙−1(R2) for γ > 12 . Then, for γ > 32 , theglobal lifespan of the local-in-time solutions is obtained by Dong-Li [13]. On theother hand, the strong ill-posedness of 2D Euler equation (γ = 0) in the borderlinespace H1(R2)∩ H˙−1(R2) is established by Bourgain-Li [4]. More precisely, theyprove that for any given compactly supported smooth initial data, a small pertur-bation in the borderline space can be always found such that the perturbed solu-tion uniquely exists in some other solution space but leaves the borderline spaceinstantaneously. Later, Elgindi-Jeong [14] gives a very delicate proof (based onKiselev-Sˇvera´k [27]) and show the ill-posedness for some special initial data onthe torus T2.In this chapter, we prove that the logarithmically regularized 2D Euler equa-tions (LE) for 0< γ < 12 are strongly ill-posed in the critical Sobolev space H1(R2)∩H˙−1(R2). The ill-posedness in the strong sense is defined as in [4]. This closes the9gap between γ = 0 (ill-posed) and γ > 12 (well-posed), and give complete answersto well/ill-posedness questions of logarithmically regularized 2D Euler equations.We consider two types of perturbations: one has non-compact support and theother is compactly supported.Theorem 2.1.1 (Non-compact case). Let 0< γ ≤ 12 and a∈C∞c (R2). Then, for anyε > 0, we can find a small perturbation ζ ∈C∞(R2) in the sense of‖ζ‖H˙1(R2)+‖ζ‖L1(R2)+‖ζ‖L∞(R2) < εsuch that for the perturbed initial data from a, we have a unique classical solutionω to (LE) ∂tω+u ·∇ω = 0, (x, t) ∈ R2× (0,1]u = ∇⊥ψ, ∆ψ = Tγω,ω|t=0 = a+ζ ,satisfying ω(·, t) ∈C∞(R2) for 0 ≤ t ≤ 1 and ω ∈C([0,1];L1(R2)∩L∞(R2)), butthe solution ω leaves the critical Sobolev space instantaneously. i.e., for each0 < T ≤ 1,‖ω‖L∞([0,T ];H˙1(R2)) =+∞.Theorem 2.1.2 (Compact case). Let 0 < γ ≤ 12 and a ∈C∞c (R2) which is odd in x2.Then, for any ε > 0, we can find a small perturbation ζ ∈Cc(R2) in the sense of‖ζ‖H˙1(R2)+‖ζ‖L∞(R2)+‖ζ‖L1(R2)+‖ζ‖H˙−1(R2) < εsuch that for the perturbed initial data from a, we have a unique solution ω :R2× [0,1]→ R in C([0,1];Cc(R2)) to (LE)∂tω+u ·∇ω = 0, (x, t) ∈ R2× (0,1]u = ∇⊥ψ, ∆ψ = Tγω,ω|t=0 = a+ζ ,satisfying L∞-norm preservation, but the solution ω leaves the critical Sobolevspace instantaneously.10The proof follows the outline of the strong ill-posedness scheme for the 2DEuler equations, developed in [4]. It consists of 3 steps: creation of the large La-grangian deformation, local inflation of the critical norm, and patching argument.More precisely, we first do local construction of the perturbation ζ by finding afamily of initial data with the large Lagrangian deformations Dφ —large in thesense of L∞-norm—in a shorter time. Then, the critical Sobolev norm inflation isinduced by the large Lagrangian deformation, so that we get a family of local solu-tions whose critical norm becomes larger in a shorter time. Finally, we sequentiallypatch the local solutions in a way of minimizing the interaction between them. Thismakes the patched solution locally behave like local solutions and hence have thecritical norm inflation property.Difficulties first arise in the local construction of the perturbation. The ve-locity u = ∇⊥∆−1Tγω in (LE) is more regular than the one in the Euler but thecritical space remains same. This makes it more difficult for local solutions to beinflated in the critical norm. Furthermore, one of the main ingredients of gettingthe larger Lagrangian deformation is missing— a pointwise estimate of the kernelsof D∇⊥∆−1Tγ . To solve these issues, we find essentially sharp pointwise lowerbounds of the kernel. What’s more, we construct local initial data having increas-ingly higher frequencies. Along these lines, the desired local construction can beachieved. Then, the successful construction of non-compactly supported perturba-tion follows as in [4], placing local solutions far from each other. However, fora compactly supported perturbation, the genuine difficulty moves to the patchingprocess of local solutions. The increasingly higher frequencies of local initial dataare likely to intensify interaction between local solutions. Moreover, in order tohave a compact support, the local solutions must be placed at an infinitesimal dis-tance from each other eventually. This enhances the interaction further. In a worsecase, the active interaction can make high frequencies of local solutions canceledout, so the norm inflation of local solutions becomes meaningless for the globalone. On the other hand, increasingly higher frequencies of local solutions mostlikely help to create the norm inflation. In order to see what really happens, a sharpcontrol of the propagation of the current local initial data is required under the pres-ence of the previously chosen ones. This can be done based on a keen analysis ofthe non-local operators. As a result, it can be shown that the existing local solution11does not destroy the norm inflation of the current local solution in a very short time.This approach is different from the one in [4] based on the perturbation argument,and makes the behaviour of the solution more clear.The chapter consists of the following sections. Based on the creation of thelarge Lagrangian deformation (Section 2.3), local critical norm inflation (Section2.4), and patching argument (Section 2.5), we get the proof of Theorem 2.1.1 inSection 2.6. Then, the compact case (Theorem 2.1.2) follows in Section Notations• For a point x ∈R2 and a positive real number R, B(x,R) is the Euclidean balldefined byB(x,R) = {y ∈ R2 : |x− y|< R}.For a set A ⊂ R2 and a positive real number R, a generalized ball B(A,R)meansB(A,R) = {y ∈ R2 : |x− y|< R for some x ∈ A}.Obviously, when A is a single point set, A= {x}, we have B(A,R) = B(x,R).• Given two sets A and B in R2, the distance between two sets is denoted bydist(A,B) := inf{|x− y| : x ∈ A and y ∈ B}.• For any function f on R2, we denote the Fourier transform of f byfˆ (k) =ˆR2f (x)e−ik·xdx, k ∈ R2,and its inverse Fourier transform byfˇ (x) =1(2pi)2ˆR2fˆ (k)eik·xdk.• For any 1 ≤ p ≤ ∞, ‖·‖Lp(R2) is the usual Lebesgue norm in R2 with itsabbreviation ‖·‖p. For any m ∈ N and 1 ≤ p ≤ ∞, ‖·‖W m,p(R2) denotes theusual Sobolev norm inR2. In the case of p= 2, we use Hm(R2) =W m,2(R2).12The homogeneous Sobolev norm is defined by‖ f‖H˙s(R2) =(ˆR2|k|2s| fˆ (k)|2dk) 12, ∀s ∈ R,which includes the definition of H˙−1(R2)-norm. We omit (R2) in the ex-pression of Sobolev norms, when the domain of a function is obvious.• Given two comparable quantities X and Y , the inequality X . Y stands forX ≤ CY for some positive constant C. In a similar way, X & Y denotesX ≥ CY for some C > 0. We write X ∼ Y when both X . Y and Y . Xhold. When the constants C in the inequalities depend on some quantitiesZ1, · · · , Zn, we use .Z1,··· ,Zn , &Z1,··· ,Zn , and ∼Z1,··· ,Zn . On the other hand, wesay X  Y if X ≤ εY for some sufficiently small ε > 0. Similarly, X  Y isdefined.Since we prove the strong ill-posedness of (LE) for each 0< γ ≤ 12 , we omit thedependence of γ below if it is not needed. Also, without mentioning, we assume0 < γ ≤ 12 .2.3 Large Lagrangian deformationIn this section, we find a family of initial data which has the large Lagrangiandeformation property. As we mentioned, one of the main ingredients is finding asharp pointwise estimate of the kernel of the operator −∂12∆−1Tγ from below. Weconsider the case Tγ(|∇|) = ln−γ(e−∆) first.Lemma 2.3.1. Let γ > 0 and K12 be the kernel of the Fourier multiplier−∂12∆−1 ln−γ(e−∆). Then, for any x = (x1,x2) ∈ R2, x1 > 0, x2 > 0, we haveK12(x1,x2)≥ Cx1x2|x|4 ln−γ(e+1|x|)e−|x|2(2.3.1)for some positive constant C depending only on γ .13Proof. Using the equalitiesˆ ∞0e−|k|2s|k|2ds = 1, for k 6= 0,1Γ(γ)ˆ ∞0e−attγdtt= a−γ , for a > 0,the Fourier transform of K12 can be written asK̂12(k) =−k1k2|k|2 ln−γ(e+ |k|2) =ˆ ∞0e−|k|2s(−k1k2) ln−γ(e+ |k|2)ds=ˆ ∞01Γ(γ)ˆ ∞0(e+ |k|2)−te−|k|2s(−k1k2)tγ dtt ds=1Γ(γ)ˆ ∞01Γ(t)ˆ ∞0e−eβˆ ∞0(−k1k2)e−|k|2(β+s)dsβ t dββ tγ dtt, ∀k 6= 0.(2.3.2)Taking the inverse Fourier transform, the kernel K12(x), for any x 6= 0, can beexpressed as an integral form:K12(x) =1Γ(γ)ˆ ∞01Γ(t)ˆ ∞0e−eβ(ˆ ∞0∂12(e(s+β )∆δ0)(x)ds)β tdββtγdtt∼γ x1x2ˆ ∞01Γ(t)ˆ ∞0e−eβ(ˆ ∞01(s+β )3e−|x|24(s+β ) ds)β tdββtγdtt=x1x2|x|4ˆ ∞0|x|2tΓ(t)ˆ ∞0e−e|x|2β˜(ˆ ∞01(s˜+ β˜ )3e− 14(s˜+β˜ ) ds˜)β˜ tdβ˜β˜tγdtt,where et∆δ0 is the usual heat kernel. The last equality easily follows from thechange of variables β = |x|2β˜ and s = |x|2s˜.Then, the integral in s˜ can be computed asˆ ∞01(s˜+ β˜ )3e− 14(s˜+β˜ ) ds˜ =ˆ ∞β˜1τ3e−14τ dτ =ˆ ∞β˜1τ(4e−14τ )′dτ=4τe−14τ∣∣∣∣∞β˜+ˆ ∞β˜4τ2e−14τ dτ= 16(1− e−14β˜ − 14β˜e− 14β˜),(2.3.3)14so that we simplify the integral form asK12(x)∼γ x1x2|x|4ˆ ∞0|x|2tΓ(t)ˆ ∞0e−e|x|2β˜(1− e−14β˜ − 14β˜e− 14β˜)β˜ tdβ˜β˜tγdtt, ∀x 6= 0.Now, for each x = (x1,x2) with x1 > 0 and x2 > 0, we find the lower bound ofthe kernel. Indeed, the desired lower bound (2.3.1) follows fromˆ ∞0|x|2tΓ(t)ˆ ∞0e−e|x|2β˜(1− e−14β˜ − 14β˜e− 14β˜)β˜ tdβ˜β˜tγdtt& e−|x|2ˆ 10|x|2tΓ(t)ˆ 1e0β˜ tdβ˜β˜tγdtt& e−|x|2ˆ 10|x|2ttΓ(t)tγdtt& e−|x|2ˆ 10|x|2ttγ dtt&γ ln−γ(e+1|x|)e−|x|2.Now, we consider the case of Tγ(|∇|) = ln−γ(e+ |∇|). To express the corre-sponding kernel as an integral form, we need the following identity.Lemma 2.3.2. (Subordination identity) For any r ≥ 0, we havee−r =1√piˆ ∞0e−τe−r24τ τ−12 dτ.Proof. By using Fourier transform, it is easy to seee−r =1piˆ ∞−∞11+θ 2eiθrdθ , ∀r ≥ 0.Since we can write11+θ 2=ˆ ∞0e−τe−τθ2dτ,the result follows from interchanging the dθ −dτ integral.Lemma 2.3.3. Let γ > 0 and K˜12 be the kernel of the multiplier −∂12∆−1 ln−γ(e+15|∇|). Then, for any x = (x1,x2) ∈ R2, x1 > 0, x2 > 0, we haveK˜12(x1,x2)≥ Cx1x2|x|4 ln−γ(e+1|x|)e−|x|2(2.3.4)for some positive constant C depending only on γ > 0.Proof. As we did in Lemma 2.3.1, the Fourier transform of K˜12 can be expressedas follows:̂˜K12(k) =−k1k2|k|2 ln−γ(e+ |k|)=1Γ(γ)ˆ ∞01Γ(t)ˆ ∞0e−eβˆ ∞0(−k1k2)e−k|β |e−|k|2sdsβ t dββ tγ dtt, ∀k 6= 0.(2.3.5)Using the identity in Lemma 2.3.2, for β ≥ 0 we havee−|k|β =1√piˆ ∞0e−τe−|k|2β24τ τ−12 dτ, (2.3.6)so that the kernel can be written as an integral form: for any x 6= 0,K˜12(x) =1√piΓ(γ)ˆ ∞01Γ(t)ˆ ∞0e−eβˆ ∞0ˆ ∞0e−τ(∂12e(β24τ +s)∆δ0)(x)τ−12 dτdsβ tdββtγdtt∼γ x1x2ˆ ∞01Γ(t)ˆ ∞0e−eβˆ ∞0e−τˆ ∞01(β 24τ + s)3 e−|x|24(β24τ +s)dsτ−12 dτβ tdββtγdtt=x1x2|x|4ˆ ∞0|x|tΓ(t)ˆ ∞0e−e|x|β˜ˆ ∞0e−τˆ ∞01(β˜ 24τ + s˜)3 e−14(β˜24τ +s˜)ds˜τ−12 dτβ˜ tdβ˜β˜tγdtt.In the last equality, we do the change of variables β = |x|β˜ and s = |x|2s˜.The integral in s˜ can be simplified asˆ ∞01(β˜ 24τ + s˜)3 e−14(β˜24τ +s˜)ds˜ = 16(1− e−τβ˜2 − τβ˜ 2e− τβ˜2), (2.3.7)16and the integral form also becomes simple,K˜12(x)∼γ x1x2|x|4ˆ ∞0|x|tΓ(t)ˆ ∞0e−e|x|β˜(ˆ ∞0e−τ(1− e−τβ˜2 − τβ˜ 2e− τβ˜2 )τ−12 dτ)β˜ tdβ˜β˜tγdtt.To get a lower bound, we first consider the integral in τ and β˜ :ˆ ∞0e−e|x|β˜(ˆ ∞0e−τ(1− e−τβ˜2 − τβ˜ 2e− τβ˜2 )τ−12 dτ)β˜ tdβ˜β˜&ˆ ∞0e−τe−√eτ|x|(ˆ √ τe0β˜ tdβ˜β˜)τ−12 dτ≥ 1t√ete−|x|eˆ 1e30e−ττt−12 dτ & 1t(t+1)e2te−|x|e , ∀x 6= 0, t > 0.Then, for each x = (x1,x2) ∈ R2 with x1 > 0 and x2 > 0, the desired lowerbound (2.3.4) of the kernel follows fromˆ ∞0|x|tΓ(t)ˆ ∞0e−e|x|β˜(ˆ ∞0e−τ(1− e−τβ˜2 − τβ˜ 2e− τβ˜2 )τ−12 dτ)β˜ tdβ˜β˜tγdtt& e−|x|eˆ 10|x|ttΓ(t)1(t+1)e2ttγdtt&γ ln−γ(e+1|x|)e−|x|2.Remark 2.3.1. By Lemma 2.3.1 and Lemma 2.3.3, we can see that the kernels of−∂12∆−1Tγ for both Tγ = ln−γ(e−∆) and Tγ = ln−γ(e+ |∇|) have the same lowerbound. Therefore, we use the combined notations Tγ and its kernel K for both casesfrom now on.Now, we are ready to estimate the Lagrangian deformation.Proposition 2.3.4. Let γ > 0. Suppose that a function g ∈ C∞c (R2) satisfies thefollowing conditions.(i) g is odd in x1 and x2.(ii) g(x1,x2)≥ 0 on {x1 ≥ 0,x2 ≥ 0}.17(iii)G≡ˆx1>0,x2>0g(x)x1x2|x|4 ln−γ(e+1|x|)e−|x|4dx > 0.Let φ be the characteristic line defined by∂tφ(x, t) = ∇⊥∆−1Tγω(φ(x, t), t)φ(x,0) = x,where ω is a smooth solution to (LE) for the initial data ω0 = g. Then, the La-grangian deformation Dφ satisfiesˆ t0e−‖Dφ(·,τ)‖4∞dτ ≤ 1CGln(1+CGt), ∀t ≥ 0 (2.3.8)for some positive constant C =C(γ). In particular, we havemax0≤τ≤t‖Dφ(·,τ)‖∞ ≥ ln14(CGtln(1+CGt)), ∀t > 0. (2.3.9)Proof. Using the parity of g, it can be easily checked that ω is odd in x1 and x2,and hence φ(x, t) = (φ1(x1,x2, t),φ2(x1,x2, t)) satisfiesφ1(0,x2, t)≡ 0, φ2(x1,0, t)≡ 0 ∀x1 ∈ R,x2 ∈ R, (2.3.10)φ(0, t)≡ 0.Also, the Frechet derivative [Du(0, t)]i j = ∂ jui(0, t) of u = ∇⊥∆−1Tγω at x = 0takes the formDu(0, t) =(λ (t) 00 −λ (t)),where λ (t) =−∂12∆−1Tγw(0, t). Then, this implies(Dφ)(0, t) =exp(´ t0 λ (τ)dτ) 00 exp(−´ t0 λ (τ)dτ) .On the other hand, by (2.3.10) and the sign preservation property of φ1 and φ2,18we obtain for any x1 ≥ 0, x2 ≥ 0, and t ≥ 0,1‖Dφ(·, t)‖∞φ1(x1,x2, t)≤ x1 ≤ φ1(x1,x2, t)‖Dφ(·, t)‖∞ ,1‖Dφ(·, t)‖∞φ2(x1,x2, t)≤ x2 ≤ φ2(x1,x2, t)‖Dφ(·, t)‖∞ .(2.3.11)Thus, for any x1 > 0, x2 > 0, and t ≥ 0,φ1φ2φ 21 +φ 22=1φ1φ2 +φ2φ1≥ 1‖Dφ‖2∞x1x2|x|2 .Recall that we denote the kernel of the operator −∂12∆−1Tγ by K. By Lemma2.3.1 and Lemma 2.3.3, for any x = (x1,x2) with x1 > 0 and x2 > 0, and t ≥ 0,K(φ(x, t))&γ(φ1φ2|φ |2)1|φ |2 ln−γ(e+1|φ |)e−|φ |2& 1‖Dφ‖4∞x1x2|x|4 ln−γ(e+‖Dφ‖∞|x|)e−‖Dφ‖2∞|x|2& 1‖Dφ‖4∞x1x2|x|4 ln−γ(e+1|x|)(1+ ln(1+‖Dφ‖∞))−γ e−14‖Dφ‖4∞e−|x|4&γ e−‖Dφ(·,t)‖4∞x1x2|x|4 ln−γ(e+1|x|)e−|x|4.Now, we estimate λ (t) from belowλ (t) =ˆR2K(y)ω(y, t)dy = 4ˆy1>0,y2>0K(y)ω(y, t)dy= 4ˆx1>0,x2>0K(φ(x, t))g(x)dx&γ e−‖Dφ(·,t)‖4∞ˆx1>0,x2>0g(x)x1x2|x|4 ln−γ(e+1|x|)e−|x|4dx= e−‖Dφ(·,t)‖4∞G.Then, since‖Dφ(·, t)‖∞ ≥ |Dφ(0, t)| ≥ exp(ˆ t0λ (τ)dτ), ∀t ≥ 019where | · | is the usual matrix norm, we have a positive constant C > 0 dependingonly on γ such that‖Dφ(·, t)‖∞ ≥ exp(14CGˆ t0e−‖Dφ(τ)‖4∞dτ), ∀t ≥ 0.This implies thatddtexp(CGˆ t0e−‖Dφ(τ)‖4∞dτ)= exp(CGˆ t0e−‖Dφ(τ)‖4∞dτ)CGe−‖Dφ(t)‖4∞≤CG‖Dφ(τ)‖4∞ e−‖Dφ(τ)‖4∞ ≤CG.Therefore, we obtainexp(CGˆ t0e−‖Dφ(τ)‖4∞dτ)≤ 1+CGt.The inequalities (2.3.8) and (2.3.9) then follow easily.Remark 2.3.2. By a slight modification of the proof, we can restrict the regionwhere the large Lagrangian deformation occurs;max0≤τ≤t‖Dφ(·,τ)‖L∞(B(0,R)) ≥ ln14(CGtln(1+CGt)), ∀0 < t ≤ 1, (2.3.12)if R > 0 satisfiessupp(g)⊂ B(0,R) and φ−1(Bg, t)⊂ B(0,R)for all 0≤ t ≤ 1, where Bg =B(0,Rg) is the smallest ball containing⋃0≤t≤1 supp(ω(·, t)).Indeed, if x is in supp(g), then φ(x, t) ⊂ supp(ω(·, t)) and |φ(x, t)| ≤ Rg when0≤ t ≤ 1. This implies that for 0≤ t ≤ 1∥∥D(φ−1)(·, t)∥∥L∞(Bg) = ∥∥(Dφ)−1(φ−1(·, t), t)∥∥L∞(Bg) ≤ ‖Dφ(·, t)‖L∞(B(0,R)) .In the inequality, we use |det(Dφ(·, t))|= 1 for any t ≥ 0. Then, a modification of20(2.3.11) holds; for x = (x1,x2) ∈ supp(g), x1 ≥ 0, x2 ≥ 0, and 0≤ t ≤ 1, we have1‖Dφ(·, t)‖L∞(B(0,R))φ1(x1,x2, t)≤ x1 ≤ φ1(x1,x2, t)‖Dφ(·, t)‖L∞(B(0,R)) ,1‖Dφ(·, t)‖L∞(B(0,R))φ2(x1,x2, t)≤ x2 ≤ φ2(x1,x2, t)‖Dφ(·, t)‖L∞(B(0,R)) .The rest of the proof is almost identical.2.4 Local critical Sobolev norm inflationIn this section, we show that the inflation of the critical Sobolev norm can be in-duced from the largeness of the Lagrangian deformation. Then, based on this, weconstruct a family of local solutions whose critical norm gets larger in a shortertime, while the critical norm of initial data gets smaller.We first recall Lemma 4.1 in [4].Lemma 2.4.1. Suppose u= u(x, t) and v= v(x, t) are smooth vector fields on R2×R. Let φ : R2×R→ R2 and φ˜ : R2×R→ R2 be the solutions to∂tφ(x, t) = u(φ(x, t), t)φ(x,0) = xand ∂t φ˜(x, t) = u(φ˜(x, t), t)+ v(φ˜(x, t), t)φ˜(x,0) = x.Then, we have positive constants C and C1 satisfyingmax0≤t≤1(∥∥(φ˜ −φ)(·, t)∥∥∞+∥∥(Dφ˜ −Dφ)(·, t)∥∥∞)≤C max0≤t≤1‖v(·, t)‖W 1,∞ · exp(C1 max0≤t≤1‖Dv(·, t)‖∞),where C depends on∥∥D2u(·, t)∥∥L∞([0,1]×R2) and ‖Du(·, t)‖L∞([0,1]×R2), and C1 is anabsolute constant.The following is the main proposition in this section.21Proposition 2.4.2. Suppose thatω is a smooth solution to (LE) with the initial dataω0 and its velocity u =−∇⊥∆−1Tγω , γ > 0, and satisfies the following properties.(i) ‖ω0‖∞+‖ω0‖1+‖ω0‖H˙−1 < ∞.(ii) There exists R0 > 0 such thatsupp(ω0)⊂ B(0,R0)and the characteristic line φ , i.e., the solution to∂tφ(x, t) = u(φ(x, t), t) R2× (0,∞)φ(x,0) = x R2,satisfies‖(Dφ)(·, t0)‖L∞(B(0,R0)) > L (2.4.1)for some 0 < t0 ≤ 1 and L > 89 ·106.Then, we can construct a new smooth solution ω˜ to (LE) for a new initial dataω˜0 which satisfies the following conditions.(i) The size of the new initial data is controlled by that of the original one,‖ω˜0‖H˙−1 ≤ 2‖ω0‖H˙−1 (2.4.2)‖ω˜0‖1 ≤ 2‖ω0‖1, ‖ω˜0‖∞ ≤ 2‖ω0‖∞ , (2.4.3)‖ω˜0‖H˙1 ≤ ‖ω0‖H˙1 +L−12 . (2.4.4)(ii) The new initial data is compactly supported,supp(ω˜0)⊂ B(0,R0). (2.4.5)(iii) The large Lagrangian deformation at t0 induces H˙1-norm inflation:‖ω˜(·, t0)‖H˙1(R2) > L13 . (2.4.6)22Proof of the Proposition.Sketch of the idea. Let φ˜ be the characteristic line corresponding to thenew smooth solution ω˜ . Then, it solves∂t φ˜(x, t) = u˜(φ˜(x, t), t) R2× (0,∞)φ˜(x,0) = x R2,where u˜ = ∇⊥∆−1Tγ ω˜ . Since ω˜(φ˜(x, t), t) = ω˜0(x), we can write the H˙1-norm ofω˜ as‖∇ω˜(·, t)‖22 =ˆR2|∇ω˜0(x) · (∇⊥φ˜2)(x, t)|2dx+ˆR2|∇ω˜0(x) · (∇⊥φ˜1)(x, t)|2dx.(2.4.7)By Lemma 2.4.1, if we choose a new initial data ω˜0 to make ‖u− u˜‖W 1,∞ small,∥∥Dφ −Dφ˜∥∥∞ also gets small. It follows that the main part in the right hand side of(2.4.7) is the one in which φ˜ is replaced by φ . Then, we can produce the H˙1-norminflation of ω˜ at t0 from largeness of the Lagrangian deformation Dφ in (2.4.1)sense. Indeed, we construct the desired new initial data by adding a perturbation,localized at the point where the large Lagrangian deformation occurs, to the origi-nal initial data.Step 1. Construction of the new initial data ω˜0.Assume‖∇ω(·, t0)‖2 ≤ L13 .Otherwise, ω˜0 = ω0 completes the proof.By the assumption (2.4.1) and the smoothness of φ , we can find xL = (x1L,x2L),x1Lx2L 6= 0, in B(0,R0) such that one of the entries of Dφ(xL, t0), say ∂2φ2(xL, t0),satisfies|∂2φ2(xL, t0)|> L.If we further use the continuity of Dφ , we can choose sufficiently small δ > 0satisfying δ min(x1L,x2L), B(xL,δ )⊂ B(0,R0), and|∂2φ2(x, t0)|> L, ∀|x− xL|< δ .23Choose Ψ to be a smooth radial bump function which is compactly supportedon the unit ball B(0,1) and satisfies Ψ ≡ 1 on B(0, 12) and 0 ≤ Ψ ≤ 1. Set Ψδ =1δΨ(x−xLδ ). By the choice of xL and δ , we note that the support of Ψδ lies on one ofthe four quadrants. Now, let b be the odd extension of Ψδ in both variables. Then,we define the new initial data ω˜0, adding a perturbationη0(x) = ω˜0(x)−ω0(x) = 120k√Lcos(kx1)b(x),to the original one ω0 where k will be chosen later sufficiently large. We can easilysee that the perturbation η0 is odd in both variables.Step 2. Check the required conditions on ω˜ .By its construction, the support of η0 is contained in B(0,R0), so that (2.4.5)holds.To get (2.4.2) and (2.4.3), we estimate the corresponding Sobolev norms of η0,‖η0‖1 ≤120k√L‖b‖1 ‖η0‖∞ ≤120k√L‖b‖∞‖η0‖H˙−1 .∥∥x̂η0∥∥∞+‖η0‖2 . 1k ,where the estimate for the negative Sobolev norm follows from the parity of η0.For sufficiently large k, both (2.4.2) and (2.4.3) hold true.Finally, (2.4.4) follows from‖b‖2 ≤ 4‖Ψδ‖2 = 4‖Ψ‖2 < 4√pi,and‖∇η0‖2 ≤120k√L(k‖b‖2+‖∇b‖2)≤1√L,provided that k is sufficiently large.Now, consider the H˙1-norm inflation of the new solution ω˜ . As we mentioned,we first show that the perturbation in the Lagrangian deformation is small. For thispurpose, we consider the perturbation of velocity in W 1,∞(R2).24Since we have‖∇(u˜−u)‖∞ .γ (‖∇ω˜‖4+‖∇ω‖4)23 ‖ω˜−ω‖132 , (2.4.8)it is enough to consider the terms on the right hand side. The terms ‖∇ω˜‖4 and‖∇ω‖4 are estimated by the usual energy method. From the equation for ω˜ , wehaveddt‖∇ω˜‖44 ≤ 4‖∇u˜‖∞ ‖∇ω˜‖44 . (2.4.9)By the log-type interpolation inequality,‖∇u˜(·, t)‖∞ . 1+‖ω˜0‖∞ log(10+‖ω˜0‖2+‖∇ω˜(·, t)‖44),we obtainmax0≤t≤1‖∇ω˜(·, t)‖4 ≤C, (2.4.10)for some constant C = C(‖∇ω˜0‖4 ,‖ω˜0‖2). Note that we can choose an upperbound C which is independent of k. Similarly, we havemax0≤t≤1‖∇ω(·, t)‖4 ≤C (2.4.11)for some positive constant C independent of k.On the other hand, from the equations for ω˜ and ω , we get the equation forη = ω− ω˜ ,∂tη+∇⊥∆−1Tγη ·∇ω+∇⊥∆−1Tγ ω˜ ·∇η = 0.Taking´ ·ηdx on both side, η satisfies12ddt‖η(·, t)‖22 ≤∥∥∥∇⊥∆−1Tγη∥∥∥4‖∇ω‖4 ‖η‖2 . ‖∇ω‖4 ‖η‖22 .Here, the last inequality follows from the Hardy-Littlewood Sobolev inequality and25the compactness of the support of η . By the Gro¨nwall inequality, we obtainmax0≤t≤1‖η(·, t)‖2 . ‖η0‖2 .1k. (2.4.12)Combining with (2.4.8), (2.4.10), and (2.4.11), the perturbation of u can be esti-mated by‖∇(u˜−u)‖∞ . k−13 .Finally, by the Gagliardo-Nirenberg interpolation inequality, for any 0≤ t ≤ 1,we have‖(u˜−u)(·, t)‖∞ . ‖∇(u˜−u)‖13∞ ‖u˜−u‖234 . k−19 ‖η‖232 . k−79 .Therefore, Lemma 2.4.1 gives the desired estimate for the perturbation of the La-grangian deformation,max0≤t≤1(∥∥(φ˜ −φ)(·, t)∥∥∞+∥∥(Dφ˜ −Dφ)(·, t)∥∥∞). k− 13 .Now, we are ready to get H˙1-norm inflation. Recall (2.4.7) and we furtherestimate its right hand side as follows.‖∇ω˜(·, t0)‖22 ≥ˆR2|∇ω˜0(x) · (∇⊥φ˜2)(x, t0)|2dx≥ 12ˆR2|∇ω˜0(x) · (∇⊥φ2)(x, t0)|2dx−O(k− 23 )≥ 14ˆR2|∇η0(x) · (∇⊥φ2)(x, t0)|2dx− 12ˆR2|∇ω0(x) · (∇⊥φ2)(x, t0)|2dx−O(k− 23 ).(2.4.13)By the assumption on ω , we haveˆR2|∇ω0(x) · (∇⊥φ2)(x, t0)|2dx≤ ‖∇ω(·, t0)‖22 ≤ L23 .26On the other hand, by the construction of the perturbation η0, we obtainˆR2|∇η0(x) · (∇⊥φ2)(x, t0)|2dx≥ 1800LˆR2|sin(kx1)b(x)∂2φ2(x, t0)|2dx−O(k−2)≥ L8001δ 2ˆ|x−xL|< 12 δ|sin(kx1)|2dx−O(k−2)≥ 126 ·102 L−O(k−1).Therefore, we get the desired norm inflation‖∇ω˜(·, t0)‖22 ≥128 ·102 L−12L23 −O(k− 23 )> L 23provided that L > 89 · 106 and k is sufficiently large. In other words, (2.4.6) isobtained.Remark 2.4.1. Based on Proposition 2.3.4 and Proposition 2.4.2, we can constructa family of initial data having H˙1-norm inflation.Choose a nonzero radial bump function ϕ ∈ C∞c (R2) satisfying 0 ≤ ϕ ≤ 1,ϕ ≡ 1 on B(0, 12), and supp(ϕ)⊂ B(0,1). Then, we define ρ ∈C∞c (R2) byρ(x) = ρ(x1,x2) = ∑a1,a2=±1a1a2ϕ(x1−a1,x2−a22−100). (2.4.14)Clearly, the function ρ is odd in both variables, andˆx1>0,x2>0ρ(x)x1x2|x|4 e−|x|4dx > 0.Now, for each 0 < γ ≤ 12 , define gA ∈C∞c (R2) bygA(x) =CA∑aA≤ j<bA1jγ ρ(2jx), 0 < γ < 12CA∑lnA≤ j<A+lnA 1√ jρ(2jx), γ = 12(2.4.15)where CA = 1√lnA1ln lnA , aA = A11−2γ , and bA = (A+ lnA)11−2γ . Note that the summa-27tions in (2.4.15) are over integer j in the range.First, gA satisfies all assumptions in Proposition 2.3.4. Obviously, gA is an oddfunction in x1 and x2, and gA(x1,x2) ≥ 0 for x1 ≥ 0 and x2 ≥ 0. Using disjointsupports of ρ(2 j·), j ∈ N, we have for A≥ e2,GA =ˆx1>0,x2>0gA(x)x1x2|x|4 ln−γ(e+1|x|)e−|x|4dx=CA∑j1jγˆx1>0,x2>0ρ(2 jx)x1x2|x|4 ln−γ(e+1|x|)e−|x|4dx=CA∑j1jγˆx1>0,x2>0x∈supp(ρ)ρ(x)x1x2|x|4 ln−γ(e+2 j|x|)e−|x|424 j dx≥CA∑j1j2γ(ˆx1>0,x2>0ρ(x)x1x2|x|4 e−|x|4dx)> 0.(2.4.16)Here, the range of summation over j depends on γ , which follows to the one in(2.4.15).Since for A 1, we have∑j1j2γ∼´ bAaA1x2γ dx =11−2γ (b1−2γA −a1−2γA ) = 11−2γ lnA, 0 < γ < 12´ A+lnAlnA1x dx = ln(A+ lnA)− ln lnA, γ = 12∼γ lnA,GA has a lower boundGA &γ√lnAln lnA.Then, by Proposition 2.3.4, for any A with A ≥ A0 for some A0 = A0(γ), we canfind tA ∈(0, 1ln lnA]such that the characteristic line φA corresponding to each initialdata gA has the large Lagrangian deformation‖DφA(·, tA)‖L∞(B(0, 12 )) > ln14 ln ln lnA. (2.4.17)Now, we induce the critical norm inflation from the large Lagrangian deformation.28Observe that all assumptions in Proposition 2.4.2 hold for ω0 = gA, t0 = tA, L =ln14 ln ln lnA, and R0 = 1, provided that A is sufficiently large. Indeed, using|φA(x, t)− x| ≤ˆ t0|∂sφA(x,s)|ds≤∥∥∥∇⊥∆−1Tγ(gA ◦φ−1A )∥∥∥L∞x,t t . ‖gA‖ 121 ‖gA‖ 12∞ tfor all x ∈ R2 and t ≥ 0, we have φ−1(BgA , t) ⊂ B(0,1) for sufficiently large A,where BgA is defined as in Remark 2.3.2. In what follows, we have a desired family{g˜A} of a new initial data which has the following properties:(i) g˜A gets small as A goes to infinity in the following sense:‖g˜A‖1 ≤ 2‖gA‖1 .1Aln4,‖g˜A‖∞ ≤ 2‖gA‖∞ ≤2√lnA‖∇g˜A‖2 ≤ ‖∇gA‖2+ ln−18 ln ln lnA≤ Cγln lnA+ ln−18 ln ln lnA(2.4.18)where Cγ is independent of A.(ii) supp(g˜A)⊂ B(0,1).(iii) The smooth solution ω˜A to (LE) for the initial data g˜A has local critical norminflation:‖∇ω˜A(·, tA)‖2 > ln112 ln ln lnA.2.5 Patching argumentIn this section, we introduce useful lemmas and a proposition for the constructionof the desired global solution from local ones. For the non-compactly supportedcase, our strategy is using a huge distance between local solutions so that theybarely interact with each other. This leads the global solution to locally behave likelocal solutions. The following proposition describes this in detail.Proposition 2.5.1. Let {ω j0} ⊂C∞c (B(0,1)) be a sequence of functions satisfying∞∑j=1(∥∥ω j0∥∥2H1 +∥∥ω j0∥∥1)+ supj∥∥ω j0∥∥∞ ≤M (2.5.1)29for some M > 1. For each γ > 0, let C0 be an absolute constant such that∥∥∥∇⊥∆−1Tγ f∥∥∥∞≤C0(‖ f‖1+‖ f‖∞).Then, we can find a sequence {x j} of centers with |x j− xk|  1 for j 6= k suchthat there exists a unique classical solution ω to (LE) for the initial dataω0(x) =∞∑j=1ω j0(x− x j) ∈ L1∩L∞∩H1∩C∞such that the following hold.(i) For any 0≤ t ≤ 1, ω(·, t) is supported in the union of disjoint balls:supp(ω(·, t))⊂∞⋃j=1B(x j,3C0M). (2.5.2)(ii) For each 0≤ t ≤ 1, ω(·, t) ∈C∞(R2), and ω ∈C([0,1];L1(R2)∩L∞(R2)).(iii) For any ε > 0, we can find a sufficiently large integer j0 = j0(ε) so that forj ≥ j0, we havemax0≤t≤1∥∥(ω−ω j)(·, t)∥∥H2(B(x j,3C0M)) < ε, (2.5.3)where a local solution ω j solves (LE) for the initial dataω j|t=0 = ω j0(·− x j).Before we prove this proposition, we consider some preliminary lemmas.Lemma 2.5.2. Suppose that f ∈ Hk∩L1 for some k ≥ 2 and g ∈ H2∩L1 satisfy‖ f‖1+‖g‖1+ sup(‖ f‖∞ ,‖g‖∞)≤M < ∞,dist(supp( f ),supp(g))≥ 100C0M > 0 (2.5.4)for some constant M > 1, and the Lebesgue measure of the support of f is boundedby some positive constant M1.30Then, the solution ω to∂tω+u ·∇ω = 0 R2× (0,1]u = ∇⊥∆−1Tγωω|t=0 = f +ghas the following properties.(i) The solution ω can be decomposed as ω = ω f +ωg such thatω f |t=0 = f , ωg|t=0 = gsupp(ω f (·, t))⊂ B(supp( f ),2C0M), (2.5.5)supp(ωg(·, t))⊂ B(supp(g),2C0M), (2.5.6)dist(supp(ω f (·, t)),supp(ωg(·, t)))≥ 90C0M, ∀0≤ t ≤ 1, (2.5.7)where C0 is defined as in Proposition 2.5.1.(ii) The Sobolev norms of ω f can be estimated bymax0≤t≤1∥∥ω f (·, t)∥∥Hk ≤C (2.5.8)for some constant C =C(‖ f‖Hk ,k,M,M1) independent of ‖g‖Hk .Proof. Define ω f and ωg by the solutions to∂tω f +u ·∇ω f = 0ω f |t=0 = f (2.5.9)and ∂tωg+u ·∇ωg = 0ωg|t=0 = g. (2.5.10)31Let φ be the characteristic line which solves∂tφ(x, t) = u(φ(x, t), t)φ(x,0) = x.Then, the equations (2.5.9) and (2.5.10) can be written asω f (φ(x, t), t) = f (x), and ωg(φ(x, t), t) = g(x).From these forms, it follows that for 1≤ p≤ ∞∥∥ω f (·, t)∥∥p = ‖ f‖p , and ‖ωg(·, t)‖p = ‖g‖p , ∀0≤ t ≤ 1,andmax0≤t≤1‖u(·, t)‖∞ ≤C0M.Since we have|φ(x, t)− x| ≤ˆ t0|∂sφ(x,s)|ds≤ max0≤s≤1‖u(·,s)‖∞ t ≤C0Mt,(2.5.5) and (2.5.6) easily follows fromsupp(ω f (·, t))⊂ φ(supp( f ), t)⊂ B(supp( f ),2C0M),supp(ωg(·, t))⊂ φ(supp(g), t)⊂ B(supp(g),2C0M), ∀0≤ t ≤ 1.Using the assumption (2.5.4) additionally, the triangle inequality impliesdist(supp(ω f (·, t)),supp(ωg(·, t)))≥ 90C0M, ∀0≤ t ≤ 1. (2.5.11)In other words, (2.5.7) is obtained.To control the Sobolev norm of ω f , we first estimate ∇⊥∆−1Tγωg when 0≤ t ≤1 and x ∈ supp(ω f (·, t)). Since the supports of ω f (·, t) and ωg(·, t) are apart from32each other for 0≤ t ≤ 1 (see (2.5.11)), we have for 0≤ t ≤ 1 and x∈ supp(ω f (·, t)),∣∣∣∂α∇⊥∆−1Tγωg(x, t)∣∣∣= ∣∣∣∣ˆ|y−x|≥90C0M ∂αH(x− y)ωg(y)dy∣∣∣∣≤ ‖∂αH‖L∞(|z|≥90C0M) ‖g‖1 ,(2.5.12)where H is the kernel of the Fourier multiplier ∇⊥∆−1Tγ . By Lemma 2.8.1, for anymulti-index α with |α| ≥ 0, H satisfies|∂αH(z)|.α,γ 1|z||α|+1 , ∀z 6= 0and thereforemax0≤t≤1maxx∈supp(ω f (·,t))|∂α∇⊥∆−1Tγωg(x, t)|.α,γ 1. (2.5.13)To get (2.5.8), we use the energy method. We consider the Sobolev normW 1,p(R2) for 2 < p≤+∞ first. From the equation (2.5.9) for ω f , we have1pddt∥∥∇ω f∥∥pp ≤ (∥∥∥D∇⊥∆−1Tγω f∥∥∥∞+∥∥∥D∇⊥∆−1Tγωg∥∥∥L∞(supp(ω f (·,t))))∥∥∇ω f∥∥pp(2.5.14)By a log-type interpolation inequality together with the Lp-norm preservation ofω f ,∥∥∥D∇⊥∆−1Tγω f (·, t)∥∥∥∞.p 1+‖ f‖∞ log(10+‖ f‖2+∥∥∇ω f (·, t)∥∥pp), ∀0≤ t ≤ 1.Combining with (2.5.13) and (2.5.14), this impliesmax0≤t≤1∥∥ω f (·, t)∥∥W 1,p(R2) ≤C(‖ f‖W 1,p(R2) , p,M).We now estimate in Hk(R2), k ≥ 2. By the commutator estimate in [34, Theo-33rem 1.9], for J = (1−∆) 12 , we getddt∥∥Jkω f∥∥2 ≤ ∥∥∥[Jk,∇⊥∆−1Tγω f ·∇]ω f∥∥∥2+∥∥∥[Jk,∇⊥∆−1Tγωg ·∇]ω f∥∥∥2.∥∥∥Jk−1D∇⊥∆−1Tγω f∥∥∥3∥∥∇ω f∥∥6+∥∥∥D∇⊥∆−1Tγω f∥∥∥∞∥∥Jkω f∥∥2+max|α|≤kmax0≤t≤1∥∥∥Dα∇⊥∆−1Tγωg(·, t)∥∥∥L∞(supp(ω f (·,t)))∥∥Jkω f∥∥2≤C∥∥Jkω f∥∥2 ,where the constant in the last inequality depends on ‖ f‖H2 , M, M1, and k.Therefore, by the Gro¨nwall inequality, we obtain (2.5.8).Lemma 2.5.3. Suppose that f is in H3(R2)∩L1(R2) with Leb(supp( f ))≤M1 forsome M1, g is in H2(R2)∩L1(R2), and they satisfy‖ f‖1+‖g‖1+ sup(‖ f‖∞ ,‖g‖∞)≤Mfor some M > 1. Let ω and ω˜ be solutions to (LE) for the initial data f +g and f ,respectively.Then, for each ε > 0, we can find sufficiently large R= R(ε,‖ f‖H3 ,M,M1)> 0such that ifdist(supp( f ),supp(g))≥ R, (2.5.15)then ω can be decomposed as ω = ω f +ωg such that ω f and ωg satisfy (2.5.5)-(2.5.7) andmax0≤t≤1∥∥(ω f − ω˜)(·, t)∥∥H2 < ε. (2.5.16)Remark 2.5.1. Similar to (2.5.5) and (2.5.6), we havesupp(ω˜(·, t))⊂ B(supp( f ),2C0M), ∀0≤ t ≤ 1, (2.5.17)where C0 is defined as in Proposition 2.5.1. It follows from max0≤t≤1 ‖u˜(·, t)‖∞ ≤34C0M for u˜ = ∇⊥∆−1Tγ ω˜ .Proof. We use the same decomposition ω = ω f +ωg in Lemma 2.5.2. Then, wehave (2.5.5) and (2.5.6). Furthermore, (2.5.7) is also obtained, provided that R ≥100C0M. In fact, using (2.5.15), we havedist(supp(ω f (·, t),supp(ωg(·, t))))≥ R−10C0M ≥ 12R, ∀0≤ t ≤ 1 (2.5.18)for sufficiently large R.To get (2.5.16), we recall the equation for ω f ,∂tω f +u ·∇ω f = 0ω f |t=0 = f .By the Gagliardo-Nirenberg inequality,∥∥(ω f − ω˜)(·, t)∥∥H2 . (∥∥ω f (·, t)∥∥H3 +‖ω˜(·, t)‖H3) 23 ∥∥(ω f − ω˜)(·, t)∥∥ 132 . (2.5.19)By Lemma 2.5.2, we obtainmax0≤t≤1∥∥ω f (·, t)∥∥H3 ≤C(‖ f‖H3 ,M,M1). (2.5.20)Also, by the usual energy method, we also have a similar inequality for ω˜max0≤t≤1‖ω˜(·, t)‖H3 ≤C(‖ f‖H3 ,M,M1). (2.5.21)Therefore, it is enough to consider ‖η(·, t)‖2 for η = ω f − ω˜ .The equation for η is∂tη+∇⊥∆−1Tγ ω˜ ·∇η+∇⊥∆−1Tγη ·∇ω f +∇⊥∆−1Tγωg ·∇ω f = 0η |t=0 = 0.35Taking´ ·ηdx on both side of the first equation and using (2.5.20), we getddt‖η(·, t)‖2 ≤∥∥∥∇⊥∆−1Tγη ·∇ω f∥∥∥2+∥∥∥∇⊥∆−1Tγωg ·∇ω f∥∥∥2.M1 ‖η‖2∥∥∇ω f∥∥6+∥∥∥∇⊥∆−1Tγωg∥∥∥L∞(supp(ω f (·,t)))∥∥∇ω f∥∥2≤C(‖η‖2+∥∥∥∇⊥∆−1Tγωg∥∥∥L∞(supp(ω f (·,t)))),for some positive constant C depending on ‖ f‖H3 , M, and M1. Then by the Gro¨nwallinequality, we havemax0≤t≤1‖η(·, t)‖2 ≤C(‖ f‖H3 ,M,M1) max0≤t≤1∥∥∥∇⊥∆−1Tγωg∥∥∥L∞(supp(ω f (·,t))). (2.5.22)Using Lemma 2.8.1 and (2.5.18), we have for any 0≤ t ≤ 1 and x∈ supp(ω f (·, t)),|∇⊥∆−1Tγωg(x, t)|= |H ∗ωg(x, t)|.ˆ|x−y|≥ 12 R1|x− y| |ωg(y, t)|dy. R−1 ‖g‖1 ≤MR−1.(2.5.23)Finally, combining (2.5.19)-(2.5.23), we can find R=R(ε,‖ f‖H3 ,M,M1)> 100C0Msufficiently large such thatmax0≤t≤1∥∥(ω f − ω˜)(·, t)∥∥H2 ≤C(‖ f‖H3 ,M,M1)R− 13 < ε.Now we are ready to prove the proposition.Proof of Proposition 2.5.1. Let ω≤n, n ∈ N, be a smooth solution to∂tω≤n+∇⊥∆−1ω≤n ·∇ω≤n = 0,ω≤n|t=0 = ∑nk=1ωk0(x− xk). . (2.5.24)Our strategy is to construct a sequence {xk}k∈N of centers such that the followinghold.(i) For each j ∈ N, {ω≤n} is Cauchy in C([0,1];H2(B(x j,3C0M))).36(ii) For any n ∈ N,supp(ω≤n(·, t))⊂∞⋃j=1B(x j,3C0M).(iii) For any n ∈ N and 1≤ j ≤ n,max0≤t≤1∥∥(ω≤n−ω j)(·, t)∥∥H2(B(x j,3C0M)) < 12 j+1 .Then, the limit solution of {ω≤n} becomes the desired one ω .Step 1 Construction of the sequence {xk}k∈N.For each j ∈ N, apply Lemma 2.5.3 for f = ω j0 and ε = 12 j+1 . Then, we canfind R j > 0 such that for any h ∈ H2∩L1 with∥∥ω j0∥∥1+‖h‖1+ sup(∥∥ω j0∥∥∞ ,‖h‖∞)≤M,dist(supp(ω j0),supp(h))≥ R j,(2.5.25)where M is given in (2.5.1), the solutions ω and ω˜ j to (LE) for the initial dataω j0+h and ω j0, respectively, satisfymax0≤t≤1∥∥(ω− ω˜ j)(·, t)∥∥H2(B(0,3C0M)) < 12 j+1 , (2.5.26)andsupp(ω(·, t))⊂ B(0,3C0M)∪B(supp(h),2C0M). (2.5.27)Here, (2.5.27) is an easy consequence of (2.5.5) and (2.5.6).We find {xn} inductively. Indeed, we can relax the conditions on {xn} as fol-lows; for any n≥ 2 in N with x1 = 0,(a) xn is located at a far distance from previously chosen points|xn− xl|>n∑i=1Ri+10C0M+2n, ∀1≤ j < n,37(b) A smooth solution ω≤n to (2.5.24) satisfiessupp(ω≤n(·, t))⊂n⋃j=1B(x j,3C0M), ∀0≤ t ≤ 1.(c) Denoting B(x j,3C0M) by B j,max0≤t≤1‖(ω≤n−ω≤n−1)(·, t)‖H2(⋃n−1j=1 B j) <12n.Then, the requirements (i) and (ii) easily follow from (c) and (b), respectively.We can also check that (a) implies (iii). For each n ∈ N and 1≤ j ≤ n, plugh(x) =n∑k=1k 6= jωk0(x− xk + x j) (2.5.28)into (2.5.25). We can easily see that (2.5.25) holds true∥∥ω j0∥∥1+‖h‖1+ sup(∥∥ω j0∥∥∞ ,‖h‖∞)≤ n∑k=1‖ωk0‖1+ sup1≤k≤n‖ωk0‖∞ ≤M,anddist(supp(ω j0),supp(h)) = dist(supp(ω j0(·− x j)),supp(h(·− x j)))≥ inf1≤k≤nk 6= jdist(B(x j,1),B(xk,1))≥ inf1≤k≤nk 6= j|x j− xk|−2≥ R j.Therefore, using the translation invariant property of (LE), we havemax0≤t≤1∥∥ω≤n(·+ x j, t)− ω˜ j(·, t)∥∥H2(B(0,3C0M)) < 12 j+1 ,which follows (iii).Now, we choose {x j} satisfying (a)-(c) by induction. At the end of each induc-tive step, we also find R˜n ≥ R˜n−1 satisfying the following condition38(d) For any g ∈ H2∩L1 with∥∥∥∥∥ n∑j=1ω j0(·− x j)∥∥∥∥∥1+‖g‖1+ sup(∥∥∥∥∥ n∑j=1ω j0(·− x j)∥∥∥∥∥∞,‖g‖∞)≤M,dist(supp(n∑j=1ω j0(·− x j)),supp(g))≥ R˜n,(2.5.29)the solution ω to (LE) for the initial data ∑nj=1ω j0(x− x j)+g satisfiessupp(ω(·, t))⊂(n⋃j=1B j)⋃B(supp(g),2C0M), ∀0≤ t ≤ 1andmax0≤t≤1‖(ω−ω≤n)(·, t)‖H2(⋃nk=1 Bk) < 12n+1 . (2.5.30)Set x1 = 0 and R˜1 = R1. We first choose x2 satisfying|x2− x1|>2∑i=1Ri+10C0M+22+ R˜1.Clearly, (a) for n= 2 is obtained. Also, j = 1 and h=w20(x−x2) satisfies (2.5.25),which implies (b)-(c) for n = 2. Here, we use ω≤1 = ω1 = ω˜1.The choice of R˜2≥ R˜1 = R1 satisfying (d) for n= 2 follows from Lemma 2.5.3;apply it to f = ω≤2|t=0 and ε = 123 .Assume that {x j}nj=1 and R˜n are given and satisfy (a)-(d). Then, we pick xn+1such that|xn+1− x j|>n+1∑i=1Ri+10C0M+2n+1+ R˜n, ∀ j = 1, · · · ,n.which follows (a). To achieve (b) and (c) for n+1, we observe that g=ω(n+1)0(x−39xn+1) satisfies (2.5.29),∥∥∥∥∥ n∑j=1ω j0(·− x j)∥∥∥∥∥1+‖g‖1+ sup(∥∥∥∥∥ n∑j=1ω j0(·− x j)∥∥∥∥∥∞,‖g‖∞)≤∞∑j=1∥∥ω j0∥∥1+ supj∥∥ω j0∥∥∞ ≤Manddist(supp(n∑j=1ω j0(x− x j)),supp(g))≥ inf1≤ j≤ndist(B(x j,1),B(xn+1,1))≥ inf1≤ j≤n|xn+1− x j|−2≥ R˜n.Then by (d) for n, the conditions (b) and (c) for n+1 hold; we havesupp(ω≤n+1(·, t))⊂(n⋃j=1B j)∪B(xn+1,2C0M+1)⊂n+1⋃j=1B jandmax0≤t≤1‖(ω≤n+1−ω≤n)(·, t)‖H2(⋃nk=1 Bk) < 12n+1 .Applying again Lemma 2.5.3 for f = ω≤n+1|t=0 = ∑n+1j=1 ω j0(x− x j) and ε =12n+2 , we can find R˜n+1 ≥ R˜n satisfying (d). Therefore, we have (a)-(d) at (n+1)thstep, so that they hold true for any n≥ 2.Step 2. Check the required conditions.By condition (i), {ω≤n} is Cauchy in C([0,1];H2(B(x j,3C0M))) for each j ∈N. On the other hand, by Lemma 2.5.2, for each j ∈ N and k ≥ 2, {ω≤n} isuniformly bounded in C([0,1];Hk(B(x j,3C0M))), so that {ω≤n} is Cauchy evenin C([0,1];Hk(B(x j,3C0M))). This implies that for each 0 ≤ t ≤ 1, we have apointwise limit solutionω(x, t) =limn→∞ω≤n(x, t) x ∈⋃∞j=1 B(x j,3C0M)0 otherwise.40Obviously, ω(·, t) ∈ C∞ and ω satisfies (2.5.2) and (2.5.3) by the conditions (ii)and (iii). Furthermore, ω ∈ C([0,1];L1(R2)∩ L∞(R2)). This is because for any0≤ t ≤ 1, we have‖ω(·, t)‖1 =∞∑j=1‖ω(·, t)‖L1(B j) =∞∑j=1limn→∞‖ω≤n(·, t)‖L1(B j) =∞∑j=1∥∥ω j0∥∥1 = ‖ω0‖1and‖ω(·, t)‖∞ = supj‖ω(·, t)‖L∞(B j) = supjlimn→∞‖ω≤n(·, t)‖L∞(B j)= supj∥∥ω j0∥∥∞ = ‖ω0‖∞ .Finally, we prove that the limit solution ω is the unique classical solution to(LE) for the initial dataω|t=0(x) =∞∑j=1ω j0(x− x j).We first show that the limit solution ω solves (LE) in the sense ofω(x, t) = ω0(x)−ˆ t0(∇⊥∆−1Tγω ·∇ω)(x,s)ds, ∀(x, t) ∈ R2× (0,1). (2.5.31)At t = 0, it is apparent that the limit solution is the same with ω0. Since ω≤nsolves (2.5.31) with ω0 = ∑nj=1ω j0(·− x j) for any n ∈ N, it is enough to prove theuniform convergence ∇⊥∆−1Tγω≤n → ∇⊥∆−1Tγω on each B(x j,3C0M)× [0,1],j ∈ N. For notational simplicity, we suppress the dependence on the variable t, ifit’s not needed. Fix j ∈ N. For n > j and x ∈ B(x j,3C0M) = B j, we have|(∆−1∇⊥Tγ(ω≤n−ω)(x)| ≤ˆ|H(x− y)||(ω≤n−ω)(y)|dy= n∑m=1m6= jˆBm+ˆB j+∞∑l=n+1ˆBl |H(x− y)||(ω≤n−ω)(y)|dy= In1 + In2 + In3 .41By the choice of the centers, we have for any x ∈ B j and y ∈ Bm, m 6= j,|x− y| ≥ |x j− xm|−6C0M ≥ 2max( j,m).This implies that In1 converges to 0, as n goes to infinity; for x ∈ B j,In1 .n∑m=1m 6= jˆBm1|x− y| |(ω≤n−ω)(y, t)|dy≤n∑m=12−m ‖(ω≤n−ω)(·, t)‖L1(Bm).n∑m=12−m ‖ω≤n−ω‖C([0,1];L∞(Bm))→ 0, as n→ ∞.In a similar way, In3 approaches to 0, as n goes to infinity;In3 .∞∑l=n+1ˆBl1|x− y| |ω(y)|dy≤∞∑l=n+12−l ‖ω0‖1→ 0, as n→ ∞.Finally, since |x− y| ≤ |x− x j|+ |y− x j| ≤ 6C0M, we obtainIn2 .M max0≤t≤1‖(ω≤n−ω)(·, t)‖L∞(B j)→ 0, as n→ ∞.Therefore, we get the uniform convergence of ∇⊥∆−1Tγω≤n and hence ω solves(LE) in the sense of (2.5.31). Using the equation, we can improve the regularity ofthe solution in time, so that ω is a classical solution to (LE).For the uniqueness of the classical solution, let ω be another classical solutionto (LE) for the same initial data. Note that the statement in Lemma 2.5.3 holds alsofor a classical solution ω for initial data f +g where g ∈C∞(R2)∩L1(R2). Then,in the same way of obtaining (2.5.30), we havemax0≤t≤1‖(ω−ω≤n)(·, t)‖H2(∪nj=1B j) <12nmax0≤t≤1‖(ω−ω≤n)(·, t)‖H2(∪nj=1B j) <12n.This follows from that g = ∑∞j=n+1ω j0(·− x j) satisfies (2.5.29) for the same M, f ,and ε in the construction of R˜n+1. Therefore, we have ω = ω . In other words, theuniqueness of the classical solution holds. This completes the proof. 422.6 Proof of Theorem 2.1.1Proof of Theorem 2.1.1. Recall the family of initial data g˜A in Remark 2.4.1. By itsconstruction, for fixed 0 < γ ≤ 12 and 0 < ε < 1, we can find a sequence {A j} suchthat for any j ∈ N, ζ j = g˜A j satisfies supp(ζ j)⊂ B(0,1) and∥∥ζ j∥∥1+∥∥ζ j∥∥∞+∥∥∇ζ j∥∥2 < ε2 j , (2.6.1)and the smooth solution ω˜ j to (LE) with initial data ζ j achieves∥∥∇ω˜ j(·, t j)∥∥2 > j (2.6.2)for some t j which converges to 0 as j→ ∞.Since the solution to (LE) is translation-invariant, in the case of supp(a) ⊂B(0,1) up to translation, we can apply Proposition 2.5.1 to ω10 = a and ω j0 = ζ jfor j ≥ 2. Then, we have a sequence {x j} j∈N of centers with x1 = 0 such that forthe initial dataω0(x) = a(x− x1)+∞∑j=2ζ j(x− x j) =: a(x)+ζ (x)we have a unique classical solution ω to (LE) and the solution satisfies ω(·, t) ∈C∞(R2) for any 0≤ t ≤ 1, ω ∈C([0,1];L1(R2)∩L∞(R2)), andmax0≤t≤1∥∥(ω−ω j)(·, t)∥∥H2(B(x j,3C0M)) < 1 (2.6.3)for sufficiently large j. Here, ω j is a smooth solution to (LE) for the initial dataζ j(x−x j), C0 is the constant defined in Proposition 2.5.1, and M > 1 is a bound ofthe initial data in the sense of1+‖a‖2H1 +‖a‖1+‖a‖∞+‖ζ‖2H1 +‖ζ‖1+‖ζ‖∞ ≤M.Note that ω j for any j ∈ N satisfiessupp(ω j)⊂ B(x j,3C0M), ω j(x, t) = ω˜ j(x− x j, t).43It is easy to see that ζ ∈C∞(R2) because of ζ j ∈C∞c (B(0,1)) and |x j−xk|  1 forj 6= k. By (2.6.1), we also get‖ζ‖H˙1(R2)+‖ζ‖1+‖ζ‖∞ ≤∞∑j=2∥∥∇ζ j∥∥2+∥∥ζ j∥∥1+∥∥ζ j∥∥∞ < ε.By direct computation, we can also see that ζ 6∈W 1,p(R2) for p > 2.On the other hand, (2.6.2), (2.6.3), and supp(ω j(·, t))⊂B(x j,3C0M), 0≤ t ≤ 1,implies that∥∥ω(·, t j)∥∥H˙1 ≥ ∥∥ω j(·, t j)∥∥H˙1(B(x j,3C0M))−∥∥(ω−ω j)(·, t j)∥∥H˙1(B(x j,3C0M))≥ ∥∥ω˜ j(·, t j)∥∥H˙1(R2)− max0≤t≤1∥∥(ω−ω j)(·, t)∥∥H˙1(B(x j,3C0M))> j−1.Therefore, the constructed perturbation ζ satisfies all requirements in Theorem2.1.1. If supp(a) 6⊂ B(0,1) up to translation, we slightly modify the proof of theProposition and obtain the same conclusion.2.7 The compact caseIn this section, we prove Theorem 2.1.2, the compact case. Unlike the non-compactcase, a large distance between local solutions cannot be used in order to minimizetheir interactions and make a global solution locally behave like local ones. Forthis reason, we adopt a different scheme; use the smallness in the L1-norm of thetail part of a global solution.The following proposition describes a simple scenario of patching.Proposition 2.7.1. Suppose that f ∈C∞c (R2) satisfiessupp( f )⊂ {(x1,x2) ∈ R2 : x1 ≤−2R0} for some R0 > 0,f (x1,x2) =− f (x1,−x2) ∀(x1,x2) ∈ R2.(2.7.1)Then, for any 0 < ε0 < R0100 , we can find δ = δ ( f ,ε0,R0) > 0, t0 = t0( f ,ε0,R0) ∈(0,ε0), and g = g( f ,ε0,R0) ∈C∞c (B(0,ε0)) such that the following holds.44(i) g satisfies‖g‖H˙1 +‖g‖∞+‖g‖1+‖g‖H˙−1 < ε0g(x1,x2) =−g(x1,−x2), ∀(x1,x2) ∈ R2.(ii) For any given h ∈C∞c (R2) withsupp(h)⊂ {(x1,x2) : x1 ≥ R0}, ‖h‖1+‖h‖∞ ≤ δ , (2.7.2)the smooth solution ω to (LE) for the initial data ω|t=0 = f + g+ h has adecompositionω = ω f +ωg+ωh, on R2× [0, t0]such thatsupp(ω f (·, t))⊂ B(supp( f ), 18R0),supp(ωg(·, t))⊂ B(0,ε0+ 18R0),supp(ωh(·, t))⊂ B(supp(h), 18R0), ∀0≤ t ≤ t0(2.7.3)and‖ωg(·, t0)‖H˙1 >1ε0. (2.7.4)To prove this proposition, we need some preliminary lemmas. The first lemmais about the finite time propagation.Lemma 2.7.2. Let Ω be a smooth solution to∂tΩ+∇⊥∆−1TγΩ ·∇Ω+(B+E−C) ·∇Ω= 0C(t) = (−∂2∆−1TγΩ(0,0, t), 0)ᵀΩ|t=0 =Ω0(2.7.5)where B, E, and Ω0 are smooth functions satisfying45•‖Ω0‖∞ ≤ B0, for some B0 > 0,supp(Ω0)⊂ B(0,R), for some R > 0, (2.7.6)• B and E are divergence-free∇ ·B = ∇ ·E = 0.• For some positive numbers B1 and B2,|B(y, t)| ≤ B1|y|, |E(y, t)| ≤ B2|y|2, ∀(y, t) ∈ R2× [0,1].Then, we can find R0 > 0 and 0 < t0 < 1 both depending only on B0, B1 and B2such that if 0 < R≤ R0, a characteristic line Φ which solves∂tΦ(y, t) = (∇⊥∆−1TγΩ+B+E−C)(Φ(y, t), t)Φ(y,0) = y (2.7.7)satisfies|Φ(y, t)| ≤ 2R, ∀|y| ≤ R, t ∈ [0, t0].In particular, the solution Ω satisfiessupp(Ω(·, t))⊂ B(0,2R), ∀0≤ t ≤ t0.Proof. From (2.7.7), we obtain∂t |Φ(y, t)| ≤ 2∥∥∥∇⊥∆−1TγΩ∥∥∥∞+B1|Φ(y, t)|+B2|Φ(y, t)|2. (2.7.8)By using Lp-norm preservation and (2.7.6), we have∥∥∥∇⊥∆−1TγΩ∥∥∥∞. ‖Ω‖121 ‖Ω‖12∞ . R‖Ω0‖∞ ≤ RB0.46Combining with (2.7.8), we can find t0 > 0 and R0 > 0 such that if 0 < R≤ R0,|Φ(y, t)| ≤ 2R, ∀|y| ≤ R, t ∈ [0, t0].Furthermore, using the characteristic, (2.7.5) can be written as Ω(Φ(y, t), t) =Ω0(y), so thatsupp(Ω(·, t))⊂Φ(supp(Ω0), t).Then, it easily follows that supp(Ω(·, t))⊂ B(0,2R) for any 0≤ t ≤ t0.Recall the definition of gA in (2.4.15). This family of initial data was used inorder to create the large Lagrangian deformation. Now, we redefine gA when γ = 12bygA(x) =1ln ln lnA1√ln lnA ∑A≤ j<A lnA1√jρ(2 jx), (2.7.9)where ρ is given as in (2.4.14). In the case of 0 < γ < 12 , we use the same gA in(2.4.15).Then, gA satisfies• supp(gA)⊂ B(0,2 ·2−A).•‖gA‖1 . 2−2A, ‖gA‖∞ .1Aγ, ‖gA‖H˙−1 . 2−A, ‖∇gA‖2 .1ln ln lnA.• ˆz1>0,z2>01|z|2 ln−γ(e+1|z|)e−|z|4gA(z)dz&√ln lnAln ln lnA.In the next Lemma, we prove that for this newly redefined family, we can createthe large Lagrangian deformation even in the presence of a compactly supportedperturbation f whose support is away from the origin. In other words, the supportof f is away from that of gA.47Lemma 2.7.3. Suppose that f satisfies (2.7.1). Let ω be a smooth solution to∂tω+∇⊥∆−1Tγω ·∇ω = 0ω|t=0 = f +gA.Then, a characteristic line φ which solves∂tφ(x, t) = ∇⊥∆−1Tγω(φ(x, t), t)φ(x,0) = x (2.7.10)satisfiesmax0≤t≤ 1ln ln lnA‖Dφ(·, t)‖L∞(B(0,10·2−A)) > ln14 ln ln lnA (2.7.11)for sufficiently large A.Proof. Suppose that (2.7.11) doesn’t hold true. i.e.,max0≤t≤ 1ln ln lnA‖Dφ(·, t)‖L∞(|x|≤10·2−A) ≤ ln14 ln ln lnA.First, we decompose the solution ω into ω f and ωg, where ωg solves∂tωg+∇⊥∆−1Tγωg ·∇ωg+∇⊥∆−1Tγω f ·∇ωg = 0ωg|t=0 = gA.Since both f and gA are odd in x2, so are ω and ωg. Also, we haveφ1(x1,−x2, t) = φ1(x1,x2, t), φ2(x1,−x2, t) =−φ2(x1,x2, t)and therefore φ2(x1,0, t) = 0 for any x1 ∈R and t ≥ 0. Let a(t) = φ1(0,0, t). Then,it satisfies a′(t) =−∂2∆−1Tγω(a(t),0, t)a(0) = 0.Similar to (2.5.5)-(2.5.7), we can easily see that the supports of ω f and ωg are48apart from each other for a short time. Indeed, on [0, tA], tA = 1ln ln lnA ,supp(ω f (·, t))⊂ B(supp( f ),18R0)⊂{x1 ≤−158 R0},supp(ωg(·, t))⊂ B(supp(gA),18R0)⊂ B(0,14R0),provided that A is sufficiently large. It follows that ∇⊥∆−1Tγω f is smooth and hasSobolev norm bounds on B(0, 14 R0)× [0, tA], where the bounds depend only on fand R0. Therefore, we can expand it at the point (a(t),0), which is in B(0, 18 R0) for0≤ t ≤ tA, to get∇⊥∆−1Tγω f (a(t)+ y1,y2, t) =(a′(t)+∂2∆−1Tγωg(a(t),0, t)0)+b(t)(−y1y2)+E(y, t)for any (a(t)+ y1,y2, t) ∈ B(0, 14 R0)× [0, tA]. Here, b(t) = ∂12∆−1Tγω f (a(t),0, t)has a bound |b(t)| ≤ B1 for some B1 = B1(R0, f ) and a divergence-free vector Ecan be chosen satisfying|E(y, t)| ≤ B2|y|2, |DE(y, t)| ≤ B2|y|, |D2E(y, t)| ≤ B2, ∀y ∈ R2,for some B2 = B2(R0, f ). In the expansion, we use the oddness of ω f in x2 and∂1∆−1Tγω f (a(t),0, t) = ∂11∆−1Tγω f (a(t),0, t) = ∂22∆−1Tγω f (a(t),0, t) = 0.We do the change of variables (x1,x2, t) = (a(t)+y1,y2, t) and denote the solu-tion in a new coordinate system (y, t) byΩ(y, t)=ωg(a(t)+y1,y2, t)=ωg(x1,x2, t).Then, the equation for Ω on R2× [0, tA] can be written as∂tΩ+(∇⊥∆−1TγΩ+B+E−C) ·∇Ω= 0Ω|t=0 = gA,where B and C areB = b(−y1y2), C =(−∂2∆−1TγΩ(0,0, t)0).49Also, we let Φ be a characteristic in a new coordinate, which solves∂tΦ(y, t) =(∇⊥∆−1TγΩ+B+E−C)(Φ(y, t), t)Φ(y,0) = y.From now on, without mentioning, we only consider t ∈ [0, tA].We can easily check that φ−1(x, t)=Φ−1(y, t) andΦ−1(0, t)= φ−1(a(t),0, t)=0. Furthermore, Φ−1 satisfiesΦ−11 (y1,y2, t) =Φ−11 (y1,−y2, t), Φ−12 (y1,y2, t) =−Φ−12 (y1,−y2, t).By Lemma 2.7.2, on the other hand, we have|Φ(y, t)| ≤ 4 ·2−A, ∀|y| ≤ 2 ·2−A,for sufficiently large A. Also, if |y| ≤ 4 · 2−A and tA is sufficiently small, by finitespeed propagation, |φ−1(a(t)+ y1,y2, t)| ≤ 10 ·2−A. It follows thatmax0≤t≤tA‖DΦ(·, t)‖L∞(|y|≤2·2−A) ≤ max0≤t≤tA∥∥D(Φ−1)(·, t)∥∥L∞(|y|≤4·2−A)≤ max0≤t≤tA‖Dφ(·, t)‖L∞(|x|≤10·2−A)≤ ln 14 ln ln lnA = MA.(2.7.12)Indeed, (DΦ(x, t))−1 = D(Φ−1)(Φ(x, t)) and (Dφ(x, t))−1 = D(φ−1)(φ(x, t)) areused in the first and second inequalities, respectively.Then, by Lemma 2.8.3 with φ =Φ−1, we havesup0≤t≤tA∥∥R11TγΩ(·, t)∥∥∞+∥∥R22TγΩ(·, t)∥∥∞ ≤ CγMA√ln lnA (2.7.13)forRi jω = ∆−1∂i jω and for some constant Cγ > 0 depending only on γ .Now, we find a lower bound of DΦ which makes a contradiction to (2.7.12).50From the equation for Φ, we get∂tDΦ(y, t) =D∇⊥∆−1TγΩ+b−1 00 1+DE(Φ(y, t), t)DΦ(y, t)DΦ(y,0) = I,(2.7.14)and the derivative of the velocity can be rewritten asD∇⊥∆−1TγΩ+b(−1 00 1)+DE=(−R12TγΩ−b 00 R12TγΩ+b)+(0 −R22TγΩR11TγΩ 0)+DE=(−R12TγΩ−b 00 R12TγΩ+b)+P.By Gro¨nwall’s inequality, we haveDΦ(y, t) = exp(´ t0 λ (y,s)−b(s)ds 00´ t0−λ (y,s)+b(s)ds)+ˆ t0exp(´ tτ λ (y,s)−b(s)ds 00´ tτ −λ (y,s)+b(s)ds)P(Φ(y,τ),τ)DΦ(y,τ)dτ(2.7.15)where λ (y, t) =−R12TγΩ(Φ(y, t), t).Since |Φ(y, t)| ≤ 4 · 2−A for |y| ≤ 2 · 2−A, A 1, we have |DE(Φ(y, t), t)| .B22−A. Combining (2.7.15) with (2.7.12) and (2.7.13), we obtain for |y| ≤ 2 ·2−A,exp∣∣∣∣ˆ t0λ (y,s)−b(s)ds∣∣∣∣≤MA+ CγM2A√ln lnA max0≤τ≤t exp(2∣∣∣∣ˆ τ0λ (y,s)−b(s)ds∣∣∣∣) .Then, by the continuation argument, we getexp∣∣∣∣ˆ t0λ (y,s)−b(s)ds∣∣∣∣≤ 2MA51for sufficiently large A, so that we can consider the second term in (2.7.15) as anerror term.The remaining analysis is similar to the proof of Proposition 2.3.4. UsingΦ(0, t) = 0, it follows that for |y| ≤ 2 ·2−AΦ(y, t) =Φ(y, t)−Φ(0, t) =ˆ 10∂∂θ[Φ(θy, t)]dθ =(ˆ 10DΦ(θy, t)dθ)y=(y1ˆ 10exp(ˆ t0λ (θy,s)−b(s)ds)dθ ,y2ˆ 10exp(−ˆ t0λ (θy,s)−b(s)ds)dθ)+ e(2.7.16)where|e(y, t)|.γ M4A√ln lnA|y|.Since 1MA M4A√ln lnAif A 1 and y1 ∼ y2 for y= (y1,y2) ∈ supp(gA), it follows thatfor sufficiently large A, Φ has a sign preserving property;Φ1(y, t)> 0, Φ2(y, t)> 0, y ∈ supp(gA)∩{y1 > 0,y2 > 0}Φ1(y, t)< 0, Φ2(y, t)> 0, y ∈ supp(gA)∩{y1 < 0,y2 > 0}Based on this, we getλ (0, t) =−R12TγΩ(Φ(0, t), t) =−R12TγΩ(0, t)=ˆR2K(−z, t)Ω(z, t)dz = 2ˆz2>0K(z, t)Ω(z, t)dz = 2ˆz2>0K(Φ(z, t), t)gA(z)dz≥ˆz1>0,z2>0K(Φ(z, t), t)gA(z)dzwhere K is the kernel of the operator−∂12∆−1Tγ . The fourth equality follows fromthe parity of K and Ω in z2. The last inequality follows from the positiveness of theintegrand on {z1 < 0,z2 > 0}.Note that if z ∈ supp(gA)∩{z1 > 0,z2 > 0}, we have12<z1z2< 252and hence by (2.7.16)110M2A<Φ1(z, t)Φ2(z, t)< 10M2A.Also, we have |z|MA ≤ |Φ(z, t)| ≤MA|z| for z ∈ supp(gA). Then, by Lemma 2.3.1 andLemma 2.3.3, we getˆz1>0,z2>0K(Φ(z, t), t)gA(z)dz&γˆz1>0,z2>0Φ1(z, t)Φ2(z, t)|Φ(z, t)|4 ln−γ(e+1|Φ(z, t)|)e−|Φ(z,t)|2gA(z)dz≥ 1M2Aˆz1>0,z2>01Φ1(z,t)Φ2(z,t) +Φ2(z,t)Φ1(z,t)· 1|z|2 ln−γ(e+MA|z|)e−M2A|z|2gA(z)dz& e34 M4AM4A(1+ ln(1+MA))γe−M4Aˆz1>0,z2>01|z|2 ln−γ(e+1|z|)e−|z|4gA(z)dz& e−M4Aˆz1>0,z2>01|z|2 ln−γ(e+1|z|)e−|z|4gA(z)dz&γ√ln lnAln ln lnAe−M4A =√ln lnA(ln ln lnA)2,provided that A 1. Therefore, we getmax0≤t≤tA‖DΦ(·, t)‖L∞(|y|≤2·2−A) ≥ max0≤t≤tA |DΦ(0, t)|≥ exp((Cγ√ln lnA(ln ln lnA)2−B1)1ln ln lnA)−1,which makes a contradiction to (2.7.12)max0≤t≤tA‖DΦ(·, t)‖L∞(|y|≤2·2−A) ≤ ln14 ln ln lnAfor sufficiently large A.Now, we give a proof of the main proposition.Proof of Proposition 2.7.1.Step 1. Critical norm inflation of a local solution.By Lemma 2.7.3, we can create the large Lagrangian deformation (2.7.11) at53the presence of f satisfying (2.7.1). Then, similar to Proposition 2.4.2, we can finda perturbed initial data g˜A ∈C∞c (|x|. 2−A) from gA such that it satisfiesg˜A(x1,x2) =−g˜A(x1,−x2),‖g˜A‖H˙1 +‖g˜A‖∞+‖g˜A‖1+‖g˜A‖H˙−1 . ln−18 ln ln lnA,and the smooth solution ω˜(A) to∂ω˜(A)+∇⊥∆−1Tγ ω˜(A) ·∇ω˜(A) = 0ω˜(A)|t=0 = f + g˜Ahas a decomposition ω˜(A) = ω˜(A)f + ω˜(A)g˜ where ω˜(A)g satisfiesmax0≤t≤ 1ln ln lnA∥∥∥∇ω˜(A)g (·, t)∥∥∥2≥ ln 112 ln ln lnA,for sufficiently large A. Indeed, ω˜(A)g solves∂ω˜(A)g +∇⊥∆−1Tγ ω˜(A) ·∇ω˜(A)g = 0ω˜(A)g |t=0 = g˜A.Then, we construct g by choosing A0 =A0(ε0) 1 such that g= g˜A0 ∈C∞c (B(0,ε0))and ω˜g = ω˜(A0)g satisfies‖g‖H˙1 +‖g‖∞+‖g‖1+‖g‖H˙−1 < ε0,andmax0≤t≤ 1ln ln lnA0‖∇ω˜g(·, t)‖2 >2ε0.In particular, we can find 0 < t0 ≤ 1ln ln lnA0 < ε0 such that‖∇ω˜g(·, t0)‖2 >2ε0. (2.7.17)Step 2. Patch a function h.54Suppose that h satisfies (2.7.2) and δ < 1. Let ω be a solution to∂tω+∇⊥∆−1Tγω ·∇ω = 0ω|t=0 = f +g+h.We decompose ω = ω f +ωg+ωh where ω f and ωg are defined as solutions to∂tω f +∇⊥∆−1Tγω ·∇ω f = 0ω|t=0 = fand ∂tωg+∇⊥∆−1Tγω ·∇ωg = 0ω|t=0 = g,respectively. Since∥∥∥∇⊥∆−1Tγω∥∥∥∞. ‖ω‖1+‖ω‖∞ = ‖ω|t=0‖1+‖ω|t=0‖∞ . 1+‖ f‖1+‖ f‖∞ ,similar to (2.5.5) and (2.5.6), we can easily check ω f , ωg and ωh satisfies (2.7.3),provided that t0 is sufficiently small. If necessary, we can adjust the choice of A0to make t0 small enough.Now, recall (2.5.23). By the assumption (2.7.2) on h and (2.7.3), we have∥∥∥∇⊥∆−1Tγωh(·, t)∥∥∥L∞(B(supp( f ), 18 R0)∪B(0,ε0+ 18 R0)).R0 ‖h‖1 ≤ δfor any 0≤ t ≤ t0. Then, by the same arguments in Lemma 2.5.3, we get‖(ωg− ω˜g)(·, t0)‖H2 ≤ max0≤t≤t0∥∥((ω f +ωg)− ω˜)(·, t)∥∥H2≤C(‖ f‖H3 ,R0,supp( f ))δ ≤1ε0,provided that δ ∈ (0,1) is sufficiently small. Combining with (2.7.17), we obtainthe desired inflation (2.7.4).Before we prove Theorem 2.1.2, we need the following lemma for the unique-55ness.Lemma 2.7.4. Suppose that f ∈C∞c (R2) with the compact support in B(0,R) forsome R > 0 and g ∈ L∞(R2)∩ H˙−1(R2) with ‖g‖∞ ≤M for some M > 0. Let ω˜ bea smooth solution to ∂tω˜+∇⊥∆−1Tγ ω˜ ·∇ω˜ = 0ω˜|t=0 = fand ω be a weak solution in C([0,1];L1(R2)∩L∞(R2)) to∂tω+∇⊥∆−1Tγω ·∇ω = 0ω|t=0 = f +gsatisfying L∞-norm preservation‖ω(·, t)‖∞ = ‖ f +g‖∞ , ∀0≤ t ≤ 1.Then, for any ε > 0, we can find a constant δ = δ (ε, f ,M) > 0 such that if‖g‖H˙−1 < δ ,max0≤t≤1‖(ω− ω˜)(·, t)‖H˙−1(R2) < ε.Furthermore, under the additional assumption g∈C∞c (B(0,R)), we have δ˜ = δ˜ (ε,R, f )>0 such that if ‖g‖∞ < δ˜ ,max0≤t≤1‖(ω− ω˜)(·, t)‖L∞(R2) < ε.Proof. The equation for η = ω− ω˜ is∂tη+∇⊥∆−1Tγη ·∇ω+∇⊥∆−1Tγ ω˜ ·∇η = 0η |t=0 = g.56Taking´R2 ·Λ−2ηdx, Λ= (−∆)12 , on both side of the equation, we have12ddt‖η‖2H˙−1(R2) ≤∣∣∣∣ˆ ω(∇⊥∆−1Tγη ·∇)Λ−2ηdx∣∣∣∣+ ∣∣∣∣ˆ η(∇⊥∆−1Tγ ω˜ ·∇)Λ−2ηdx∣∣∣∣≤ ‖ω‖∞ ‖η‖2H˙−1(R2)+∥∥Λ−1η∥∥2∥∥∥[Λ,∇⊥∆−1Tγ ω˜ ·∇]Λ−2η∥∥∥2. (‖ f‖∞+‖g‖∞+∥∥∥D∇⊥∆−1Tγ ω˜∥∥∥∞)‖η‖2H˙−1 .Here, the second inequality follows fromˆΛ−1η(∇⊥∆−1Tγ ω˜ ·∇)Λ−1ηdx = 12ˆ(∇⊥∆−1Tγ ω˜ ·∇)|Λ−1η |2dx = 0and the third one from the commutator estimate‖Λ(lm)− l(Λm)‖2 . ‖Dl‖∞ ‖m‖2 .By the Gro¨nwall inequality, we havemax0≤t≤1‖η(·, t)‖H˙−1(R2) ≤ ‖g‖H˙−1(R2) exp(C(‖ f‖∞+‖ f‖W 1,4(R2)+M)).Therefore, for given ε > 0, we can find the desired δ = δ (ε, f ,M).Now, we further assume that g is in C∞c (B(0,R)). Then, the weak solution ωbecomes a smooth solution. The equation for η can be rewritten as∂tη+∇⊥∆−1Tγω ·∇η+∇⊥∆−1Tγη ·∇ω˜ = 0,so that we have‖η(·, t)‖∞ ≤ ‖g‖∞+ˆ t0∥∥∥(∇⊥∆−1Tγη)(·,s)∥∥∥∞‖∇ω˜(·,s)‖∞ ds. (2.7.18)By the usual energy estimate, we havemax0≤t≤1‖∇ω˜(·, t)‖∞ . f 1.Using f ,g ∈ C∞c (B(0,R)) and Lebesgue measure preservation of the supports ω57and ω˜ ,∥∥∥(∇⊥∆−1Tγη)(·,s)∥∥∥∞. ‖η(·, t)‖121 ‖η(·, t)‖12∞≤ (|supp(ω(·,0))|+ |supp(ω˜(·,0))|) 12 ‖η(·, t)‖∞. R‖η(·, t)‖∞ .Then, combining with (2.7.18) and using Gro¨nwall inequality, we havemax0≤t≤1‖η(·, t)‖∞ ≤C(R, f )‖g‖∞ .This completes the proof.Finally, we find the compactly supported perturbation in our main theorem.Proof of Theorem 2.1.2.Fix 0 < ε < 1200 . Without loss of generality, we may assume the support ofthe given initial data lies on {x = (x1,x2) : x1 ≤ −1}∩B(0,R) for some R ≥ 10.(Otherwise, using the translation invariant property of the solution, we apply theproof for a suitably translated initial data in x1 direction. Note that the translatedone is still odd in x2.) Let {xn = (x1n,0)} be a sequence of centres withx11 = 0, x1n =n−1∑j=112 jfor n≥ 2.Now, we construct sequences {ζn}n∈N⊂C∞c (B(0,2−(n+1))), {(δn, δ˜n, tn)}n∈N⊂R3+ such that for any n ∈ N,• ζn is odd in x2 and satisfies‖ζn‖ ≡ ‖ζn‖H˙1 +‖ζn‖∞+‖ζn‖1+‖ζn‖H˙−1 < min(ε2n,δn−12n−1,δ˜n−12n−1),where δ0 = δ˜0 = 1.58• for any h ∈C∞c (R2) withsupp(h)⊂ {x = (x1,x2) : x1 ≥ 12n+1 }‖h‖1+‖h‖∞ ≤ δn,(2.7.19)a smooth solution ω to (LE) for the initial dataω|t=0(x) = a(x+ xn)+n−1∑j=1ζ j(x− x j + xn)+ζn(x)+h(x)has a decompositionω = ω≤n−1+ωn+ωhsuch that the supports of ω≤n−1, ωn, and ωh are disjoint for t ∈ [0, tn], and‖ωn(·, tn)‖H˙1 > 2n.• {δn} and {δ˜n} are decreasing sequences. Also, tn converges to 0.• for any g satisfying ‖g‖∞ ≤ 1 and ‖g‖H˙−1 ≤ δ˜n,max0≤t≤1‖(ω˜− ω˜≤n)(·, t)‖H˙−1 <12n. (2.7.20)where ω˜ ∈ C([0,1];L1(R2)∩ L∞(R2)) is a weak solution having L∞-normpreservation and ω˜≤n is a smooth solution to (LE) for initial dataω˜|t=0(x) = a(x)+n∑j=1ζ j(x− x j)+g, ω˜≤n|t=0 = a(x)+n∑j=1ζ j(x− x j).(2.7.21)Furthermore, if g ∈C∞c (B(0,R)) with ‖g‖∞ ≤ δ˜n, we havemax0≤t≤1‖(ω˜− ω˜≤n)(·, t)‖∞ <12n.The construction is based on induction. First, we choose ζ1, and (δ1, δ˜1, t1). By59Proposition 2.7.1 withf = a(x) = a(x+ x1), R0 =14, ε0 =ε2,there exist an smooth function ζ1 odd in x2 and compactly supported in B(0, 14),and positive constants 0 < δ1 < δ0 and 0 < t1 < 12 which satisfy the following:•‖ζ1‖< ε2• If h ∈C∞c (R2) satisfiessupp(h)⊂ {x = (x1,x2) ∈ R2 : x1 ≥ 14},‖h‖1+‖h‖∞ ≤ δ1,a smooth solution ω to (LE) for the initial dataω|t=0 = a(x)+ζ1(x)+h(x)has a decompositionω = ωa+ω1+ωh, on R2× [0, t1]such that the supports of ωa, ω1, and ωh are disjoint for t ∈ [0, t1] and‖ω1(·, t1)‖H˙1 > 2.Then, we apply Lemma 2.7.4 for f = a+ζ1, R = R, M = 1, and ε = 12 , so that weobtain 0 < δ˜1 ≤ δ˜0 such that if ‖g‖∞ ≤ 1, and ‖g‖H˙−1 ≤ δ˜1, then we havemax0≤t≤1‖(ω˜− ω˜≤1)(·, t)‖H˙−1(R2) <12,where ω˜ and ω˜≤1 are solutions to (LE) for the initial dataω˜|t=0 = a+ζ1+g, ω˜≤1|t=0 = a+ζ1.60Furthermore, if g ∈C∞c (B(0,R)) satisfies ‖g‖∞ ≤ δ˜1, then we havemax0≤t≤1‖(ω˜− ω˜≤1)(·, t)‖∞ <12,Therefore, we obtain the desired ζ1 and (δ1, δ˜1, t1).Assume that we have {ζ j}nj=1 and {(δ j, δ˜ j, t j)}nj=1 satisfying all conditionsabove. Then, applying Proposition 2.7.1 forf = a(x+xn+1)+n∑j=1ζ j(x−x j+xn+1), R0 = 12n+2 , ε0 =min(ε2n+1,δn2n,δ˜n2n),we can find ζn+1 ∈C∞c (B(0,2−(n+2))) odd in x2, and 0 < δn+1 ≤ δn and 0 < tn+1 <12n+1 such that•‖ζn+1‖< min(ε2n+1,δn2n,δ˜n2n).• for any h ∈C∞c (R2) withsupp(h)⊂ {x = (x1,x2) : x1 ≥ 12n+2 }‖h‖1+‖h‖∞ ≤ δn+1,the smooth solution ω to (LE) for the initial dataω|t=0(x) = a(x+ xn+1)+n∑j=1ζ j(x− x j + xn+1)+ζn+1(x)+h(x)has a decompositionω = ω≤n+ωn+1+ωh, on R2× [0, tn+1]such that the supports of ω≤n, ωn+1, and ωh are disjoint for t ∈ [0, tn+1], and‖ωn+1(·, tn+1)‖H˙1 > 2n+1.61Once we obtain ζn+1, applying Lemma 2.7.4 for f (x) = a(x)+∑n+1j=1 ζ j(x−x j),R= R, M = 1, and ε = 2−(n+1), we can find 0 < δ˜n+1 ≤ δ˜n such that for any g with‖g‖∞ ≤ 1 and ‖g‖H˙−1 ≤ δ˜n+1, we havemax0≤t≤1‖(ω˜− ω˜≤n+1)(·, t)‖∞ <12n+1,where ω˜ and ω˜≤n+1 solves (LE) for the initial dataω˜|t=0 = a+n+1∑j=1ζ j(·− x j)+g, ω˜≤n+1|t=0 = a+n+1∑j=1ζ j(·− x j).If g further satisfies g ∈C∞c (B(0,R)) and ‖g‖∞ ≤ δ˜n+1, we getmax0≤t≤1‖(ω˜− ω˜≤n+1)(·, t)‖∞ <12n+1.Therefore, by the induction argument, we obtain the desired sequences {ζn}, {(δn, δ˜n, tn)}.Now, we set the perturbation asζ (x) =∞∑j=1ζ j(x− x j).Obviously, the perturbation satisfies‖ζ‖H˙1 +‖ζ‖∞+‖ζ‖1+‖ζ‖H˙−1 = ‖ζ‖ ≤∞∑j=1∥∥ζ j∥∥< ε.Since ζn+1(·− xn+1) ∈C∞c (B(0,R)) and ‖ζn+1‖∞ ≤ ‖ζn+1‖ ≤ δ˜n, we plug g =ζn+1(·− xn+1) into (2.7.21) to getmax0≤t≤1‖(ω˜≤n+1− ω˜≤n)(·, t)‖∞ <12n. (2.7.22)Indeed, for any n ∈ N, ζn(· − xn) ∈ C∞c (B(0,R)), and by finite speed propagationwe have ω˜≤n ∈ C([0,1]× B(0,R∗)) for some finite number R∗. Then, (2.7.22)implies that {ω˜≤n} is Cauchy in C([0,1]×B(0,R∗)), and hence we have its limitω ∈ C([0,1];Cc(R2)). On the other hand, since the L∞-norm of ω˜≤n is preserved62for any n ∈ N, so is that of ω .Now, we check thatω is the unique weak solution in C([0,1];L1(R2)∩L∞(R2))to the equation (LE) for the initial dataω|t=0 = a+ζ , (2.7.23)having L∞-norm preservation. Since ω˜≤n is smooth solution to (LE), it satisfies forany ϕ ∈C1([0,1];C1c (R2)) and n ∈ N,ˆR2ω˜≤n(x,1)ϕ(x,1)dx=ˆR2ω˜≤n(x,0)ϕ(x,0)dx+ˆ 10ˆR2(∂sϕ+∇⊥∆−1Tγ ω˜≤n ·∇ϕ)ω˜≤ndxds.Sending n to infinity, ω solves (LE) in a weak sense. Then the uniqueness followsfrom (2.7.20). Indeed, for any weak solution ω ∈ C1([0,1];L1(R2)∩L∞(R2)) to(LE) for the same initial data with ω having L∞-norm preservation, we havemax0≤t≤1‖ω− ω˜≤n(·, t)‖H˙−1(R2) <12n,for sufficiently large n. Here, we use sup j∥∥ζ j∥∥∞ ≤ 1 and∞∑j=n+1∥∥ζ j(·− x j)∥∥H˙−1(R2) < ∞∑j=n+1δ˜ j−12 j−1≤ δ˜n∞∑j=n+112 j−1≤ δ˜n.Therefore, if the weak solution is not unique, i.e., ω 6= ω , then it makes a contra-diction tomax0≤t≤1‖(ω−ω)(·, t)‖H˙−1(R2) <12n−1, ∀n ∈ N.Therefore, we obtain uniqueness.Finally, since h = ∑∞j=n+1 ζ j(x− x j + xn) satisfies the conditions (2.7.19), wehave‖ω(·, tn)‖H˙1 ≥ ‖ωn(·, tn)‖H˙1 > 2n. (2.7.24)Indeed, in Proposition 2.7.1, the assumption h ∈ C∞c (R2) can be dropped if wehave a unique weak solution ω ∈ C([0,1];L1(R2)∩ L∞(R2)) to (LE) with initial63data ω|t=0 = f +g+h. This leads to (2.7.24).Using the continuity of ‖ωn(·, t)‖H1 , we have a short time interval [t ln, trn], t ln ≤tn ≤ trn such that trn converges to 0 and‖ω(·, t)‖H˙1 > n, ∀t ln ≤ t ≤ trn.This implies the desired critical Sobolev norm inflation.2.8 Analysis of the velocityIn this section, we provide proofs of some inequalities for self-containedness.2.8.1 Kernel for the velocityIn this section, we estimate the kernel H in the velocity u = ∇⊥∆−1Tγω = H ∗ω .Lemma 2.8.1. Let γ > 0 and H is the kernel of the multiplier ∇⊥∆−1Tγ , where Tγis eitherTγ = ln−γ(e−∆), or Tγ = ln−γ(e+ |∇|).Then, for each α with |α| ≥ 0, we have|∂αH(x)|.α 1|x||α|+1 , ∀x 6= 0. (2.8.1)Proof. By a similar argument in Lemma 2.3.1 and Lemma 2.3.3, we have an ex-plicit expression of the kernel H∆ of the multiplier ∇⊥∆−1 ln−γ(e−∆),H∆(x) =CΓ(γ)x⊥|x|2ˆ ∞01Γ(t)ˆ ∞0e−eβ (1− e− |x|24β )β tdββtγdtt=:x⊥|x|2 Hr(x)where x⊥ = (−x2,x1) for some absolute constant C > 0.64Also, the kernel H˜|∇| of the multiplier ∇⊥∆−1 ln−γ(e+ |∇|) isH˜|∇| =C˜Γ(γ)x⊥|x|2ˆ ∞01Γ(t)ˆ ∞0e−τˆ ∞0e−eβ(1− e−τ|x|2β2)β tdββτ−12 dτtγdtt,=:x⊥|x|2 H˜r(x)for some constant C˜ > 0.Using |tne−t | ≤C(n) for any t ≥ 0, we have for each |α| ≥ 0,|∂α(1− e− |x|24β )|.α 1|x||α| ∀x 6= 0, β > 0,where the constant in the inequality is independent of β . Since1Γ(γ)ˆ ∞01Γ(t)ˆ ∞0e−eββ tdββtγdtt=1Γ(γ)ˆ ∞0e−ttγdtt= 1,we can easily get|∂αHr(x)|.α 1|x||α| .On the other hand, we have1Γ(γ)ˆ ∞01Γ(t)ˆ ∞0e−τˆ ∞0e−eββ tdββτ−12 dτtγdtt=ˆ ∞0e−ττ−12 dτ . 1.Therefore, we also obtain|∂αH˜r(x)|.α 1|x||α| .Finally, since for each |α| ≥ 0, we have∣∣∣∣∂α( xi|x|2)∣∣∣∣.α 1|x||α|+1 , ∀x 6= 0,the desired estimate (2.8.1) follows easily.652.8.2 Operator norm of Tγ on LpLemma 2.8.2. Let γ > 0 and f ∈C∞c (R2). For any 1≤ p≤ ∞, we have∥∥Tγ f∥∥p ≤ ‖ f‖p .Proof. Let Kγ be the kernel for Tγ . In other words, Tγ f = Kγ ∗ f . Then, by Young’sinequality, it is enough to show that∥∥Kγ∥∥1 = 1. First, consider Tγ = ln−γ(e−∆).Since we haveln−γ(e+ |ξ |2) = 1Γ(γ)ˆ ∞01Γ(t)ˆ ∞0e−eβ e−|ξ |2ββ tdββtγdtt,we take the inverse Fourier transform to get the corresponding kernelKγ(x) =1Γ(γ)ˆ ∞01Γ(t)ˆ ∞0e−eβ eβ∆δ0(x)β tdββtγdtt.Therefore using∥∥eβ∆δ0∥∥1 = 1, we can easily get ∥∥Kγ∥∥1 = 1. Here, et∆δ0 is theusual heat kernel.Similarly, when Tγ = ln−γ(e+ |∇|), the integral expression of the kernel isKγ(x) =1Γ(γ)ˆ ∞01Γ(t)ˆ ∞0e−eβ1√piˆ ∞0e−τeβ4τ ∆δ0(x)τ−12 dτβ tdββtγdtt,and hence again∥∥∥e β4τ ∆δ0∥∥∥1= 1 implies∥∥Kγ∥∥1 = Estimate for ∆−1∂iiTγWIn this section, we estimate RiiTγ(gA ◦ φA) = ∆−1∂iiTγ(gA ◦ φA) in the L∞-norm.Here, gA is defined as in (2.4.15) for 0 < γ < 12 and (2.7.9) for γ =12 . The functionφA is a bi-Lipschitz function having certain properties. More precisely, we obtainthe following lemma by a slight modification of the proof of Lemma 3.2 in [4].Lemma 2.8.3. Let {gA} be a family of functions defined as in (2.4.15) for 0< γ < 12and (2.7.9) for γ = 12 . Suppose that φA = (φ1A,φ 2A) : R2 → R2 is a bi-Lipschitzfunction such that66• φA(0) = 0.• φ 1A(y1,−y2) = φ 1A(y1,y2) and φ 2A(y1,−y2) =−φ 2A(y1,y2).• For some integer mA ≥ 1,‖DφA‖L∞(|y|≤4·2−A) ≤ 2mA ,∥∥D(φ−1A )∥∥L∞(|y|≤2·2−A) ≤ 2mA . (2.8.2)• |det(DφA)|= |det(D(φ−1A ))|= 1.• If |φA(y)| ≤ 2 ·2−A, then |y| ≤ 4 ·2−A.Then, we have∥∥R11Tγ(gA ◦φA)∥∥∞+∥∥R22Tγ(gA ◦φA)∥∥∞ .γ 2mA√ln lnA . (2.8.3)Proof. Recall the definition of gA,gA(y) =CA∑aA≤ j<bA1jγ ρ(2jy), 0 < γ < 121ln ln lnA1√ln lnA ∑A≤ j<A lnA1√jρ(2jy), γ = 12where CA = 1√lnA1ln lnA , aA = A11−2γ , and bA = (A+ lnA)11−2γ . Here, ρ is an oddfunction in both variables and satisfies 12 ≤ |x| ≤ 2 for x ∈ supp(ρ). (See (2.4.14))We first considerRiiTγ(ρ j ◦φA) for j≥A, where ρ j = ρ(2 j·). For convenience,we drop the index A in gA, φA and mA below. Denote the kernel for the operatorRiiTγ by Kii for i = 1,2 and fix y ∈ R2 \ {0} with |y| ∼ 2−l for some l. Note thatthe kernel Kii, i = 1,2, satisfies |Kii(y)|. 1|y|2 for y 6= 0.Case 1. 2 j 2l−m.By the assumption on φ , for x with |φ(x)| ≤ 2 · 2−A, we have |x| ≤ 4 · 2−A.Then, using φ(0) = 0 and (2.8.2), x with 2− j−1 ≤ |φ(x)| ≤ 2− j+1 satisfies2− j+m & |x|& 2− j−m. (2.8.4)67Therefore, if y and z satisfy φ(y− z) ∈ supp(ρ j), we have 2− j−1 ≤ |φ(y− z)| ≤2− j+1 and hence 2−l  2− j−m . |y− z| . 2− j+m. Combining with |y| ∼ 2−l , forsuch y and z, we get2− j−m . |z|. 2− j+m.Now, we estimateRiiTγ(ρ j ◦φ) for i = 1,2.|RiiTγ(ρ j ◦φ)(y)|=∣∣∣∣ˆ (ρ j ◦φ)(y− z)Kii(z)dy∣∣∣∣≤ˆ2− j−m.|z|.2− j+m|(ρ j ◦φ)(y− z)− (ρ j ◦φ)(−z)||Kii(z)|dy. |y|∥∥∇(ρ j ◦φ)∥∥∞ˆ2− j−m.|z|.2− j+m1|z|2 dz. 2−l+m+ jm.In the first inequality, we use φ(y− z) ∈ supp(ρ j) andRiiTγ(ρ j ◦φ)(0) =ˆc≤|z|≤C(ρ j ◦φ)(−z)Kii(z)dz = 0for any arbitrary constants 0 < c <C <+∞. This is because φ 1 and Kii for i = 1,2are even in z2, while φ 2, and ρ are odd in z2.Case 2. 2 j 2l+mBy (2.8.4) with 2−l 2− j+m, we have |z| ∼ 2−l when φ(y− z) ∈ supp(ρ j) and|y| ∼ 2−l . This implies that for i = 1,2|RiiTγ(ρ j ◦φ)(y)| ≤ ‖Kii‖L∞(|y|∼2−l)∥∥ρ j ◦φ∥∥1 . 4l− j.Case 3. 2l−m . 2 j . 2l+m∥∥RiiTγ(ρ j ◦φ)∥∥∞ . ∥∥ρ j ◦φ∥∥ 122 ∥∥∇(ρ j ◦φ)∥∥ 12∞ . ∥∥ρ j∥∥ 122 ∥∥∇ρ j∥∥ 12∞ 2 m2 . 2 m2 .68Combining all the cases, we have∑j∥∥RiiTγ(ρ j ◦φ)∥∥∞ . 2 m2 m+m. 2 m2 m, i = 1,2.Then, (2.8.3) easily follows.69Chapter 3Global Navier-Stokes flows fornon-decaying initial data withslowly decaying oscillation3.1 IntroductionIn this chapter, we consider the incompressible Navier-Stokes equations∂tv−∆v+(v ·∇)v+∇p = 0divv = 0v|t=0 = v0(NS)in R3× (0,T ) for 0 < T ≤ ∞. These equations describe the flow of incompress-ible viscous fluids, so the solution v : R3× (0,T )→ R3 and p : R3× (0,T )→ Rrepresent the flow velocity and the pressure, respectively.For an initial datum with finite kinetic energy, v0 ∈ L2(R3), the existence ofa time-global weak solution dates back to Leray [33]. This solution has a finiteglobal energy, i.e, it satisfies the energy inequality:‖v(·, t)‖2L2(R3)+2‖∇v‖2L2(0,t;L2(R3)) ≤ ‖v0‖2L2(R3) , ∀t > 0. (3.1.1)70In Hopf [20], this result is extended to smooth bounded domains with the Dirichletboundary condition. We say v is a Leray-Hopf weak solution to (NS) in Ω× (0,T )for a domain Ω⊂ R3, ifv ∈ L∞(0,T ;L2σ (Ω))∩L2(0,T ;H10,σ (Ω))∩Cwk([0,T );L2σ (Ω))satisfies the weak form of (NS) and the energy inequality (3.1.1).However, when a fluid fills an unbounded domain, it is possible to have finitelocal energy but infinite global energy. One such example is a fluid with constantvelocity. There are also many interesting non-decaying infinite energy flows liketime-dependent spatially periodic flows (flows on torus) and two-and-a-half dimen-sional flows; see [37, Section 2.3.1] and [16]. Can we get global existence for suchdata? To analyze the motion of such fluids, one may consider the class L2uloc for thevelocity field v0 in R3 whose kinetic energy is uniformly locally bounded. Here,for 1≤ q≤ ∞, we denote by Lquloc the space of functions in R3 with‖v0‖Lquloc := supx0∈R3‖v0‖Lq(B(x0,1)) < ∞.In [31], Lemarie´-Rieusset introduced the class of local energy solutions for initialdata v0 ∈ L2uloc (see Section 3.3 for details). He proved the short time existence forinitial data in L2uloc, and the global in time existence for v0 ∈ E2, those initial datain L2uloc which further satisfy the spatial decay conditionlim|x0|→∞ˆB(x0,1)|v0|2dx = 0. (3.1.2)Then, Kikuchi-Seregin [26] added more details to the results in [31], especially thecareful treatment of the pressure. They also allowed a force term g in (NS) whichsatisfies divg = 0 andlim|x0|→∞ˆ T0ˆB(x0,1)|g(x, t)|2dxdt = 0, ∀T > 0.Recently, Maekawa-Miura-Prange [35] generalized this result to the half-spaceR3+.The treatment of the pressure in [35] is even more complicated.71One key difficulty in the study of infinite energy solutions is the estimates ofthe pressure. While finite energy solutions have enough decay at spatial infinityand one may often get the pressure from the equation p = (−∆)−1∂i∂ j(viv j), thisis not applicable to infinite energy solutions because of their slow (or no) spatialdecay.To estimate the pressure, the definition of a local energy solution in [26] in-cludes a locally-defined pressure decomposition near each point in R3, see condi-tion (v) in Definition 3.3.1. (It is already in [31] but not part of the definition.) In[21]-[22], on the other hand, Jia and Sˇvera´k use a slightly different definition byreplacing the decomposition condition by the spatial decay of the velocitylim|x0|→∞ˆ R20ˆB(x0,R)|v(x, t)|2dxdt = 0, ∀R > 0. (3.1.3)Under the decay assumption (3.1.2) on initial data, these two definitions can beshown to be equivalent; see [23, 35]. However, for general non-decaying initialdata, the decay condition (3.1.3) is not expected, while the decomposition conditionstill works. For this reason, we follow the definition of Kikuchi-Seregin [26] in thischapter.A new feature in the study of infinite energy solutions with non-decaying initialdata is the abundance of parasitic solutions,v(x, t) = f (t), p(x, t) =− f ′(t) · xfor a smooth vector function f (t). They solve the Navier-Stokes equations withinitial data f (0). If we choose f1(t) 6= f2(t) with f1(0) = f2(0), the correspondingparasitic solutions give two different local energy solutions with the same initialdata. Such solutions have non-decaying initial data, and can be shown to fail thepressure decomposition condition. More generally, if (v, p) is a solution to (NS),then the following parasitic transformu(x, t) = v(y, t)+q′(t), pi(x, t) = p(y, t)−q′′(t) · y, y = x−q(t) (3.1.4)gives another solution (u,pi) to (NS) with the same initial data v0 for any vector72function q(t) satisfying q(0) = q′(0) = 0.We now summarize the known existence results in R3. In addition to the weaksolution approach based on the a priori bound (3.1.1) following Leray and Hopf,another fruitful approach is the theory of mild solutions, treating the nonlinearterm as a source term of the nonhomogeneous Stokes system. In the frameworkof Lq(R3), there exist short time mild solutions in Lq(R3) when 3 ≤ q ≤ ∞ ([15,18, 24]). When q = 3, these solutions exist for all time for sufficiently small initialdata in L3(R3); see [24]. Similar small data global existence results hold for manyother spaces of similar scaling property, such as L3weak, Morrey spaces Mp,3−p,negative Besov spaces B˙3/q−1q,∞ , 3 < q <∞, and the Koch-Tataru space BMO−1; Seee.g. [1, 2, 8, 19, 25, 28, 29].For any data v0 ∈ Lq(R3), 2< q< 3, Caldero´n [7] constructed a global solution.His strategy is to first decompose v0 = a0 + b0 with small a0 ∈ L3(R3) and largeb0 ∈ L2(R3). A solution is then obtained as v = a+ b, where a is a global smallmild solution of (NS) in L3(R3) with a(0) = a0, and b is a global weak solution ofthe a-perturbed Navier-Stokes equations in the energy class with b(0) = b0.This idea is then used by Lemarie´-Rieusset [31] to construct global local energysolutions for v0 ∈ E2; also see Kikuchi-Seregin [26].We now summarize the known existence results for non-decaying initial data.For the local existence, many mild solution existence theorems mentioned earlierallow non-decaying data. The most relevant to us are Giga-Inui-Matsui [18] forinitial data in L∞(R3) and BUC(R3), and Maekawa-Terasawa [36] for initial datain the closure of⋃p>3 Lpuloc in L3uloc-norm, and any small initial data in L3uloc. Small-ness is needed for L3uloc data even for short time existence.When it comes to the global existence for non-decaying data, a solution the-ory for perturbations of constant vectors seems straightforward. The only otherresult we are aware of is the recent paper Maremonti-Shimizu [38], which provedthe global existence of weak solutions for initial data v0 in L∞(R3)∩C0(R3)W˙1,q,3 < q < ∞. In particular, they assume ∇v0 ∈ Lq(R3). Their strategy is to decom-pose the solution v = U +w, U = ∑nk=1 vk, where v1 solves the Stokes equationswith the given initial data, and vk+1, k ≥ 1, solves the linearized Navier-Stokesequations with the force f k = −vk ·∇vk and homogeneous initial data. The forcef 1 ∈ Lq(0,T ;Lq(R3)) thanks to the assumption on v0. In each iteration, we get an73additional decay of the force f k. The perturbation w is then solved in the frame-work of weak solutions. The paper [38] motivated this chapter.We now state our main theorem. Denote the average of a function v in a setE ⊂ R3 by (v)E = 1|E|´E v(x)dx. We denote w ∈ E2σ if w ∈ E2 and divw = 0.Theorem 3.1.1. For any vector field v0 ∈ E2σ +L3uloc satisfying divv0 = 0 andlim|x0|→∞ˆB(x0,1)|v0− (v0)B(x0,1)|dx = 0, (3.1.5)we can find a time-global local energy solution (v, p) to the Navier-Stokes equa-tions (NS) in R3× (0,∞), in the sense of Definition 3.3.1.Our main assumption is the “oscillation decay” condition (3.1.5). Note that allv0 ∈ L2uloc satisfying (3.1.2) also satisfy (3.1.5). Furthermore, for v0 ∈ L2uloc, eitherv0 ∈ E1 or ∇v0 ∈ E1 implies the condition (3.1.5). Here Eq for 1 ≤ q ≤ ∞ is thespace of functions in Lquloc whose Lq-norm in a ball B1(x0) goes to zero as |x0| goesto infinity. In particular, our result generalizes the global existence for decayinginitial data v0 ∈ E2 in [31] and [26]. It also extends [38] for v0 ∈ L∞ and ∇v0 ∈ Lq.The condition v0 ∈ E2σ +L3uloc gives us more regularity on the nondecaying partof v0. We do not know if it is necessary for the global existence, but it is essentialfor our proof, and enables us to prove that for small t > 0,‖w(t)χR‖L2uloc . (t120 +‖w0χR‖L2uloc), (3.1.6)where χR(x) is a cut-off function supported in |x|> R, we decompose v0 = w0+u0with w0 ∈ E2σ and u0 ∈ L3uloc, and w(t) = v(t)−et∆u0 with w(0) =w0. This estimateshows that ‖w(t)χR‖L2uloc vanishes as t→ 0+ and R→ ∞.The idea of our proof is as follows. First, we construct a local energy solutionin a short time. For v0 ∈ L2uloc, this is done in [31] but not in [26]. However, we usea slightly revised approximation scheme to make all statements about the pressureeasy to verify. In our scheme, we not only mollify the non-linear term as in [33]and [31], but also insert a cut-off function, so that the non-linear term (v ·∇)v isreplaced by (Jε(v) ·∇)(vΦε), whereJε is a mollification of scale ε and Φε is aradial bump function supported in the ball B(0,2ε−1).74Once we have a local-in-time local energy solution, we need some smallnessto extend the solution globally in time. To this end, we decompose the solutionas v = V +w where V (t) = et∆u0 solves the heat equation. The main effort is toshow that w(t) ∈ E2 for all t and w(t) ∈ E6 for almost all t. The proof is similarto the decay estimates in [26, 31] and we try to do local energy estimate for wχR.The background V has no spatial decay, but we can show the decay of ∇V (x, t) inL∞(BcR× (t0,∞)) as R→ ∞ for any t0 > 0. This decay is not uniform up to t0 = 0as u0 is rather rough. We need a new decomposition formula of the pressure, sothat in the intermediate regions we can show the decay of the pressure using thedecay of ∇V . Because the decay of ∇V is not up to t0 = 0, we need to do the localenergy estimate in the time interval [t0,T ), 0 < t0 1. This forces us to prove theestimate (3.1.6), and the strong local energy inequality for w away from t = 0.Once we have shown w(t)∈ E6 for almost all t < T , we can extend the solutionas in [31] and [26]. However, we avoid using the strong-weak uniqueness as in[26, 31], and choose to verify the definition of local energy solutions directly as in[35].The rest of the chapter consists of the following sections. In Section 3.2, wediscuss the properties of the heat flow et∆u0, especially the decay of its gradient atspatial infinity assuming (3.1.5). In Section 3.3, we recall the definition of localenergy solutions as in [26] and use our revised approximation scheme to find alocal energy solution local-in-time. In Section 3.4, we find a new pressure decom-position formula suitable of using the decay of ∇V , prove the estimate (3.1.6) andthe strong local energy inequality, and then do the local energy estimate of wχR,which implies w(t) ∈ E6 for almost all t. In Section 3.5, we construct the desiredtime-global local energy solution.3.2 Notations and preliminaries3.2.1 NotationGiven two comparable quantities X and Y , the inequality X .Y stands for X ≤CYfor some positive constant C. In a similar way, & denotes ≥ C for some C > 0.We write X ∼ Y if X . Y and Y . X . Furthermore, in the case that a constant C75in X ≤ CY depends on some quantities Z1, · · · , Zn, we write X .Z1,··· ,Zn Y . Thenotations &Z1,··· ,Zn and ∼Z1,··· ,Zn are similarly defined.For a point x ∈ R3 and a positive real number r, B(x,r) is the Euclidean ball inR3 centered at x with a radius r,B(x,r) = Br(x) = {y ∈ R3 : |y− x|< r}.When x = 0, we denote Br = B(0,r). For a point x ∈ R3 and r > 0, we denote theopen cube centered at x with a side length 2r asQ(x,r) = Qr(x) ={y ∈ R3 : maxi=1,2,3|yi− xi|< r}.We denote the mollification Jε(v) = v ∗ ηε , ε > 0, where the mollifier isηε(x) = ε−3η( xε)and η is a fixed nonnegative radial bump function in C∞c (R3)supported in B(0,1) satisfying´η dx = 1.Various test functions in this chapter are defined by rescaling and translating anon-negative radially decreasing bump function Φ satisfying Φ= 1 on B(0,1) andsupp(Φ)⊂ B(0, 32).For k ∈ N∪{0,∞}, let Ckc(R3) be the subset of functions in Ck(R3) with com-pact supports, andCkc,σ (R3) ={u ∈Ckc(R3;R3) : divu = 0}.3.2.2 Uniformly locally integrable spacesTo consider infinite energy flows, we work in the spaces Lquloc, 1 ≤ q ≤ ∞, andU s,p(t0, t) for 1≤ s, p≤ ∞ and 0≤ t0 < t ≤ ∞, defined byLquloc ={u ∈ L1loc(R3) : ‖u‖Lquloc = supx0∈R3‖u‖Lq(B1(x0)) <+∞}76andU s,p(t0, t)={u ∈ L1loc(R3× (t0, t)) : ‖u‖U s,p(t0,t) = supx0∈R3‖u‖Ls(t0,t;Lp(B1(x0))) <+∞}.When t0 = 0, we simply use Us,pT =Us,p(0,T ). Note that U∞,p(t0, t)=L∞(t0, t;Lpuloc),1≤ p≤ ∞, but for general 1≤ s < ∞ and 1≤ p≤ ∞, U s,p(t0, t) and Ls(t0, t;Lpuloc)are not equivalent norms. Indeed, we can only guarantee that‖u‖U s,p(t0,t) ≤ ‖u‖Ls(t0,t;Lpuloc) , (3.2.1)but not the inequality of the other direction.Example 3.2.1. Fix 1 ≤ s < ∞ and p ∈ [1,∞]. Let xk be a sequence in R3 withdisjoint B1(xk), k ∈ N, and let tk = t0 + 2−k. Define a function u by u(x,τ) = 2k/son B1(xk)× (t0, tk), k ∈N, and u(x,τ) = 0 otherwise. It is defined independently ofp. We have u ∈U s,p(t0, t), butˆ t1t0‖u(·,τ)‖sLpuloc dτ =∞∑k=1ˆ tktk+1cp2kdτ =∞∑k=112cp = ∞,and hence u 6∈ Ls(t0, t;Lpuloc).We define a local energy space E (t0, t) byE (t0, t) ={u ∈ L2loc([t0, t]×R3;R3) : divu = 0, ‖u‖E (t0,t) <+∞}, (3.2.2)where‖u‖E (t0,t) := ‖u‖U∞,2(t0,t)+‖∇u‖U2,2(t0,t) .When t0 = 0, we use the abbreviation ET = E (0,T ).The spaces E p and Gp(t0, t), 1 ≤ p ≤ ∞, are defined by an additional decaycondition at infinity,E p := { f ∈ Lpuloc : ‖ f‖Lp(B(x0,1))→ 0, as |x0| → ∞},77andGp(t0, t) := {u ∈U p,p(t0, t) : ‖u‖Lp([t0,t]×B(x0,1))→ 0, as |x0| → ∞}.We let Lpuloc,σ , Epσ and Gpσ (t0, t) denote divergence-free vector fields with compo-nents in Lpuloc, Ep and Gp(t0, t), respectively.The space E p, 1≤ p < ∞, can be characterized as C∞c (R3)Lpuloc . The analogousstatement for E pσ is true.Lemma 3.2.2. ([26, Appendix]) Suppose that f ∈ E pσ for some 1 ≤ p < ∞. Then,for any ε > 0, we can find f ε ∈C∞c,σ (R3) such that‖ f − f ε‖Lpuloc < ε.3.2.3 Heat and Oseen kernels on LqulocNow, we study the operators et∆ and et∆P∇· on Lquloc. Here P denotes the Helmholtzprojection in R3. Both are defined as convolution operatorset∆ f = Ht ∗ f , and et∆Pi j∂kFjk = ∂kSi j ∗Fjk,where Ht and Si j are the heat kernel and the Oseen tensor, respectively,Ht(x) =1√4pit3exp(−|x|24t),andSi j(x, t) = Ht(x)δi j +14pi∂ 2∂xi∂x jˆR3Ht(y)|x− y|dy.In this paper, we use (divF)i = (∇ ·F)i = ∂ jFji. Note that the Oseen tensor satisfiesthe following pointwise estimates|∇lx∂ kt S(x, t)| ≤Ck,l(|x|+√t)−3−l−2k. (3.2.3)We have the following estimates.78Lemma 3.2.3 (Remark 3.2 in [36]). For 1 ≤ q ≤ p ≤ ∞, the following holds. Forany vector field f and any 2-tensor F in R3,∥∥∥∂αt ∂ βx et∆ f∥∥∥Lpuloc. 1t |α|+|β |2(1+1t32(1q− 1p))‖ f‖Lquloc ,∥∥∥∂αt ∂ βx et∆P∇ ·F∥∥∥Lpuloc. 1t |α|+|β |2 +12(1+1t32(1q− 1p))‖F‖Lquloc .Note p = ∞ is allowed, with L∞uloc = L∞.Lemma 3.2.4. For any T > 0, if f ∈ L2uloc and F ∈U2,2T , then we have∥∥et∆ f∥∥ET. (1+T 12 )‖ f‖L2uloc ,∥∥∥∥ˆ t0e(t−s)∆P∇ ·F(s)ds∥∥∥∥ET. (1+T )‖F‖U2,2T .Recall ‖u‖ET = ‖u‖U∞,2T +‖∇u‖U2,2T . Similar estimates can be found in the proofof [32, Theorem 14.1]. We give a slightly revised proof here for completeness.Proof. Fix x0 ∈ R3 and let φx0(x) =Φ( x−x02). We decompose f and F asf = fφx0 + f (1−φx0) = f1+ f2andF = Fφx0 +F(1−φx0) = F1+F2.Since f1 ∈ L2(R3) and F1 ∈ L2(0,T ;L2(R3)), by the usual energy estimates forthe heat equation and the Stokes system, we get∥∥et∆ f1∥∥ET . ‖ f1‖2 . ‖ f‖L2uloc (3.2.4)and ∥∥∥∥ˆ t0e(t−s)∆P∇ ·F1(s)ds∥∥∥∥ET. ‖F1‖L2(0,T ;L2(R3)) . ‖F‖U2,2T . (3.2.5)79On the other hand, by Lemma 3.2.3,∥∥et∆ f2∥∥U∞,2T = ∥∥et∆ f2∥∥L∞(0,T ;L2uloc) . ‖ f2‖L2uloc . ‖ f‖L2uloc .Together with (3.2.4), we get∥∥et∆ f∥∥U∞,2T . ‖ f‖L2uloc . (3.2.6)(This also follows from Lemma 3.2.3.) By the heat kernel estimates,∥∥∇et∆ f2∥∥L2((0,T )×B(x0,1)) . T 12 ∥∥∇et∆ f2∥∥L∞((0,T )×B(x0,1)). T 12ˆB(x0,2)c1|x0− y|4 | f2(y)|dy≤ T 12∞∑k=1ˆB(x0,2k+1)\B(x0,2k)1|x0− y|4 | f (y)|dy. T 12∞∑k=1124kˆB(x0,2k+1)| f (y)|dy.We may cover B(x0,2k+1) by⋃Jkj=1 B(xkj,1) with Jk bounded by C023k for someconstant C0 > 0. Then∥∥∇et∆ f2∥∥L2((0,T )×B(x0,1)) . T 12 ∞∑k=1124kJk∑j=1ˆB(xkj ,1)| f (y)|dy. T 12 ‖ f‖L2uloc .Together with (3.2.4), we get∥∥∇et∆ f∥∥L2((0,T )×B(x0,1)) . (1+T 12 )‖ f‖L2uloc .Taking supremum in x0, we obtain∥∥∇et∆ f∥∥U2,2T . (1+T 12 )‖ f‖L2uloc .This and (3.2.6) show the first bound of the lemma,∥∥et∆ f∥∥ET. (1+T 12 )‖ f‖L2uloc .Denote ΨF(t) =´ t0 e(t−s)∆P∇ ·F(s)ds. By the pointwise estimates (3.2.3) for80the Oseen tensor, we have‖ΨF2‖L∞(0,T ;L2(B(x0,1))) .ˆ t0ˆB(x0,2)c1|x0− y|4 |F2(y,s)|dyds≤∞∑k=1124kˆ t0ˆB(x0,2k+1)|F(y,s)|dyds≤∞∑k=1124kJk∑j=1ˆ t0ˆB(xkj ,1)|F(y,s)|dyds. ‖F‖U1,1T . T12 ‖F‖U2,2Tand‖∇ΨF2‖L2((0,T )×B(x0,1)) . T12 ‖∇ΨF2‖L∞((0,T )×B(x0,1)). T 12ˆ t0ˆB(x0,2)c1|x0− y|5 |F2(y,s)|dyds≤ T 12∞∑k=1125kˆ t0ˆB(x0,2k+1)|F(y,s)|dyds. T ‖F‖U2,2T .Combined with (3.2.5), we have‖ΨF‖L∞(0,T ;L2(B(x0,1))) . (1+T12 )‖F‖U2,2Tand‖∇ΨF‖L2((0,T )×B(x0,1)) . (1+T )‖F‖U2,2T .Finally, we take suprema in x0 to get∥∥∥∥ˆ t0e(t−s)∆P∇ ·F(s)ds∥∥∥∥ET. (1+T )‖F‖U2,2T .This is the second bound of the lemma.813.2.4 Heat kernel on L1uloc with decaying oscillationIn this subsection, we investigate how the decaying oscillation assumption (3.1.5)on initial data affects the heat flow. Recall(u)Qr(x) = Qr(x)u(y)dy =1|Qr(x)|ˆQr(x)u(y)dy.Lemma 3.2.5. Suppose that u ∈ L1uloc(R3) satisfieslim|x0|→∞ˆQ1(x0)|u− (u)Q1(x0)|dx = 0. (3.2.7)Then, for any r > 0, we havelim|x0|→∞ˆQr(x0)|u− (u)Qr(x0)|dx = 0, (3.2.8)andlim|x0|→∞supy∈Q2r(x0)|(u)Qr(y)− (u)Qr(x0)|= 0. (3.2.9)Proof. First note that (u)Qr(x) is finite for any x ∈ R3 and r > 0. Indeed,|(u)Qr(x)| ≤Cr ‖u‖L1ulocfor a constant Cr independent of x, Cr <C for r > 1, and Cr ∼ r−3 for r 1.Fix x0 ∈ R3 and r > 0. For any constant c ∈ R, we get Qr(x0)|u− (u)Qr(x0)|dx≤ Qr(x0)|u− c|+ |(u)Qr(x0)− c|dx= Qr(x0)|u− c|dx+∣∣∣∣ Qr(x0)(u− c)dx∣∣∣∣≤ 2 Qr(x0)|u− c|dx.82Then, for Qr = Qr(x1)⊂ QR(x0), R > r, we get Qr|u− (u)Qr |dx≤ 2 Qr|u− (u)QR(x0)|dx≤2R3r3 QR(x0)|u− (u)QR(x0)|dx.(3.2.10)With x0 = x1 and R = 1 in (3.2.10), (3.2.7) implies (3.2.8) for all r ∈ (0,1).If y ∈ Q2r(x0), thenQr(x0)∪Qr(y)⊂ QR(x1), x1 = 12(x0+ y), R≥ 2r.Thus,∣∣(u)Qr(x0)− (u)Qr(y)∣∣≤ ∣∣∣∣ Qr(x0)u− (u)QR(x1)dx∣∣∣∣+ ∣∣∣∣ Qr(y)u− (u)QR(x1)dx∣∣∣∣≤ Qr(x0)|u− (u)QR(x1)|dx+ Qr(y)|u− (u)QR(x1)|dx≤ 2R3r3 QR(x1)|u− (u)QR(x1)|dx.(3.2.11)With R = 1, this and (3.2.7) imply (3.2.9) for all r ∈ (0, 12 ].Now, for any Qr(x0) with r > 1, choose the smallest integer N > 2r and letρ = r/N < 12 . We can find a set S = Sx0,r of N3 points such that {Qρ(z) : z ∈ S} aredisjoint andQr(x0) =⋃z∈SQρ(z).For any z,z′ ∈ S, we can connect them by points z j in S, j = 0,1, . . . ,N, such thatz0 = z, zN = z′, and z j ∈ Q2ρ(z j−1), j = 1, . . . ,N. We allow z j+1 = z j for some j.Thus|(u)Qρ (z)− (u)Qρ (z′)| ≤N∑j=1|(u)Qρ (z j)− (u)Qρ (z j−1)|,and hencemaxz,z′∈Sx0 ,r|(u)Qρ (z)− (u)Qρ (z′)|= o(1) as |x0| → ∞ (3.2.12)83by (3.2.9) as ρ ∈ (0, 12). We have Qr(x0)|u− (u)Qr(x0)|dx=∑z∈SN−3 Qρ (z)|u− (u)Qr(x0)|dx≤∑z∈SN−3( Qρ (z)|u− (u)Qρ (z)|+ |(u)Qr(x0)− (u)Qρ (z)|dx)≤(∑z∈SN−3 Qρ (z)|u− (u)Qρ (z)|dx)+maxz,z′∈S|(u)Qρ (z)− (u)Qρ (z′)|= o(1) as |x0| → ∞by (3.2.8) and (3.2.12) for ρ ∈ (0, 12). This shows (3.2.8) for all r > 1.Finally, (3.2.9) for r > 1/2 follows from (3.2.8) and (3.2.11).The following lemma says that decaying oscillation over cubes is equivalent todecaying oscillation over balls.Lemma 3.2.6. Suppose u ∈ L1uloc. Then u satisfies (3.2.7) if and only iflim|x0|→∞ˆB1(x0)|u− (u)B1(x0)|dx = 0. (3.2.13)Proof. Let ρ = 3−1/2. We have Qρ(x0)⊂ B1(x0)⊂Q1(x0). Similar to the proof of(3.2.10), we haveˆB1(x0)|u− (u)B1(x0)|dx≤CˆQ1(x0)|u− (u)Q1(x0)|dxand hence (3.2.13) follows from (3.2.7). Similarly, we also haveˆQρ (x0)|u− (u)Qρ (x0)|dx≤CˆB1(x0)|u− (u)B1(x0)|dxand hence (3.2.8) for r = ρ follows from (3.2.13). Then v(x) = u(ρx) satisfies(3.2.7). By Lemma 3.2.5, v satisfies (3.2.8) for any r > 0, and we get (3.2.7) foru.84Lemma 3.2.7. Suppose v0 ∈ L1uloc andˆQ(x0,1)|v0− (v0)Q(x0,1)| → 0, as |x0| → ∞.Let V = et∆v0. Then (∇V )(t0) ∈ C0(R3) for every t0 > 0. Furthermore, for anyt0 > 0, we havesupt>t0‖∇V (·, t)‖L∞(B(x0,1))→ 0, as |x0| → ∞. (3.2.14)Proof. For k ∈ Z3, let Σk denote the set of its neighbor integer points,Σk = Z3∩Q(k,1.01)\{k}.Letak = (v0)Q1(k), bk = maxk′∈Σk|ak′−ak|, ck =ˆQ1(k)|v0(x)−ak|dx.By the assumption, ck→ 0 as |k| → ∞ and by Lemma 3.2.5, bk→ 0 as |k| → ∞.Choose a nonnegative φ ∈C∞c (R3) with suppφ ⊂ Q1(0) and∑k∈Z3φk(x) = 1 ∀x ∈ R3, φk(x) = φ(x− k).Definev1(x) = ∑k∈Z3akφk(x).Since |ak|. ‖v0‖L1uloc , v1 is in L∞(R3). For x ∈ Q1(k), it can be written asv1(x) = ak + ∑k′∈Σk(ak′−ak)φk′(x).ThusˆQ1(k)|v0(x)− v1(x)|dx≤ˆQ1(k)|v0(x)−ak|dx+ ∑k′∈ΣkˆQ1(k)|ak−ak′ |φk′(x)dx≤ ck +Cbk,(3.2.15)85andsupx∈Q1(k)|∇v1(x)| ≤ supx∈Q1(k)∑k′∈Σk|ak′−ak| · |∇φk′(x)| ≤Cbk. (3.2.16)Let ψR(x) =Φ( xR). We decompose∇V (x, t) =ˆ∇Ht(x− y)v0(y)(1−ψR(x− y))dy+ˆ∇Ht(x− y)[v0(y)− v1(y)]ψR(x− y)dy+ˆ∇Ht(x− y)v1(y)ψR(x− y)dy = I1+ I2+ I3.By integration by parts, we can rewrite I3,I3 =ˆHt(x−y)∇v1(y)ψR(x−y)dy−ˆHt(x−y)v1(y)(∇ψR)(x−y)dy= I31+ I32.Fix ε > 0 and consider t > t0 > 0. Since for any t > 0 and x ∈ R3, we have|I1|.ˆB(x,R)c|x− y|5t52e−|x−y|24t1|x− y|4 |v0(y)|dy.ˆB(x,R)c1|x− y|4 |v0(y)|dy.1R‖v0‖L1uloc ,and|I32|. ‖Ht‖1 ‖v1‖∞ ‖∇ψR‖∞ .1R‖v0‖L1uloc ,we can choose sufficiently large R > 0 such that|I1, I32|< ε.The integrands of both I2 and I31 are supported in |y−x| ≤ 2R. If |x|> 2ρ withρ > 2R and |y− x| ≤ 2R, then |y| ≥ |x|− |x− y|> ρ . Let1>ρ(y) = 1 for |y|> ρ, and 1>ρ(y) = 0 for |y| ≤ ρ.86We have|I2| ≤∥∥|∇Ht | ∗ |v0− v1|1>ρ∥∥L∞(R3) . t− 120 (1+ t− 320 )∥∥|v0− v1|1>ρ∥∥L1ulocby Lemma 3.2.3, and|I31| ≤∥∥et∆(|∇v1|1>ρ)∥∥L∞(R3) . ∥∥|∇v1|1>ρ∥∥L∞(R3) .If we take ρ sufficiently large, by (3.2.15) and (3.2.16), we have |I2|+ |I31| ≤ 2ε .Since for any t > t0 and ε > 0, we can choose ρ > 0 such thatsupt>t0‖∇V (·, t)‖L∞(B(0,2ρ)c) < 4ε,we get (3.2.14).3.3 Local existenceIn this section, we recall the definition of local energy solutions and prove theirtime-local existence using a revised approximation scheme. Note that we do notassume spatial decay of initial data for the time-local existence.As mentioned in the introduction, we follow the definition in Kikuchi-Seregin[26].Definition 3.3.1 (local energy solution). Let v0 ∈ L2uloc with divv0 = 0. A pair(v, p) of functions is a local energy solution to the Navier-Stokes equations (NS)with initial data v0 in R3× (0,T ), 0 < T < ∞, if it satisfies the following.(i) v ∈ ET , defined in (3.2.2), and p ∈ L32loc([0,T )×R3).(ii) (v, p) solves the Navier-Stokes equations (NS) in the distributional sense.(iii) For any compactly supported function ϕ ∈ L2(R3), the function ´R3 v(x, t) ·ϕ(x)dx of time is continuous on [0,T ]. Furthermore, for any compact setK ⊂ R3,‖v(·, t)− v0‖L2(K)→ 0, as t→ 0+.87(iv) (v, p) satisfies the local energy inequality (LEI) for any t ∈ (0,T ):ˆR3|v|2ξ (x, t)dx+2ˆ t0ˆR3|∇v|2ξ dxds≤ˆ t0ˆR3|v|2(∂sξ +∆ξ )+(|v|2+2p)(v ·∇)ξ dxds,(3.3.1)for all non-negative smooth functions ξ ∈C∞c ((0,T )×R3).(v) For each x0 ∈ R3, we can find cx0 ∈ L32 (0,T ) such thatp(x, t) = p̂x0(x, t)+ cx0(t), in L32 (B(x0, 32)× (0,T )), (3.3.2)wherep̂x0(x, t) =−13|v(x, t)|2+p.v.ˆB(x0,2)Ki j(x− y)viv j(y, t)dy+ˆB(x0,2)c(Ki j(x− y)−Ki j(x0− y))viv j(y, t)dy(3.3.3)for K(x) = 14pi|x| and Ki j = ∂i jK.We say the pair (v, p) is a local energy solution to (NS) in R3×(0,∞) if it is a localenergy solution to (NS) in R3× (0,T ) for all 0 < T < ∞.For an initial data v0 ∈ L2uloc whose local kinetic energy is uniformly bounded,we reprove the local existence of a local energy solution of [31, Chapt 32].Theorem 3.3.2 (Local existence). Let v0 ∈ L2uloc with divv0 = 0. IfT ≤ ε11+‖v0‖4L2ulocfor some small constant ε1 > 0, we can find a local energy solution (v, p) on R3×(0,T ) to the Navier-Stokes equations (NS) for the initial data v0, satisfying ‖v‖ET ≤C‖v0‖L2uloc .Note that we do not assume v0 ∈ E2, i.e., we do not assume spatial decay of v0.Although the local existence theorem is proved in [31, Chapt 32], a few details are88missing there, in particular those related to the pressure. These details are given in[26] for the case v0 ∈ E2. Here we treat the general case v0 ∈ L2uloc.Recall the definitions of Jε(·) and Φ in Section 3.2 and let Φε(x) = Φ(εx),ε > 0. To prove Theorem 3.3.2, we consider approximate solutions (vε , pε) to thelocalized-mollified Navier-Stokes equations∂tvε −∆vε +(Jε(vε) ·∇)(vεΦε)+∇pε = 0divvε = 0vε |t=0 = v0(3.3.4)in R3× (0,T ).Since v0 ∈ L2uloc has no decay, it cannot be approximated by L2-functions, aswas done in [26] when v0 ∈ E2. Hence the approximation solution vε cannot beconstructed in the energy class L∞(0,T ;L2(R3))∩L2(0,T ; H˙1(R3)), and has to beconstructed in ET directly.Compared to [26, 31], our mollified nonlinearity has an additional localizationfactor Φε . It makes the decay of the Duhamel term apparent when the approxima-tion solutions have no decay.We first construct a mild solution vε of (3.3.4) in ET .Lemma 3.3.3. For each 0< ε < 1 and v0 with ‖v0‖L2uloc ≤B, if 0<T <min(1,cε3B−2),we can find a unique solution v = vε to the integral form of (3.3.4)v(t) = et∆v0−ˆ t0e(t−s)∆P∇ · (Jε(v)⊗ vΦε)(s)ds (3.3.5)satisfying‖v‖ET ≤ 2C0B,where c > 0 and C0 > 1 are absolute constants and (a⊗b) jk = a jbk.Proof. Let Ψ(v) be the map defined by the right side of (3.3.5) for v ∈ ET . By89Lemma 3.2.4 and T ≤ 1,‖Ψ(v)‖ET . ‖v0‖L2uloc +‖Jε(v)⊗ vΦε‖U2,2T. ‖v0‖L2uloc +‖Jε(v)‖L∞(0,T ;L∞(R3)) ‖v‖U2,2T. ‖v0‖L2uloc + ε− 32√T ‖v‖2U∞,2T .Thus‖Ψ(v)‖ET ≤C0 ‖v0‖L2uloc +C1ε− 32√T ‖v‖2ET ,for some constants C0,C1 > 0. Similarly, for v,u ∈ ET ,‖Ψ(v)−Ψ(u)‖ET ≤C1ε−32√T(‖v‖ET +‖u‖ET )‖v−u‖ET .By the Picard contraction theorem, if T satisfiesT <ε364(C0C1B)2= cε3B−2,then we can always find a unique fixed point v ∈ ET of v = Ψ(v), i.e., (3.3.5),satisfying‖v‖ET ≤ 2C0B.Lemma 3.3.4. Let v0 ∈ L2uloc with divv0 = 0. For each ε ∈ (0,1), we can find vε inET and pε in L∞(0,T ;L2(R3)) for some positive T = T (ε,‖v0‖L2uloc) which solvesthe localized-mollified Navier-Stokes equations (3.3.4) in the sense of distributions,and limt→0+ ‖vε(t)− v0‖L2(E) = 0 for any compact subset E of R3.Proof. By Lemma 3.3.3, there is a mild solution vε ∈ ET of (3.3.5) for some T =T (ε,‖v0‖L2uloc). Apparently,∥∥vε − et∆v0∥∥U∞,2t =∥∥∥∥ˆ t0e(t−s)∆P∇ · (Jε(v)⊗ vΦε)(s)ds∥∥∥∥U∞,2t. ‖Jε(v)⊗ vΦε‖U2,2t . ε− 32√t ‖v‖2U∞,2T .Also, for any compact subset E of R3, we have∥∥et∆v0− v0∥∥L2(E)→ 0 as t goes to900; by Lebesgue’s convergence theorem∥∥et∆v0− v0∥∥L2(E) ≤ 1(4pi) 32ˆe−|z|24∥∥v0(·−√tz)− v0∥∥L2(E)dz→ 0,as t → 0+. Then, it follows that limt→0+ ‖vε(t)− v0‖L2(E) = 0 for any compactsubset E of R3.Note that et∆v0 with v0 ∈ L2uloc solves the heat equation in the distributionalsense. Also, using divv0 = 0, we can easily see that divet∆v0 = 0.On the other hand,Jε(vε) ∈ L∞(R3× [0,T ]) and vε ∈ ET implyJε(vε)⊗ vεΦε ∈ L∞(0,T ;L2(R3))and hence by the classical theory, wε = vε −V and pε defined bypε = (−∆)−1∂i∂ j(Jε(vεi )vεjΦε) ∈ L∞(0,T ;L2(R3)). (3.3.6)solves the Stokes system with the source term ∇ · (Jε(vε)⊗vεΦε) in the distribu-tion sense.By adding the heat equation for V with divV = 0 and the Stokes system for(wε , pε), vε =V +wε satisfies∂tvε −∆vε +(Jε(vε) ·∇)(vεΦε)+∇pε = 0in the sense of distribution.To extract a limit solution from the family (vε , pε) of approximation solutions,we need a uniform bound of (vε , pε) on a uniform time interval [0,T ], T > 0.Lemma 3.3.5. For each ε ∈ (0,1), let (vε , pε) be the solution on R3× [0,Tε ], forsome Tε > 0, to the localized-mollified Navier-Stokes equations (3.3.4) constructedin Lemma 3.3.4. There is a small constant ε1 > 0, independent of ε and ‖v0‖2L2uloc ,such that, if Tε ≤ T0 = ε1(1+‖v0‖4L2uloc)−1, then vε is uniformly bounded‖vε‖ETε ≤C‖v0‖L2uloc , (3.3.7)91where the constant C on the right hand side is independent of ε and Tε .Proof. Let φx0 = Φ(·− x0) be a smooth cut-off function supported around x0. Forthe convenience, we drop the index x0. Starting from vε ∈ ETε and pε ∈ L∞TεL2,and using the interior regularity theory for perturbed Stokes system with smoothcoefficients, we have‖vε ,∂tvε ,∇vε ,∆vε‖L∞((δ ,Tε )×R3) <+∞for any δ ∈ (0,Tε). Using 2vεψ with ψ ∈ C∞c ((0,Tε)×R3) as a test function in(3.3.4), we get2ˆ Tε0ˆ|∇vε |2ψdxds =ˆ Tε0ˆ|vε |2(∂sψ+∆ψ)dxds+ˆ Tε0ˆ|vε |2Φε(Jε(vε) ·∇)ψdxds+2ˆ Tε0ˆpεvε ·∇ψdxds−ˆ Tε0ˆ|vε |2ψ(Jε(vε) ·∇)Φεdxds.Using limt→0+ ‖vε(t)− v0‖L2(Bn) = 0 for any n ∈ N (Lemma 3.3.4), we can showˆ|vε |2ψ(x, t)dx+2ˆ t0ˆ|∇vε |2ψdxds =ˆ|v0|2ψ(·,0)dx+ˆ t0ˆ|vε |2(∂sψ+∆ψ)dxds+ˆ t0ˆ|vε |2Φε(Jε(vε) ·∇)ψdxds+2ˆ t0ˆpεvε ·∇ψdxds−ˆ t0ˆ|vε |2ψ(Jε(vε) ·∇)Φεdxds(3.3.8)for any ψ ∈C∞c ([0,Tε)×R3) and 0 < t < Tε .We suppress the index ε in vε and pε , and take ψ(x,s) = φ(x)2θ(s) where92θ(s) ∈C∞c ([0,Tε)) and θ(s) = 1 on [0, t] to get‖v(t)φ‖22+2‖|∇v|φ‖2L2([0,t]×R3).‖v0‖2L2uloc +∣∣∣∣ˆ t0ˆ|v|2|∆φ 2|dxds∣∣∣∣+ ∣∣∣∣ˆ t0ˆ|v|2φ 2(Jε(v) ·∇)Φεdxds∣∣∣∣+∣∣∣∣ˆ t0ˆ|v|2Φε(Jε(v) ·∇)φ 2dxds∣∣∣∣+ ∣∣∣∣ˆ t0ˆ2 p̂(v ·∇)φ 2dxds∣∣∣∣=‖v0‖2L2uloc + I1+ I2+ I3+ I4,(3.3.9)where p̂= p̂εx0 will be defined later in (3.3.11) as a function satisfying∇(p− p̂) = 0on B(x0, 32)× (0,T ).The bounds of I1, I2 and I3 can be easily obtained by Ho¨lder inequalities,I1 . ‖v‖2U2,2t , and I2, I3 . ‖v‖3U3,3t. (3.3.10)Here we have used |∇Φε |. ε ≤ 1.On the other hand, I4 can be estimated asI4 . ‖ p̂‖L 32 ([0,t]×B(x0, 32 )) ‖v‖U3,3t .Now, we define p̂ε on B(x0, 32)× [0,T ] byp̂ε(x, t) =− 13Jε(vε) · vεΦε(x, t)+p.v.ˆB(x0,2)Ki j(x− y)Jε(vεi )vεj(y, t)Φε(y)dy+ˆB(x0,2)c(Ki j(x− y)−Ki j(x0− y))Jε(vεi )vεj(y, t)Φε(y)dy= p̂1+ p̂2+ p̂3.(3.3.11)Comparing the above with (3.3.6) for pε , which has the singular integral formpε(x, t) =−13Jε(vε) · vε(x, t)Φε(x)+p.v.ˆKi j(x− y)Jε(vεi )vεj(y, t)Φε(y)dy,93we see that p− p̂ depends only on t, and hence ∇p̂ = ∇p on B(x0, 32)× [0,T ].Then, we take the L32 ([0, t]×B(x0, 32))-norm for each term to get∥∥p̂1∥∥L32 ([0,t]×B(x0, 32 )). ‖v‖2U3,3t ,and∥∥p̂2∥∥L32 ([0,t]×B(x0, 32 ))≤ ∥∥p̂2∥∥L32 ([0,t]×R3) .∥∥Jε(vi)v jΦε∥∥L 32 ([0,t]×B(x0,2)) . ‖v‖2U3,3t .The second inequality for p̂2 follows from the Calderon-Zygmund theorem. Fi-nally, using|Ki j(x− y)−Ki j(x0− y)|. |x− x0||x0− y|4for x ∈ B(x0, 32) and y ∈ B(x0,2)c, we have∥∥p̂3∥∥L32 ([0,t]×B(x0, 32 )).∥∥∥∥ˆB(x0,2)c1|x0− y|4Jε(vi)v j(y,s)Φε(y)dy∥∥∥∥L32 (0,t).∥∥∥∥∥ ∞∑k=1 124kˆB(x0,2k+1)|Jε(vi)v j|(y,s)dy∥∥∥∥∥L32 (0,t)≤∞∑k=1124k∥∥∥∥∥ Jk∑j=1ˆB(xkj ,1)|Jε(vi)v j|(y,s)dy∥∥∥∥∥L32 (0,t).∞∑k=1Jk24k∥∥Jε(vi)v j∥∥U32 ,32t. ‖v‖2U3,3t .Above we have taken B(x0,2k+1)⊂ ∪Jkj=1B(xkj,1) with Jk . 23k.Therefore, we get‖p̂‖L32 ([0,t]×B(x0, 32 )). ‖v‖2U3,3t (3.3.12)andI4 . ‖v‖3U3,3t .Combining this with (3.3.10) and taking supremum on (3.3.9) over {x0 ∈ R3},94we have‖v(t)‖2L2uloc +2‖∇v‖2U2,2t. ‖v0‖2L2uloc +ˆ t0‖v(s)‖2L2uloc ds+‖v‖3U3.3t.Then, using the interpolation inequality and Young’s inequality,‖v‖3U3,3t . ‖v‖3/2U6,2t‖v‖3/2U2,6t. ‖v‖6L6(0,t;L2uloc)+‖v‖2L2(0,t;L2uloc)+‖∇v‖2U2,2t ,we get‖v(t)‖2L2uloc +‖∇v‖2U2,2t. ‖v0‖2L2uloc +ˆ t0‖v(s)‖2L2uloc ds+ˆ t0‖v(s)‖6L2uloc ds. (3.3.13)Finally, we apply the Gro¨nwall inequality, so that there is a small ε1 > 0 suchthat, if vε exists on [0,T ] for T ≤ T0, T0 = ε1(1+‖v0‖4L2uloc)−1, then we havesup0<t<T‖vε(t)‖L2uloc . ‖v0‖L2uloc(1−Ct ‖v0‖4L2ulocmin(1,‖v0‖L2uloc)4)− 14≤ ‖v0‖L2uloc (1−Cε1)− 14 .Together with (3.3.13), this completes the proof.Lemma 3.3.6. The distributional solutions {(vε , pε)}0<ε<1 of (3.3.4) constructedin Lemma 3.3.4 can be extended to the uniform time interval [0,T0], where T0 is asin Lemma 3.3.5.Proof. We will prove it by iteration. For the convenience, we fix 0 < ε < 1 anddrop the index ε in vε and pε . Denote the uniform bound in Lemma 3.3.5 byB =C(‖v0‖L2uloc), B≥ ‖v0‖L2uloc .If an initial data v(t0) satisfies ‖v(·, t0)‖L2uloc ≤ B, by Lemma 3.3.3, we get S =S(ε,B)> 0 and a unique solution v(x, t+ t0) on R3× [0,S] to (3.3.5) satisfying‖v(t+ t0)‖ES ≤ 2C0B.95Now, we start the iteration scheme. Since ‖v0‖L2uloc ≤ B, a unique solution vexists in ES to (3.3.5). By Lemma 3.3.4 and Lemma 3.3.5, v satisfies‖v‖ES ≤ B.Then, we choose τ ∈ (34 S,S), so that ‖v(τ)‖L2uloc ≤B, and hence we obtain a solutionv˜ ∈ E (τ,τ+S) tov˜(t) = e(t−τ)∆v|t=τ +ˆ tτe(t−s)∆P∇ ·Nε(v˜)(s)ds,where we denote Nε(v) =Jε(v)⊗ vΦε .Denote the glued solution by u(x, t) = v(x, t)1[0,τ](t)+ v˜(x, t)1(τ,τ+S](t), where1E is a characteristic function of a set E ⊂ [0,∞). We claim that it solves (3.3.5) in(0,τ+S); it is obvious for t ∈ (0,τ], and for t ∈ (τ,τ+S],u(t) = v˜(t)=e(t−τ)∆(eτ∆v0+ˆ τ0e(τ−s)∆P∇ ·Nε(v)(s)ds)+ˆ tτe(t−s)∆P∇ ·Nε(v˜)(s)ds= et∆v0+ˆ τ0e(t−s)∆P∇ ·Nε(v)(s)ds+ˆ tτe(t−s)∆P∇ ·Nε(v˜)(s)ds= et∆v0+ˆ t0e(t−s)∆P∇ ·Nε(u)(s)ds.By Lemma 3.3.5 again, it satisfies‖u‖E (0,τ+S) ≤ B.By uniqueness, we get u = v for 0≤ t ≤ S. In other words, u is an extension of v.Repeat this until the extended solution exists on [0,T0]. Since at each iteration,we can extend the time interval by at least 34 S, in a finite number of iterations, wehave a distributional solution (vε , pε) of (3.3.4) on R3× [0,T0].Proof of Theorem 3.3.2. For 0< ε 1, let (vε , pε) be the distributional solution tothe localized-mollified Navier-Stokes equations (3.3.4) on R3× [0,T ] constructedin Lemmas 3.3.4 and 3.3.6, where T = T (‖v0‖L2uloc) is independent of ε . By Lemma963.3.5,‖vε‖ET ≤C(‖v0‖L2uloc).We then define pε ∈ L32loc([0,T ]×R3) bypε(x, t) =−13Jε(vε) · vε(x, t)Φε(x)+p.v.ˆB2Ki j(x− y)Nεi j(y, t)dy+p.v.ˆBc2(Ki j(x− y)−Ki j(−y))Nεi j(y, t)dy,Nεi j(y, t) =Jε(vεi )vεj(y, t)Φε(y).(3.3.14)Because Nεi j ∈L∞(0,T ;L2(R3)), the right side of (3.3.14) is defined in L∞(0,T ;L2(R3))+L∞(0,T ). Note that ∇(pε − pε) = 0 because(pε − pε)(t) =ˆBc2Ki j(−y)Jε(vεi )vεj(y, t)Φε(y)dy ∈ L32 (0,T ).Therefore, (vε , pε) is another distributional solution to the localized-mollified equa-tions (3.3.4). We will show that for each n ∈ N, pε has a bound independent of εin L32 ([0,T ]×B2n). We drop the index ε in vε and pε for a moment.For n ∈ N, we rewrite (3.3.14) for x ∈ B2n as follows.p(x, t) =− 13Jε(v) · v(x, t)Φε(x)+p.v.ˆB2Ki j(x− y)Nεi j(y, t)dy+(p.v.ˆB2n+1\B2+p.v.ˆBc2n+1)(Ki j(x− y)−Ki j(−y))Nεi j(y, t)dy= p1+ p2+ p3+ p4.All pi are defined in L∞(0,T ;L2)+L∞(0,T ).By Lemma 3.3.5, we have∥∥Nεi j∥∥U32 ,32T. ‖Jε(v)‖U3,3T ‖v‖U3,3T ≤C(‖v0‖L2uloc), (3.3.15)97and ∥∥Nεi j∥∥L 32 ([0,T ]×B2n ) . 22n ‖Jε(v)‖U3,3T ‖v‖U3,3T ≤C(n,‖v0‖L2uloc), ∀n ∈ N.(3.3.16)Then, the bound of p1 can be obtained since‖p1‖L 32 ([0,T ]×B2n ) .3∑i=1‖Nεii‖L 32 ([0,T ]×B2n ) .Using the Calderon-Zygmund theorem, we get‖p2‖L 32 ([0,T ]×B2n ) .∥∥Nεi j∥∥L 32 ([0,T ]×B2) ,and‖p31‖L 32 ([0,T ]×B2n ) .∥∥Nεi j∥∥L 32 ([0,T ]×B2n+1 ) ,wherep31(x, t) = p.v.ˆB2n+1\B2Ki j(x− y)Jε(vi)v j(y, t)Φε(y)dy.On the other hand, p32 = p3− p31 satisfies‖p32‖L 32 ([0,T ]×B2n ) . 22n∥∥∥∥ 1|y|3∥∥∥∥L3(B2n+1\B2)∥∥Nεi j∥∥L 32 ([0,T ]×B2n+1 ). 22n∥∥Nεi j∥∥L 32 ([0,T ]×B2n+1 ) .Since for x ∈ B2n and y ∈ Bc2n+1 , we have|Ki j(x− y)−Ki j(−y)|. |x||y|4 .2n|y|4 ,the bound of p4 can be obtained as‖p4‖L 32 ([0,T ]×B2n ) . 22n ‖p4‖L 32 (0,T ;L∞(B2n )) . 23n∥∥∥∥∥ˆBc2n+11|y|4 |Nεi j|(y, t)dy∥∥∥∥∥L32 (0,T ). 23n∞∑k=n+1124k∥∥Nεi j∥∥L 32 (0,T ;L1(B2k+1 )) .n ∥∥Nεi j∥∥U 32 ,1T .98Adding the estimates and using (3.3.15)-(3.3.16), we get for each n ∈ N,‖pε‖L32 ([0,T ]×B2n )≤C(n,‖v0‖L2uloc). (3.3.17)Now, we find a limit solution of (vε , pε) up to subsequence on each [0,T ]×B2n ,n ∈ N.First, construct the solution v on the compact set [0,T ]× B2. By uniformbounds on vε and the compactness argument, we can extract a sequence v1,k from{vε} satisfyingv1,k ∗⇀ v1 in L∞(0,T ;L2(B2)),v1,k ⇀ v1 in L2(0,T ;H1(B2)),v1,k→ v1 in L3(0,T ;L3(B2)),J1,k(v1,k)→ v1 in L3(0,T ;L3(B2−)),as k→ ∞. Let v = v1 on [0,T ]×B2.Then, we extend v to [0,T ]×B4 as follows. In a similar way to getting v1,we can find a subsequence {(v2,k, p2,k)}k∈N of {(v1,k, p1,k)}k∈N which satisfies thefollowing convergence:v2,k ∗⇀ v2 in L∞(0,T ;L2(B4)),v2,k ⇀ v2 in L2(0,T ;H1(B4)),v2,k→ v2 in L3(0,T ;L3(B4)),J2,k(v2,k)→ v2 in L3(0,T ;L3(B4−)),as k→∞. Here, we can easily check that v2 = v1 on [0,T ]×B2, so that v= v2 is thedesired extension. By repeating this argument, we can construct a sequence {vn,k}and its limit v. Indeed, by the diagonal argument, v can be approximated byv(k) =vk,k [0,T ]×B2k ,0 otherwise , ∀k ∈ NMore precisely, on each [0,T ]×B2n , {v(k)}∞k=n enjoys the same convergence prop-99erties as above. This follows from that {vm, j} j∈N, m ≥ n is a subsequence of{vn, j} j∈N. Indeed, for each vk,k, k ≥ n, we can find jk ≥ k such thatvk,k = vn, jk .Then, by its construction, for each n ∈ N, {v(k)}∞k=n satisfiesv(k) ∗⇀ v in L∞(0,T ;L2(B2n)), (3.3.18)v(k) ⇀ v in L2(0,T ;H1(B2n)), (3.3.19)v(k)→ v in L3(0,T ;L3(B2n)), (3.3.20)J(k)(v(k))→ v in L3(0,T ;L3(B2n−)) (3.3.21)as k→ ∞. Furthermore, since vε are uniformly bounded in ET , we can easily seethat v ∈ ET and v ∈U3,3T ,‖v‖ET +‖v‖U3,3T ≤C(‖v0‖L2uloc).Now, we construct a pressure p corresponding to v. Using (3.3.14), we definep(k) byp(k)(x, t) =− 13J(k)(v(k)) · v(k)(x, t)Φ(k)(x)+p.v.ˆB2Ki j(x− y)J(k)(v(k)i )v(k)j (y, t)Φ(k)(y)dy+p.v.ˆBc2(Ki j(x− y)−Ki j(−y))J(k)(v(k)i )v(k)j (y, t)Φ(k)(y)dy.(3.3.22)where Φ(k) =Φεk for εk satisfying vk,k = vεk . Also definep(x, t) = limn→∞ pn(x, t) (3.3.23)where pn(x, t) is defined for |x|< 2n bypn(x, t) =− 13|v(x, t)|2+p.v.ˆB2Ki j(x− y)viv j(y, t)dy+ pn3+ pn4, (3.3.24)100withpn3(x, t) = p.v.ˆB2n+1\B2(Ki j(x− y)−Ki j(−y))viv j(y, t)dy,pn4(x, t) =ˆBc2n+1(Ki j(x− y)−Ki j(−y))viv j(y, t)dy.The first two terms in pn are defined in U32 ,32T . By estimates similar to those for pε ,we get pn3, pn4 ∈ L3/2((0,T )×B2n) andpn3+ pn4 = pn+13 + pn+14 , in L3/2((0,T )×B2n)Thus pn(x, t) is independent of n for n > log2 |x|.Our goal is to show that the strong convergences (3.3.20)-(3.3.21) of {v(k)}givesp(k)→ p in L 32 ([0,T ]×B2n), for each n ∈ N, (3.3.25)Let N(k)i j =J(k)(v(k)i )v(k)j Φ(k) and Ni j = viv j. For any fixed R > 0, we have∥∥∥N(k)i j −Ni j∥∥∥L 32 ([0,T ]×BR)≤∥∥∥(J(k)(v(k)i )− vi)v(k)j Φ(k)∥∥∥L 32 ([0,T ]×BR)+∥∥∥vi(v(k)j − v j)Φ(k)∥∥∥L 32 ([0,T ]×BR)+∥∥viv j(1−Φ(k))∥∥L 32 ([0,T ]×BR).∥∥∥J(k)(v(k))− v∥∥∥L3([0,T ]×BR)∥∥∥v(k)∥∥∥L3([0,T ]×BR)+∥∥∥v(k)− v∥∥∥L3([0,T ]×BR)‖v‖L3([0,T ]×BR)+∥∥|v|2(1−Φ(k))∥∥L 32 ([0,T ]×BR) −→ 0(3.3.26)by (3.3.20), (3.3.21), and Lebesgue dominated convergence theorem. Then, it pro-101vides the convergence of p(k) to p: On [0,T ]×B2n , for m > n,p(k)− p =− 13tr(N(k)−N)+p.v.ˆB2Ki j(·− y)(N(k)i j −Ni j)(y)dy+[p.v.ˆB2n+1\B2+ˆB2m\B2n+1+ˆBc2m](Ki j(·− y)−Ki j(−y))(N(k)i j −Ni j)(y)dy= q1+q2+q3+q4+q5.In a similar way to getting (3.3.17), we have‖q1,q2,q3‖L 32 ([0,T ]×B2n ) .n∥∥∥N(k)−N∥∥∥L32 ([0,T ]×B2n+1 ),and‖q4‖L 32 ([0,T ]×B2n ) .∥∥∥N(k)−N∥∥∥L32 ([0,T ]×B2m ),On the other hand, using|Ki j(x− y)−Ki j(−y)|. |x||y|4we obtain‖q5‖L 32 ([0,T ]×B2n ) .23n2m(‖v‖2U3,3T +∥∥∥J(k)(v(k))∥∥∥U3,3T∥∥∥v(k)∥∥∥U3,3T)≤C(n,‖v0‖L2uloc ,T )12m.Therefore, for fixed n, if we choose sufficiently large m, we can make q5 verysmall in L32 ([0,T ]×B2n) and then for sufficiently large k, q1, q2, q3, and q4 alsobecome very small in L32 ([0,T ]×B2n) because of (3.3.26). This gives the desiredconvergence (3.3.25) of p(k) to p.Now, we check that (v, p) is a local energy solution. It is easy to prove that(v, p) solves the Navier-Stokes equation in the distributional sense by using thedistributional form of (3.3.4) for (v(k), p(k)) and the convergence (3.3.18)-(3.3.21)102and (3.3.25)-(3.3.26). For example, for any ξ ∈C∞c ((0,T )×R3;R3),ˆ T0ˆv(k) ·∂tξdxdt→ˆ T0ˆv ·∂tξdxdtˆ T0ˆJ(k)(v(k))(v(k)Φ(k)) : ∇ξdxdt→ˆ T0ˆv⊗ v : ∇ξdxdt as k→ ∞.Since we haveˆ t0ˆ(∆v− (v ·∇)v−∇p) ·φdxdt≤∣∣∣∣ˆ t0ˆ∇v ·∇φdxdt∣∣∣∣+ ∣∣∣∣ˆ t0ˆv(v ·∇)φdxdt∣∣∣∣+ ∣∣∣∣ˆ t0ˆpdivφdxdt∣∣∣∣.‖∇v‖L2(0,T ;L2(B2n )) ‖∇φ‖L2(0,T ;L2(R3))+(‖v‖2L3(0,T ;L3(B2n ))+‖p‖L 32 (0,T ;L 32 (B2n )))‖∇φ‖L3(0,T ;L3(R3))≤C(n,T,‖v0‖L2uloc)‖∇φ‖L3(0,T ;L3(R3)) ,for any φ ∈C∞c ([0,T ]×B2n), n ∈ N, it follows that∂tv = ∆v− (v ·∇)v−∇p ∈ Xnfor any n ∈ N, where Xn is the dual space of L3(0,T ;W 1,30 (B2n)).With this bound of ∂tv, for each n∈N, we may redefine v(t) on a measure-zerosubset Σn of [0,T ] such that the functiont 7−→ˆR3v(x, t) ·ζ (x)dx (3.3.27)is continuous for any vector ζ ∈ C∞c (B2n). Redefine v(t) recursively for all n sothat (3.3.27) is true for any ζ ∈C∞c (R3). It is then true for any ζ ∈ L2(R3) with acompact support using v ∈ L∞(0,T ;L2uloc).Furthermore, consider the local energy equality (3.3.8) for (v(k), p(k)) on thetime interval (0,T ) for a non-negativeψ ∈C∞c ([0,T )×R3). The first term´ |v(k)|2ψ(x,T )dxvanishes. Taking limit infimum as k goes to infinity, and using the weak conver-gence (3.3.19) and the strong convergence (3.3.20)-(3.3.21) and (3.3.25)-(3.3.26),103we get2ˆ T0ˆ|∇v|2ψ dxds≤ˆ|v0|2ψ(·,0)dx+ˆ T0ˆ|v|2(∂sψ+∆ψ)+(|v|2+2 p̂)(v ·∇)ψ dxds,(3.3.28)for any non-negative ψ ∈C∞c ([0,T )×R3).Then, for any t ∈ (0,T ) and non-negative ϕ ∈C∞c ([0,T )×R3), take ψ(x,s) =ϕ(x,s)θε(s), ε 1, where θε(s) = θ( s−tε)for some θ ∈C∞(R) such that θ(s) = 1for s ≤ 0 and θ(s) = 0 for s ≥ 1, and θ ′(s) ≤ 0 for all s. Note that θε(s) = 1 fors≤ t and θε(s) = 0 for s≥ t+ ε . Sending ε → 0 and usingˆ|v(t)|2ϕ dx≤ liminfε→0ˆ t0ˆ|v|2ϕ(−θ ′ε)dxdsdue to the weak local L2-continuity (3.3.27), we getˆ|v(t)|2ϕ dx+2ˆ t0ˆ|∇v|2ϕ dxds≤ˆ|v0|2ϕ(·,0)dx+ˆ t0ˆ {|v|2(∂sϕ+∆ϕ)+(|v|2+2 p̂)(v ·∇)ϕ}dxds(3.3.29)for any t ∈ (0,T ) and non-negative ϕ ∈C∞c ([0,T )×R3). The local energy inequal-ity (3.3.1) is a special case of (3.3.29) for test functions vanishing at t = 0.Sending t→ 0+ in (3.3.29) we get limsupt→0+´ |v(t)|2ϕ dx≤ ´ |v0|2ϕ(·,0)dxfor any non-negative ϕ ∈C∞c . Together with the weak continuity (3.3.27), we getlimt→0+´Bn|v(x, t)− v0(x)|2dx = 0 for any n ∈ N.Finally, we consider the decomposition of the pressure. Recall that the pres-sure p is defined recursively by (3.3.23)-(3.3.24). For any x0 ∈ R3 define p̂x0 ∈104L32 ([0,T ]×B(x0, 32)) by (3.3.3), i.e.,p̂x0(x, t) =−13|v|2(x, t)+p.v.ˆB(x0,2)Ki j(x− y)viv j(y, t)dy+ˆB(x0,2)c(Ki j(x− y)−Ki j(x0− y))viv j(y, t)dy.Let cx0 = p− p̂x0 . If B(x0, 32)⊂ B2n , thencx0(t) =ˆB2n+1\B(x0,2)Ki j(x0− y)viv j(y, t)dy−ˆB2n+1\B2Ki j(−y)viv j(y, t)dy+ˆBc2n+1(Ki j(x0− y)−Ki j(−y))viv j(y, t)dy.(3.3.30)Note that cx0 ∈ L3/2(0,T ), and cx0(t) is independent of x ∈ B(x0, 32), n, and T .Therefore, we get the desired decomposition (3.3.2) of the pressure.Remark 3.3.1. Our approach in this section is similar to that in Kikuchi-Seregin[26]. However, there are two significant differences:1. Since we include initial data v0 not in E2, we add an additional localizationfactor Φ(k) to the nonlinearity in the localized-mollified equations (3.3.4).Our approximation solutions vε live in L2uloc and are no longer in the usualenergy class.2. The pressure p and cx0 are implicit in [26], but are explicit in this chapter. Wefirst specify the formula (3.3.23) of the pressure and then justify the strongconvergence and decomposition. In particular, our cx0(t) is given by (3.3.30)and independent of T .Remark 3.3.2. Estimate (3.3.12) and its proof for p̂εx0 are not limited to our approx-imation solutions. They are in fact also valid for any local energy solution (v, p) in(0,T ) with local pressure p̂x0 given by (3.3.3), that is,‖ p̂x0‖L 32 ([0,t]×B(x0, 32 )) ≤C‖v‖2U3,3t, ∀t < T, (3.3.31)105with a constant C independent of t,T .3.4 Spatial decay estimatesRecall that our initial data v0 ∈E2σ+L3uloc,σ . In Sections 3.4 and 3.5, we decomposev0 = w0+u0, w0 ∈ E2σ , u0 ∈ L3uloc,σ . (3.4.1)Our goal in this section is to show that, although the solution v has no spatial decay,its difference from the linear flow, w = v−V , V (t) = et∆u0, does decay due to thedecay of the oscillation of u0. Here, the oscillation decay of u0 follows from that ofv0 and w0 ∈ E2. The main task is to show that the contribution from the nonlinearsource term(V ·∇)V = ∇ · (V ⊗V )has decay, although V itself does not. On the other hand, we also need the decayof the pressure. However, p̂x0 given by (3.3.3) does not decay. Thus we need adifferent decomposition of the pressure p near each point x0 ∈ R3.Lemma 3.4.1 (New pressure decomposition). Let v0 = w0+u0 with w0 ∈ E2σ andu0 ∈ L3uloc,σ . Let (v, p) be any local energy solution of (NS) with initial data v0 inR3× (0,T ), 0 < T < ∞. Then, for each x0 ∈ R3, we can find qx0 ∈ L32 (0,T ) suchthatp(x, t) = qpx0(x, t)+qx0(t) in L 32 ((0,T )×B(x0, 32))106whereqpx0 =− 13(|w|2+2w ·V )+p.v.ˆB(x0,2)Ki j(·− y)(wiw j +Viw j +wiVj)(y)dy+ˆB(x0,2)c(Ki j(·− y)−Ki j(x0− y))(wiw j +Viw j +wiVj)(y)dy+ˆKi(·− y)[(V ·∇)Vi]ρ2(y)dy+ˆ(Ki j(·− y)−Ki j(x0− y))ViVj(1−ρ2)(y)dy+ˆ(Ki(·− y)−Ki(x0− y))ViVj(∂ jρ2)(y)dy.(3.4.2)Here, w = v−V , V (t) = et∆u0, Ki = ∂iK, Ki j = ∂i jK, K(x) = 14pi|x| , and ρ2 =Φ( ·−x02 ).Proof. Consider (x, t) ∈ B(x0, 32)× (0,T ). Let Fi j = wiw j +Viw j +wiVj and Gi j =ViVj. Substituting v =V +w in (3.3.3), we getp̂x0 = pFx0 + pGx0pFx0 =−13trF +p.v.ˆB(x0,2)Ki j(·− y)Fi j(y)dy+ˆB(x0,2)c(Ki j(·− y)−Ki j(x0− y))Fi j(y)dy(3.4.3)andpGx0 =−13trG+p.v.ˆB(x0,2)Ki j(·− y)Gi j(y)dy+ˆB(x0,2)c(Ki j(·− y)−Ki j(x0− y))Gi j[ρ2+(1−ρ2)](y)dy=−13trG+p.v.ˆKi j(·− y)Gi jρ2(y)dy+ pGx0,far+ q˜x0(t),107wherepGx0,far =ˆ(Ki j(·− y)−Ki j(x0− y))Gi j(1−ρ2)(y)dy,q˜x0(t) =−ˆB(x0,2)cKi j(x0− y)Gi jρ2(y)dy.Integrating by parts the principle value integral, we getpGx0 =ˆKi(·− y)∂ j[Gi jρ2(y)]dy+ pGx0,far+ q˜x0(t).Note ∂ j[Gi jρ2] = (V ·∇Vi)ρ2+Gi j∂ jρ2. Denoteq̂x0(t) =ˆKi(x0− y)ViVj(∂ jρ2)(y)dy.We getp̂x0(x, t) = pFx0 +ˆKi(·− y)(V ·∇)Viρ2(y)dy+ pGx0,far+ q˜x0(t)+ˆ(Ki(·− y)−Ki(x0− y))ViVj(∂ jρ2)(y)dy+ q̂x0(t)= qpx0(x, t)+ q˜x0(t)+ q̂x0(t).(3.4.4)Thus we have p(x, t) = qpx0(x, t)+qx0(t) withqx0(t) = cx0(t)+ q˜x0(t)+ q̂x0(t).Note that using ‖G‖U∞,1T ≤ ‖V‖2U∞,2Tand |x0− y|> 2 for y ∈ supp(∂ jρ2), we have‖q˜x0(t)‖L∞(0,T )+‖q̂x0(t)‖L∞(0,T ) .∥∥∥∥ˆB(x0,3)\B(x0,2)|Gi j|(y)dy∥∥∥∥L∞(0,T ). ‖G‖L∞(0,T ;L1(B(x0,3))) . ‖V‖2U∞,2T .(3.4.5)Since q˜x0(t)+ q̂x0(t) is in L3/2(0,T ), so is qx0(t).Although ∇V has spatial decay, it is not uniform in t. Thus, to show the spatialdecay of w, we will first show (3.1.6), i.e., the smallness of w in L2uloc at far distance108for a short time in Lemma 3.4.5. For that we need Lemmas 3.4.2, 3.4.3 and 3.4.4.Lemma 3.4.2. For u0 ∈ L3(R3), if 2s + 3q = 1 and 3≤ q < 9, then∥∥et∆u0∥∥Ls(0,∞;Lq(R3)) ≤Cq ‖u0‖L3(R3) .This is proved in Giga [17].Lemma 3.4.3. Suppose u0 ∈ L2uloc and u0 ∈ L3(B(x0,3)). Then, V = et∆u0 satisfies‖V‖L8(0,T ;L4(B(x0, 32 ))) . ‖u0‖L3(B(x0,3))+T18 ‖u0‖L2uloc . (3.4.6)Proof. Let φ(x) =Φ( x−x02 ). Decomposeu0 = u0φ +u0(1−φ) =: u1+u2.By Lemma 3.4.2,∥∥et∆u1∥∥L8(0,T ;L4(B(x0, 32 ))) ≤ ∥∥et∆u1∥∥L8(0,T ;L4(R3)) . ‖u1‖L3(R3) ≤ ‖u0‖L3(B(x0,3)) .(3.4.7)On the other hand, we have∥∥et∆u2∥∥L8(0,T ;L4(B(x0, 32 ))) . ∥∥∇et∆u2∥∥L8(0,T ;L2(B(x0, 32 )))+∥∥et∆u2∥∥L8(0,T ;L2(B(x0, 32 ))) .Obviously,∥∥et∆u2∥∥L8(0,T ;L2(B(x0, 32 ))) . T 18 ∥∥et∆u2∥∥L∞(0,T ;L2uloc) . T 18 ‖u2‖L2uloc .109Using supp(u2)⊂ B(x0,2)c and heat kernel estimate, we get∥∥∇et∆u2∥∥L8(0,T ;L2(B(x0, 32 ))) . T 18 ∥∥∇et∆u2∥∥L∞((0,T )×B(x0, 32 )). T 18ˆB(x0,2)c1|x0− y|4 |u0(y)|dy. T 18∞∑k=1ˆB(x0,2k+1)\B(x0,2k)124k|u0(y)|dy. T 18 ‖u0‖L2uloc .Therefore, we obtain∥∥et∆u2∥∥L8(0,T ;L4(B(x0, 32 ))) . T 18 ‖u0‖L2uloc .Together with (3.4.7), we get (3.4.6).The perturbation w= v−V , V (t) = et∆u0, satisfies the perturbed Navier-Stokesequations in the sense of distributions,∂tw−∆w+(V +w) ·∇(V +w)+∇p = 0divw = 0w|t=0 = w0.(3.4.8)It also satisfies the following local energy inequality for test functions supportedaway from t = 0.Lemma 3.4.4 (Local energy inequality for w). Let v0,u0 ∈ L2uloc,σ . Let (v, p) beany local energy solution of (NS) with initial data v0 in R3× (0,T ), 0 < T < ∞.Then w(t) = v(t)− et∆u0 satisfiesˆ|w|2ϕ(x, t)dx+2ˆ t0ˆ|∇w|2ϕ dxds≤ˆ t0ˆ|w|2(∂sϕ+∆ϕ+ v ·∇ϕ)dxds+ˆ t0ˆ2pw ·∇ϕ dxds+ˆ t0ˆ2V · (v ·∇)(wϕ)dxds,(3.4.9)110for any non-negative ϕ ∈C∞c ((0,T )×R3) and any t ∈ (0,T ).Note that ϕ vanishes near t = 0. If ϕ does not vanish near t = 0, the last integralin (3.4.9) may not be defined.Proof. Recall that we have the local energy inequality (3.3.1) for (v, p). The equiv-alent form for (w, p) is exactly (3.4.9). Indeed, (3.3.1) and (3.4.9) are equivalentbecause they differ by an equality which is the sum of the weak form of the V -equation with 2vϕ as the test function and the weak form of the w-equation (3.4.8)with 2Vϕ as the test function, after suitable integration by parts. This equalitycan be proved because V and ∇V are in L∞loc(0,T ;L∞(R3)), and ϕ has a compactsupport in space-time.For r > 0, letχr(x) = 1−Φ(xr),so that χr(x) = 1 for |x| ≥ 2r and χr(x) = 0 for |x| ≤ r.Lemma 3.4.5. Let v0 = w0 + u0 with w0 ∈ E2σ and u0 ∈ L3uloc,σ . Let (v, p) be anylocal energy solution of (NS) with initial data v0 in R3× (0,T ), 0 < T < ∞. Then,there exist T0 = T0(‖v0‖L2uloc) ∈ (0,1) and C0 = C0(‖w0‖L2uloc ,‖u0‖L3uloc) > 0 suchthat w(t) = v(t)− et∆u0 satisfies‖w(t)χR‖L2uloc ≤C0(t120 +‖w0χR‖L2uloc), (3.4.10)for any R > 0 and any t ∈ (0,T1), T1 = min(T0,T ).In this lemma, we do not assume the oscillation decay.Proof. By Lemma 3.2.4 and similar to (3.3.7), we can find T0 = T0(‖v0‖2L2uloc) ∈(0,1) such that, for T1 = min(T0,T ),‖w‖ET1 +‖V‖ET1 . ‖w0‖L2uloc +‖u0‖L2uloc .By interpolation, it follows that for any 2≤ s≤∞, and 2≤ q≤ 6 satisfying 2s + 3q =11132 , we have‖w‖U s,qT1 +‖V‖U s,qT1 . ‖w0‖L2uloc +‖u0‖L2uloc .On the other hand, by Lemma 3.4.3, for any t ∈ (0,1),‖V‖U8,4t . ‖u0‖L3uloc .Let A = ‖w0‖L2uloc + ‖u0‖L3uloc . Then, both inequalities can be combined for t ≤ T1as‖w‖Et +‖V‖Et +‖w‖U 103 , 103t+‖V‖U103 ,103t+‖V‖U8,4t . A. (3.4.11)Fix x0 ∈ R3 and R > 0, and letφx0 =Φ(·− x0), ξ = φ 2x0χ2R. (3.4.12)Fix Θ ∈C∞(R), Θ′ ≥ 0, Θ(t) = 1 for t > 2, and Θ(t) = 0 for t < 1. Define θε ∈C∞c (0,T ) for sufficiently small ε > 0 byθε(s) =Θ( sε)−Θ(s−T +3εε).Thus θε(s) = 1 in (2ε,T −2ε) and θε(s) = 0 outside of (ε,T − ε). We now con-sider the local energy inequality (3.4.9) for w with ϕ(x,s) = ξ (x)θε(s). We mayreplace p by p̂x0 in (3.4.9) as supp ξ ⊂ B(x0, 32) and˜cx0(t)w ·∇ξ dxdt = 0. Wenow take ε → 0+. Since ‖v(t)− v0‖L2(B2(x0))→ 0 and ‖V (t)−u0‖L2(B2(x0))→ 0 ast→ 0+, we get ˆ 2ε0ˆ|w|2ξ (θε)′ dxds→ˆ|w0|2ξdx.The last term in (3.4.9) converges by Lebesgue dominated convergence theoremusingˆ t0ˆ|V · (v ·∇)(wξ )|dxds. ‖V‖L8(0,T ;L4(B(x0, 32 ))) ‖v‖U8/3,4T (‖∇w‖U2,2T +‖w‖U2,2T ),112where the right hand side of the inequality is bounded independently of ε .In the limit ε → 0+, for any t ∈ (0,T ), we getˆ|w|2(x, t)ξ (x)dx+2ˆ t0ˆ|∇w|2ξ dxds≤ˆ|w0|2ξ dx+ˆ t0ˆ|w|2(∆ξ + v ·∇ξ )dxds+ˆ t0ˆ2 p̂x0w ·∇ξ dxds+ˆ t0ˆ2V · (v ·∇)(wξ )dxds,(3.4.13)for ξ given by (3.4.12). Now, we consider t ≤ T1. Using (3.4.11), we haveˆ t0ˆ|w|2∆ξdxds. ‖w‖2U2,2t . A2t,ˆ t0ˆ|w|2(v ·∇)ξdxds. ‖v‖U3,3t ‖w‖2U3,3t. A3t 110 .For the convenience, we suppress the indexes x0 and R in φx0 , p̂x0 and χR. Byadditionally using (3.3.31),ˆ t0ˆp̂w ·∇ξdxds.ˆ t0ˆB(x0, 32 )|p̂||w|dxds. ‖ p̂‖L32 ([0,t]×B(x0, 32 ))‖w‖U3,3t. ‖v‖2U3,3t ‖w‖U3,3t . A3t110 .To estimate the last term in (3.4.13), we decompose it asˆ t0ˆV · (v ·∇)(wξ )dxds = I1+ I2+ I3=ˆ t0ˆξV · (V ·∇)wdxds+ˆ t0ˆξV · (w ·∇)wdxds+ˆ t0ˆV ·w(v ·∇)ξ dxds.We have|I1|. ‖V‖2L4(0,T ;L4(supp(ξ ))) ‖∇w‖U2,2t . A3t14 .113On the other hand, by Poincare´ inequality, we haveˆ t0‖wφχ‖2L6 ds.ˆ t0‖∇(wφχ)‖2L2 ds+ˆ t0‖wφχ‖2L2 ds.ˆ t0‖|∇w|φχ‖2L2 ds+‖w‖2U2,2t ,which follows that (using Young’s inequality abc≤ εa2+ εb8/3+C(ε)c8)|I2| ≤ˆ t0‖|∇w|φχ‖L2 ‖wφχ‖L4 ‖V‖L4(supp(ξ )) ds≤ˆ t0‖|∇w|φχ‖L2 ‖wφχ‖34L6 ‖wφχ‖14L2 ‖V‖L4(supp(ξ )) ds≤ εˆ t0(‖|∇w|φχ‖2L2 +‖wφχ‖2L6)ds+C(ε)ˆ t0‖V‖8L4(supp(ξ )) ‖wφχ‖2L2 ds≤ 1100ˆ t0‖|∇w|φχ‖2L2 ds+A2t+Cˆ t0‖V‖8L4(supp(ξ )) ‖wφχ‖2L2 dsby choosing suitable ε . It is easy to control I3:|I3|. t 110 ‖V‖U103 ,103t‖v‖U103 ,103t‖w‖U103 ,103t. A3t 110 .Therefore, we obtain∣∣∣∣ˆ t0ˆV · (v ·∇)(wξ )dxds∣∣∣∣≤‖|∇w|φχ‖2L2([0,t]×R3)+C(1+A3)(t110 +ˆ t0‖V‖8L4(supp(ξ )) ‖wφχ‖2L2 ds),for some absolute constant C. Finally, we combine all the estimates to get from(3.4.13) that‖w(t)φχ‖2L2(R3)+‖|∇w|φχ‖2L2([0,t]×R3). ‖w0χR‖2L2uloc +(1+A3)(t110 +ˆ t0‖V‖8L4(supp(ξ )) ‖wφχ‖2L2 ds)114Note that ‖w(t)φχ‖2L2(R3) is lower semicontinuous in t as wφ is weakly L2-continuous in t. By Gro¨nwall’s inequality and (3.4.11), we have‖w(t)φχ‖2L2(R3) ≤C20(‖w0χR‖2L2uloc + t110 ),for some C0 =C0(A)> 0. Taking supremum in x0, we get‖w(t)χR‖L2uloc ≤C0(t120 +‖w0χR‖L2uloc).This finishes the proof of Lemma 3.4.5.Lemma 3.4.6 (Strong local energy inequality). Let (v, p) be a local energy solutionin R3× (0,T ) to the Navier-Stokes equations (NS) for the initial data v0 ∈ L2ulocconstructed in Theorem 3.3.2, as the limit of approximation solutions (v(k), p(k)) of(3.3.4). Then there is a subset Σ ⊂ (0,T ) of zero Lebesgue measure such that, forany t0 ∈ (0,T )\Σ and any t ∈ (t0,T ), we haveˆ|v|2ϕ(x, t)dx+2ˆ tt0ˆ|∇v|2ϕ dxds≤ˆ|v|2ϕ(x, t0)dx+ˆ tt0ˆ {|v|2(∂sϕ+∆ϕ)+(|v|2+2p)v ·∇ϕ}dxds,(3.4.14)for any ϕ ∈C∞c (R3× [t0,T )). If, furthermore, for some u0 ∈ L2uloc,σ , V (t) = et∆u0and w = v−V , then for any t0 ∈ (0,T )\Σ and any t ∈ (t0,T ), we haveˆ|w|2ϕ(x, t)dx+2ˆ tt0ˆ|∇w|2ϕ dxds≤ˆ|w|2ϕ(x, t0)dx+ˆ tt0ˆ|w|2(∂sϕ+∆ϕ+ v ·∇ϕ)dxds+ˆ tt0ˆ2pw ·∇ϕ dxds−ˆ tt0ˆ2(v ·∇)V ·wϕ dxds,(3.4.15)for any ϕ ∈C∞c (R3× [t0,T )).This lemma is not for general local energy solutions, but only for those con-115structed by the approximation (3.3.4). Also note that (3.4.14) is true for t0 = 0since it becomes (3.3.1), but (3.4.15) is unclear for t0 = 0 since the last integral in(3.4.15) may not be defined without further assumptions; Compare it with (3.4.13).Proof. For any n ∈ N, the approximations v(k) satisfylimk→∞∥∥∥v(k)− v∥∥∥L2(0,T ;L2(Bn))= 0.Thus there is a set Σn ⊂ (0,T ) of zero Lebesgue measure such thatlimk→∞∥∥∥v(k)(t)− v(t)∥∥∥L2(Bn)= 0, ∀t ∈ [0,T )\Σn.LetΣ= ∪∞n=1Σn, |Σ|= 0.We getlimk→∞∥∥∥v(k)(t)− v(t)∥∥∥L2(Bn)= 0, ∀t ∈ [0,T )\Σ, ∀n ∈ N. (3.4.16)The local energy equality of (v(k), p(k)) in [t0,T ] is derived similarly to (3.3.8)2ˆ Tt0ˆ|∇v(k)|2ψdxds =ˆ|v(k)|2ψ(x, t0)dx+ˆ Tt0ˆ|v(k)|2(∂sψ+∆ψ)dxds+ˆ Tt0ˆ|v(k)|2Φ(k)(J(k)(v(k)) ·∇)ψdxds+ˆ Tt0ˆ2p(k)v(k) ·∇ψ dxds−ˆ Tt0ˆ|v(k)|2ψ(J(k)(v(k)) ·∇)Φ(k)dxds,(3.4.17)for any ψ ∈C∞c (R3× [0,T )). By (3.4.16), we havelimk→∞ˆ|v(k)|2ψ(x, t0)dx =ˆ|v|2ψ(x, t0)dx116for t0 ∈ [0,T )\Σ. Taking limit infimum k→ ∞ in (3.4.17), we get2ˆ Tt0ˆ|∇v|2ψ dxds≤ˆ|v|2ψ(x, t0)dx+ˆ Tt0ˆ {|v|2(∂sψ+∆ψ)+(|v|2+2p)v ·∇ψ}dxds.By the same argument for (3.3.29), we get (3.4.14) from the above.Finally, inequality (3.4.15) for t0 > 0 is equivalent to (3.4.14) by the sameargument of Lemma 3.4.4. We have integrated by parts the last term in (3.4.15),which is valid since ∇V ∈ L∞(R3× (t0, t)).We now prove the main result of this section.Proposition 3.4.7 (Decay of w and qp). Let v0 = w0 + u0 with w0 ∈ E2σ and u0 ∈L3uloc,σ , andlim|x0|→∞ˆB(x0,1)|v0− (v0)B(x0,1)|dx = 0.Let (v, p) be a local energy solution in R3× (0,T ) to the Navier-Stokes equations(NS) for the initial data v0 ∈ L2uloc constructed in Theorem 3.3.2, as the limit ofapproximation solutions (v(k), p(k)) of (3.3.4). Let w = v−V for V (t) = et∆u0.Then, w and qpx0 , defined in Lemma 3.4.1, decay at spatial infinity: For any t1 ∈(0,T ),lim|x0|→∞(‖w‖L∞t L2x∩L3(Qx0 )+‖∇w‖L2(Qx0 )+‖qpx0‖L 32 (Qx0 ))= 0, (3.4.18)where Qx0 = B(x0,32)× (t1,T ).Note that we do not assert uniform decay up to t1 = 0. We assume the ap-proximation (3.3.4) only to ensure the conclusion of Lemma 3.4.6, the strong localenergy inequality.Proof. Choose A = A(‖w0‖L2uloc ,‖u0‖L2uloc ,T ) such that‖w‖ET +‖V‖ET +‖w‖U s,qT +‖V‖U s,qT . A,117for any 2≤ s≤ ∞, and 2≤ q≤ 6 satisfying 2s + 3q = 32 .Fix x0 ∈ Z3 and R ∈ N. Let φx0 =Φ(·− x0), χR(x) = 1−Φ( xR), andξ = φ 2x0χ2R.For the convenience, we suppress the indexes x0 and R in φx0 , qpx0 and χR.Let Σ be the subset of (0,T ) defined in Lemma 3.4.6. For any t0 ∈ (0, t1) \Σand t ∈ (t0,T ), choose θ(t)∈C∞c (0,T ) with θ = 1 on [t0, t]. Let ϕ(x, t) = ξ (x)θ(t).By (3.4.15) of Lemma 3.4.6, using t0 6∈ Σ, we haveˆ|w(x, t)|2ξ (x)dx+2ˆ tt0ˆ|∇w|2ξ dxds≤ˆ|w(x, t0)|2ξ (x)dx+ˆ tt0ˆ|w|2(∆ξ +(v ·∇)ξ )dxds+2ˆ tt0ˆ qpx0w ·∇ξ dxds−2ˆ tt0ˆ(v ·∇)V ·wξ dxds.(3.4.19)Above we have replaced p by qpx0 using ˜ qx0(t)w ·∇ξ dxds = 0.By the choice of ξ , we can easily see thatˆ|w(·, t)|2ξdx+2ˆ tt0ˆ|∇w|2ξdxds≥‖w(·, t)χ‖2L2(B(x0,1))+2‖|∇w|χ‖2L2([t0,t]×B(x0,1)) ,ˆ|w(·, t0)|2ξdx. ‖w(·, t0)χ‖2L2uloc ,ˆ tt0ˆ|w|2∆ξdxds. ‖wχ‖2U2,2(t0,t)+1R‖w‖2U2,2T ,andˆ tt0ˆ|w|2(v ·∇)ξdxds. ‖v‖U3,3T ‖wχ‖2U3,3(t0,t)+1R‖v‖U3,3T ‖w‖2U3,3T.A ‖wχ‖2U3,3(t0,t)+1R.118The last term can be also estimated by∣∣∣∣ˆ tt0ˆ(v ·∇)V ·wξdxds∣∣∣∣. ‖|∇V |χ‖U∞,3(t0,T ) ‖v‖U2,6T ‖wχ‖U2,2(t0,t).A ‖wχ‖2U2,2(t0,t)+‖|∇V |χ‖2U∞,3(t0,T ) .The only remaining term is the one with pressure. Noteˆ tt0ˆ qpw ·∇ξdxds. ˆ tt0ˆB(x0, 32 )|qp||w|χ2dxds+ 1Rˆ tt0ˆB(x0, 32 )|qp||w|χdxds. ‖qpχ‖L32 ([t0,t]×B(x0, 32 ))‖wχ‖U3,3(t0,t)+1R‖qp‖L32 ([0,T ]×B(x0, 32 ))‖wχ‖U3,3T .For the second term, we can use a bound uniform in x0‖qpx0‖L 32 ([0,t]×B(x0, 32 )) ≤C‖v‖2U3,3t +C(T )‖V‖2U∞,2T ,which follows from (3.3.31), (3.4.4) and (3.4.5). For the first term, although theother factor ‖wχ‖U3,3(t0,t) also has decay, it is larger than the left side of (3.4.19) byitself. Hence we need to estimate ‖qpχ‖L32 ([t0,t]×B(x0, 32 ))and show its decay.Let Fi j = wiw j +wiVj +w jVi and Gi j = ViVj. The local pressure qp defined inLemma 3.4.1 can be further decomposed asqp(x, t) = pF + pG,1+ pG,2+ pG,3where pF = pFx0 is defined as in (3.4.3),pG,1 =ˆKi(·− y)[∂ jGi j]ρ2(y)dy,pG,2 =ˆ(Ki j(·− y)−Ki j(x0− y))Gi j(ρτ −ρ2)(y)dy−ˆ(Ki(·− y)−Ki(x0− y))Gi j∂ j(ρτ −ρ2)(y)dy,119for ρτ =Φ( ·−x0τ), τ > 4, andpG,3 =ˆ(Ki j(·− y)−Ki j(x0− y))Gi j(1−ρτ)(y)dy+ˆ(Ki(·− y)−Ki(x0− y))Gi j∂ jρτ(y)dy.Recall pF = pFx0pF =−13trF +p.v.ˆB(x0,2)Ki j(·− y)Fi j(y)dy+ˆB(x0,2)c(Ki j(·− y)−Ki j(x0− y))Fi j(y)dy= pF,1+ pF,2+ pF,3.We estimate pF,iχ , i = 1,2,3. Obviously, we have∥∥pF,1χ∥∥L32 ([t0,t]×B(x0, 32 )). ‖Fχ‖U32 ,32 (t0,t).Using the Lp-norm preservation of Riesz transfroms and ‖∇χ‖∞ . 1R ,∥∥pF,2χ∥∥L32 ([t0,t]×B(x0, 32 ))≤∥∥∥∥p.v.ˆB(x0,2)Ki j(·− y)Fi j(y)χ(y)dy∥∥∥∥L32 ([t0,t]×B(x0, 32 ))+∥∥∥∥p.v.ˆB(x0,2)Ki j(·− y)Fi j(y)(χ(·)−χ(y))dy∥∥∥∥L32 ([t0,t]×B(x0, 32 )).‖Fχ‖U32 ,32 (t0,t)+1R∥∥∥∥ˆB(x0,2)1| ·−y|2 |Fi j(y)|dy∥∥∥∥L32 (t0,t;L3(R3))).‖Fχ‖U32 ,32 (t0,t)+1R‖F‖U32 ,32 (t0,t).The last inequality follows from the Riesz potential estimate. Since|χ(x)−χ(y)| ≤ ‖∇χ‖∞ |x− y|.1√R120for x ∈ B(x0, 32) and y ∈ B(x0,√R), and|x− y| ≥ |x0− y|− |x− x0| ≥ 14 |x0− y|for x ∈ B(x0, 32) and y ∈ B(x0,2)c, we get∥∥pF,3χ∥∥L32 ([t0,t]×B(x0, 32 ))≤∥∥∥∥ˆB(x0,√R)\B(x0,2)1| ·−y|4 Fi jχ(y)dy∥∥∥∥L32 ([t0,t]×B(x0, 32 ))+∥∥∥∥ˆB(x0,√R)\B(x0,2)1| ·−y|4 Fi j(y)(χ(·)−χ(y))dy∥∥∥∥L32 ([t0,t]×B(x0, 32 ))+∥∥∥∥ˆB(x0,√R)c1| ·−y|4 Fi j(y)dyχ∥∥∥∥L32 ([t0,t]×B(x0, 32 )).Thus∥∥pF,3χ∥∥L32 ([t0,t]×B(x0, 32 )).∞∑k=1∥∥∥∥ˆB(x0,2k+1)\B(x0,2k)1|x0− y|4 |Fi jχ(y)|dy∥∥∥∥L32 (t0,t;L∞(B(x0, 32 )))+1√R∞∑k=1∥∥∥∥ˆB(x0,2k+1)\B(x0,2k)1|x0− y|4 |Fi j(y)|dy∥∥∥∥L∞([t0,t]×B(x0, 32 ))+∞∑blog2√Rc∥∥∥∥ˆB(x0,2k+1)\B(x0,2k)1|x0− y|4 |Fi j(y)|dy∥∥∥∥L32 ([t0,t]×B(x0, 32 )). ‖Fχ‖L32 ([t0,t]×B(x0, 32 ))+1√R‖F‖U∞,1(t0,t) .Combining the estimates for pF,iχ , i = 1,2,3, we obtain∥∥pFχ∥∥L32 ([t0,t]×B(x0, 32 )).T ‖Fχ‖U 32 , 32 (t0,t)+1R‖F‖U32 ,32 (t0,t)+1√R‖F‖U∞,1(t0,t).A,T ‖wχ‖U3,3(t0,t)+1√R.121Now, we consider the pG,i’s. Since for x ∈ B(x0, 32), pG,1 satisfies|pG,1χ(x, t)| ≤ˆ|x0−y|≤3|(∇K)(x− y)||V ||∇V |(y, t)(|χ(y)|+ |χ(x)−χ(y)|)dy.ˆB3(x0)1|x− y|2 ||V ||∇V (y, t)|χ(y)dy+1RˆB3(x0)1|x− y| |V ||∇V (y, t)|dyusing |χ(x)− χ(y)| . ‖∇χ‖∞ |x− y|, the estimate for pG,1χ can be obtained fromYoung’s convolution inequality;∥∥pG,1χ∥∥L32 ([t0,t]×B(x0, 32 )).T∥∥∥∥ˆ|x0−y|≤3 1| ·−y|2 ||V ||∇V |(y, t)χ(y)dy∥∥∥∥L2([t0,t]×R3)+1R∥∥∥∥ˆ|x0−y|≤3 1| ·−y| |V ||∇V |(y, t)dy∥∥∥∥L2013 (t0,t;L307 (R3)).∥∥∥∥ 1| · |2∥∥∥∥32 ,∞‖|∇V |χ‖L∞t (t0,T ;L32 (B(x0,3)))‖V‖L2(0,T ;L6(B(x0,3)))+1R∥∥∥∥ 1| · |∥∥∥∥3,∞‖V‖L203 (0,T ;L52 (B(x0,3)))‖∇V‖U2,2T.A,T ‖|∇V |χ‖U∞, 32 (t0,T )+1R.By integration by parts, for x ∈ B(x0, 32), pG,2 can be rewritten aspG,2 =ˆ(Ki(·− y)−Ki(x0− y))Vi∂ jVj(y, t)(ρτ −ρ2)(y)dyand then it satisfies|pG,2χ(x, t)|.ˆ2<|x0−y|≤2τ1|x0− y|3 |V ||∇V |(y, t)(|χ(y)|+ |χ(x)−χ(y)|)dy.mτ∑i=1ˆBi+1\Bi1|x0− y|3 |V ||∇V |(y, t)(|χ(y)|+ τR)dy,122where mτ = dln(2τ)/ ln2e and Bi = B(x0,2i). Taking L2(t0, t) on it, we have∥∥pG,2χ∥∥L2(t0,t;L∞(B(x0, 32 ))) .∥∥∥∥∥mτ∑i=1ˆBi+1\Bi1|x0− y|3 |V ||∇V |(y, t)(|χ(y)|+ τR)dy∥∥∥∥∥L2(t0,t).mτ∑i=1123i(‖V |∇V |χ‖L2(t0,t;L1(Bi+1))+τR‖V |∇V |‖L2(t0,t;L1(Bi+1))).mτ∑i=1((‖|V ||∇V |χ‖U2,1(t0,T )+τR‖|V ||∇V |‖U2,1T).T lnτ ‖V‖U∞,2T ‖|∇V |χ‖U∞,2(t0,T )+τ lnτR‖V‖U∞,2T ‖∇V‖U2,2T .Lastly,|pG,3(x, t)| ≤ˆ|x0−y|≥τ|V (y, t)|2|x0− y|4 dy+1τˆτ≤|x0−y|≤2τ|V (y, t)|2|x0− y|3 dy≤1τ‖V‖2U∞,2T .Hence ∥∥pG,3χ∥∥L32 ([t0,t]×B(x0, 32 ))≤ ∥∥pG,3∥∥L32 ([t0,t]×B(x0, 32 )).A,T1τTo summarize, we have shown3∑i=1∥∥pG,iχ∥∥L32 ([t0,t]×B(x0, 32 )).A,T lnτ ‖|∇V |χ‖U∞,2(t0,T )+τ lnτR+1τ,and therefore‖qpχ‖L32 ([t0,t]×B(x0, 32 )).A,T ‖wχ‖U3,3(t0,t)+1√R+ lnτ ‖|∇V |χ‖U∞,2(t0,T )+τ lnτR+1τ.(3.4.20)Finally, combining all estimates and then taking supremum on (3.4.19) overx0 ∈ R3, we obtain‖w(·, t)χ‖2L2uloc +2‖|∇w|χ‖2U2,2(t0,t).A,T ‖w(·, t0)χ‖2L2uloc +‖wχ‖2U2,2(t0,t)+‖wχ‖2U3,3(t0,t)+(lnτ)2 ‖|∇V |χ‖2U∞,3(t0,T )+(τ lnτ)2R2+1τ2+1R.(3.4.21)123Using the estimates‖wχ‖2U3,3(t0,t) . ‖wχ‖U6,2(t0,t)(‖wχ‖U2,2(t0,t)+‖|∇w|χ‖U2,2(t0,t)+1R‖w‖U2,2T),(3.4.22)and Lemma 3.4.5, it becomes‖w(·, t)χ‖2L2uloc +‖|∇w|χ‖2U2,2(t0,t).A,T,C0 t1100 +‖w0χ‖2L2uloc +‖wχ‖2L6(t0,t;L2uloc)+(lnτ)2 ‖|∇V |χ‖2U∞,3(t0,T )+(τ lnτ)2R2+1τ2+1R,(3.4.23)where C0 is defined as in Lemma 3.4.5.Note that ‖w(·, t)χ‖2L2uloc is lower semicontinuous in t as w is weakly L2(Bn)-continuous in t for any n. By the Gro¨nwall inequality, we have‖wχ‖2L6(t0,T ;L2uloc) .A,T,C0 t1100 +‖w0χ‖2L2uloc+(lnτ)2 ‖|∇V |χ‖2U∞,3(t0,T )+(τ lnτ)2R2+1τ2+1R.(3.4.24)We now prove (3.4.18). Fix t1 ∈ (0,T ). For every n ∈ N we can choose t0 =t0(n) ∈ (0, t1)\Σ satisfyingt1100 <1n .At the same time, we pick τ = τ(n) > 4 satisfying τ−2 ≤ 1/n. After t0 and τ arefixed, we can make all the remaining terms small by choosing R=R(n,‖v0‖L2uloc , t0,τ)sufficiently large:‖w0χR‖2L2uloc +(lnτ)2 ‖|∇V |χR‖2U∞,3(t0,T )+(τ lnτ)2R2+1R≤ 1n.Here, the smallness of the second term follows from ∇V decay (Lemma 3.2.7),using the oscillation decay of v0. In conclusion, by (3.4.24), for each n ∈ N, we124can find t0, τ and R 1 so that‖wχR‖2L6(t0,T ;L2uloc) .A,T,C01n.By (3.4.23),‖wχR‖2L∞(t0,T ;L2uloc)+‖|∇w|χR‖U2,2(t0,T ) .A,T,C01n.By (3.4.22),‖wχR‖2U3,3(t0,T ) .A,T,C01n.Restricted to the original time interval (t1,T ), the perturbation w satisfieslimR→∞‖wχR‖U3,3(t1,T ) = 0,limR→∞‖wχR‖2L∞(t1,T ;L2uloc)+‖|∇w|χR‖U2,2(t1,T ) = 0.Using (3.4.20), we also havelimR→∞supx0∈R3‖qpx0χR‖L 32 (B(x0, 32 )×(t1,T )) = 0.This completes the proof of Proposition 3.4.7.Corollary 3.4.8. Under the same assumptions of Proposition 3.4.7, the perturbedNavier-Stokes flow w = v− et∆u0 satisfies w(t) ∈ E p(R3) for almost all t ∈ (0,T ]for any 3≤ p≤ 6.Proof. By Proposition 3.4.7, for any fixed x0 ∈ R3 and t1 ∈ (0,T ), the perturbedlocal energy solution w to the Navier-Stokes equations satisfies‖w‖L3(B3/2(x0)×(t1,T ))+‖qpx0‖L3/2(B3/2(x0)×(t1,T ))→ 0 as |x0| → ∞.Recall that V ∈ C1([δ ,∞)×R3) for any δ > 0. Then, by the Caffarelli-Kohn-Nirenberg criteria [6], for any t2 ∈ (t1,T ], we can find R0 > 0 such that if |x0| ≥ R0,‖w‖L∞([t2,T ]×B1(x0)) . ‖w‖L3(B3/2(x0)×(t1,T ))+‖qpx0‖1/2L3/2(B3/2(x0)×(t1,T )) ,125and the constant in the inequality is independent of x0. Moreover, ‖w‖L∞([t2,T ]×B1(x0))→0 as |x0| → ∞. Although the system (3.4.8) satisfied by w is not the original (NS),similar proof works since V ∈C1. See [22, Theorem 2.1] for more singular V ∈ Lm,m > 1, but without the source term V ·∇V .On the other hand, w ∈ ET implies thatw ∈ Ls(0,T ;Lp(BR0))for any s ∈ [2,∞] and p ∈ [2,6] with 2s + 3p = 32 , and therefore w(t) ∈ E p for a.e. t ∈(0,T ].3.5 Global existenceIn this section, we prove Theorem 3.1.1. We first give the following decay esti-mates.Lemma 3.5.1. Let (v, p) be a local energy solution in R3× [t0,T ], 0 < t0 < T <∞,to the Navier-Stokes equations (NS) for the initial datav|t=t0 = w∗+ et0∆u0where w∗ ∈ E2σ and u0 ∈ L3uloc,σ satisfies the oscillation decay (3.1.5). Let V (t) =et∆u0. Then, the perturbation w = v−V also decays at infinity:‖w‖L3([t0,T ]×B(x0,1))+‖qpx0‖L 32 ([t0,T ]×B(x0,1))→ 0,and‖w‖L∞(t0,T ;L2(B(x0,1)))+‖∇w‖L2(t0,T ;L2(B(x0,1)))→ 0, as |x0| → ∞.Remark. This T is arbitrarily large, unlike the existence time given in the localexistence theorem, Theorem 3.3.2. We assume w∗ ∈ E2, and we have V ∈C1(R3×[t0,T ]). We no longer need Lemma 3.4.5 nor the strong local energy inequality.Proof. The proof is almost the same as that of Proposition 3.4.7 except for the wayto estimate ‖w(·, t0)χR‖L2uloc in (3.4.21). Indeed, limR→∞ ‖w(·, t0)χR‖L2uloc = 0 by the126assumption w(·, t0) = w∗ ∈ E2.Now, we prove the main theorem.Proof of Theorem 3.1.1. Let (v, p) be a local energy solution to the Navier-Stokesequations in R3× [0,T0], 0 < T0 < ∞, for the initial data v|t=0 = v0, constructed inTheorem 3.3.2. By Corollary 3.4.8, there exists t0 ∈ (0,T0), arbitrarily close to T0,with w(t0) = v(t0)− et0∆u0 ∈ E4. Then, by Lemma 3.2.2, for any small δ > 0, wecan decomposew(t0) =W0+h0,where W0 ∈C∞c,σ (R3) and h0 ∈ E4(R3) with ‖h0‖L4uloc < δ .To construct a local energy solution (v˜, p˜) to (NS) for t ≥ t0 with initial datav˜|t=t0 = v(t0), we decompose (v˜, p˜) asv˜ =V +h+W, p˜ = ph+ pWwhere V (t) = et∆u0, (h, ph) satisfies∂th−∆h+∇ph =−H ·∇H, H =V +h,divh = 0, h|t=t0 = h0, (3.5.1)so that H solves (NS) with H(t0) = et0∆u0+h0, and (W, pW ) satisfies∂tW −∆W +∇pW =−[(H +W ) ·∇]W − (W ·∇)H,divW = 0, W |t=t0 =W0. (3.5.2)Our strategy is to first find, for each ε > 0, a distributional solution (hε , pεh) anda Leray-Hopf weak solution (W ε , pεW ) to ε-approximations of (3.5.1) and (3.5.2)for t ∈ I for some S = S(δ ,V ) > 0 uniform in ε . Then, we prove that they have alimit (v˜, p˜) which is a desired local energy solution to (NS) on I. By gluing twosolutions v and v˜ at t = t0, we can get the extended local energy solution on thetime interval [0, t0 + S]. Repeating this process, we get a time-global local energysolution. The detailed proof is given below.127Step 1. Construction of approximation solutionsLet I = (t0, t0 +S) for some small S ∈ (0,1) to be decided. For 0 < ε < 1, wefirst consider the fixed point problem forΨ(h) = e(t−t0)∆h0−ˆ tt0e(t−s)∆P∇ · (JεH⊗HΦε)(s)ds, H =V +h, (3.5.3)where Jε(H) = H ∗ ηε is the mollification of scale ε and Φε(x) = Φ(εx) is alocalization factor of scale ε−1. We will solve for a fixed point h = hε in theBanach spaceF =Ft0,S := {h ∈U∞,4(I) : (t− t0)38 h(·, t) ∈ L∞(I×R3)}for some small S > 0 with‖h‖F := ‖h‖U∞,4(I)+∥∥∥(t− t0) 38 h(t)∥∥∥L∞(I×R3).Denote M = ‖V‖L∞(I×R3) . (1+ t−4/30 )‖v0‖L2uloc . By Lemma 3.2.3, we have‖Ψh‖U∞,4(I) . ‖h0‖L4uloc +S18 ‖h‖2U∞,4 +S12 M ‖h‖U∞,4 +S12 ‖V‖2L∞(I;L8uloc). ‖h0‖L4uloc +S18 ‖h‖2F +S12 M ‖h‖F +S12 M2,and for t ∈ I,‖Ψh(t)‖L∞(R3) . (t− t0)−38 ‖h0‖L4uloc +ˆ tt0|t− s|−1/2(‖h(s)‖2L∞+M2)ds. (t− t0)− 38 ‖h0‖L4uloc +(t− t0)−1/4 ‖h‖2F +(t− t0)1/2M2.Therefore, we get‖Ψh‖F . ‖h0‖L4uloc +S18 ‖h‖2F +S12 M ‖h‖F +S12 M2.Similarly we can show‖Ψh1−Ψh2‖F .{S18 (‖h1‖F +‖h2‖F )+S12 M}‖h1−h2‖F .128By the Picard contraction theorem, we can find S= S(δ ,‖V‖L∞(I×R3))∈ (0,1) suchthat a unique fixed point (mild solution) hε to (3.5.3) exists inFt0,S with‖hε‖F ≤Cδ , ∀0 < ε < 1. (3.5.4)We also have the uniform bound‖hε‖E (I) . ‖JεHε ⊗HεΦε‖U2,2(I) . ‖hε‖2F +‖V‖2U4,4(I) . δ 2+M2. (3.5.5)Now, we define Hε =V +hε and the pressure pεh bypεh =−13JεHε ·HεΦε +p.v.ˆB2Ki j(·− y)(JεHε)iHεj Φε(y, t)dy+p.v.ˆBc2(Ki j(·− y)−Ki j(−y))(JεHε)iHεj Φε(y, t)dy.It is well defined thanks to the localization factor Φε . For each R > 0, we have auniform bound‖pεh‖L 32 (I×BR) ≤C(R) (3.5.6)in a similar way to getting (3.3.17). The pair (hε , pεh) solves, with Hε =V +hε ,∂thε −∆hε +∇pεh =−(JεHε ·∇)(HεΦε),divhε = 0, hε |t=t0 = h0 ∈ L4uloc (3.5.7)in R3× I in the distributional sense.We next consider the equation for W =W ε ,∂tW −∆W +∇pW = f εWf εW :=−Jε(Hε +W ) ·∇W −JεW ·∇Hε ,divW = 0, W |t=t0 =W0 ∈C∞c,σ .(3.5.8)Note that (3.5.8) is a mollified and perturbed (NS), and has no localization factorΦε .129Using W ε itself as a test function, we can get an a priori estimate: for t ∈ I,‖W (t)‖2L2(R3)+2‖∇W‖2L2([t0,t]×R3) ≤ ‖W0‖2L2(R3)+¨f εW ·W.Note that˜Jε(H+W )·∇W ·Wdxdt = 0 and−˜(JεW ·∇)hε ·Wdxdt =˜(JεW ·∇)W ·hεdxdt. Also recall that‖hεW‖L2(Q) . ‖hε‖L∞(I;L3uloc) (‖∇W‖L2(Q)+‖W‖L2(Q))for Q = [t0, t]×R3. Its proof can be found in [26, page 162]. Thus¨f εW ·W =¨(JεW ·∇)W ·hε −¨(JεW ·∇)V ·W≤C‖∇W‖L2(Q) δ (‖∇W‖L2(Q)+‖W‖L2(Q))+M1 ‖W‖2L2(Q) .where M1 = ‖∇V‖L∞(I×R3). By choosing δ sufficiently small, we conclude‖W (t)‖2L2(R3)+‖∇W‖2L2([t0,t]×R3) ≤ ‖W0‖2L2(R3)+C(1+M1)‖W‖2L2(Q) .By the Gro¨nwall inequality (using that ‖W (t)‖2L2(R3) is lower semicontinuous),we obtain‖W ε‖2L∞(I;L2(R3))+‖∇W ε‖2L2(I×R3) ≤C(M1)‖W0‖2L2(R3) . (3.5.9)With this uniform a priori bound, for each 0 < ε < 1, we can use the Galerkinmethod to construct a Leray-Hopf weak solution W ε on I×R3 to (3.5.8).Define Fεi j =Jε(Wε +Hε)iW εj +(JεWε)iHεj . We have the uniform bound∥∥Fεi j∥∥U3/2,3/2(I) ≤C‖|V |+ |hε |+ |W ε |‖2U3,3(I) ≤C(M,M1,‖W0‖L2(R3)).130Define pεW (x, t) = limn→∞ pε,nW (x, t), and pε,nW (x, t) is defined for |x|< 2n bypε,nW (x, t) =−13trFεi j(x, t)+p.v.ˆB2(0)Ki j(x− y)Fεi j(y, t)dy+(p.v.ˆB2n+1\B2+ˆBc2n+1)(Ki j(x− y)−Ki j(−y))Fεi j(y, t)dy.For each R > 0, we have a uniform bound‖pεW‖L 32 (I×BR) ≤C(R,M,M1,‖W0‖L2(R3)). (3.5.10)By the usual theory for the nonhomogeneous Stokes system inR3, the pair (W ε , pεW )solves (3.5.8) in the distributional sense.We now definevε = Hε +W ε =V +hε +W ε , pε = pεh + pεW .Summing (3.5.7) and (3.5.8), the pair (vε , pε) solves in the distributional sense∂tvε −∆vε +∇pε =−Jεvε ·∇vε +Eε ,Eε =JεHε ·∇(Hε(1−Φε)),divvε = 0, vε |t=t0 = v(t0).(3.5.11)Thanks to the mollification, hε and W ε have higher local integrability by the usualregularity theory. Thus we can test (3.5.11) by 2vεξ , ξ ∈C∞c ([t0, t0+S)×R3), andintegrate by parts to get the identity2ˆIˆ|∇vε |2ξ dxds =ˆ|v|2ξ (x, t0)dx+ˆIˆ|vε |2(∂sξ +∆ξ )+(|vε |2Jεvε +2pεvε) ·∇ξ +Eε ·2vεξ dxds.(3.5.12)Note that v in´ |v|2ξ (x, t0)dx is the original solution in [0,T ).Step 2. A local energy solution on I = (t0, t0+S)131We now show that (vε , pε) has a weak limit (v˜, p˜) which is a local energy solu-tion on I. Recall the uniform bounds (3.5.4), (3.5.5), (3.5.6), (3.5.9), and (3.5.10)for hε , pεh,Wε and pεW . As in the proof of Theorem 3.3.2, from the uniform esti-mates and the compactness argument, we can find a subsequence (v(k), p(k)), k ∈N,from (vε , pε)which converges to some (v˜, p˜) in the following sense: for each n∈N,v(k) ∗⇀ v˜ in L∞(I;L2(B2n)),v(k) ⇀ v˜ in L2(I;H1(B2n)),v(k),J(k)v(k)→ v˜ in L3(I×B2n),p(k)→ p˜ in L 32 (I×B2n),where p˜(x, t) = limn→∞ p˜n(x, t), and p˜n(x, t) is defined for |x|< 2n byp˜n(x, t) =− 13|v˜(x, t)|2+p.v.ˆB2Ki j(x− y)v˜iv˜ j(y, t)dy+(p.v.ˆB2n+1\B2+ˆBc2n+1)(Ki j(x− y)−Ki j(−y))v˜iv˜ j(y, t)dy.Taking the limit of the weak form of (3.5.11), we obtain that (v˜, p˜) satisfiesthe weak form of (NS) for the initial data v˜|t=t0 = v(t0). Furthermore, the limitof (3.5.12) gives us the local energy inequality: For any ξ ∈C∞c ([t0, t0 +S)×R3),ξ ≥ 0, we have2ˆIˆ|∇v˜|2ξ dxds≤ˆ|v|2ξ (x, t0)dx+ˆIˆ|v˜|2(∂sξ +∆ξ )+(|v˜|2+2p˜)v˜ ·∇ξ dxds.(3.5.13)Here we have used that˜E(k) ·v(k)ξ =˜J(k)H(k) ·∇(H(k)(1−Φ(k))) ·v(k)ξ = 0for k sufficiently large. In a way similar to the proof of Theorem 3.3.2, we getthe local pressure decomposition for p˜, weak local L2-continuity of v˜(t), and localL2-convergence to the initial data. We also get (3.5.13) with the time interval Ireplaced by [t0, t] and an additional term´ |v˜|2ξ (x, t)dx in the left side.We have shown that (v˜, p˜) is a local energy solution on R3× I with initial datav˜|t=t0 = v(t0).132Step 3. To extend to a time-global local energy solution.We first prove that the combined solutionu = v1[0,t0]+ v˜1I, q = p1[0,t0]+ p˜1Iis a local energy solution on the extended time interval [0,T1] = [0, t0 + S]. It isobvious that u and q are bounded in ET1 and L32loc([0,T1]×R3), respectively andq satisfies the decomposition at each point x0 ∈ R3. Since we have for any ζ ∈C∞c ([t0,T1)×R3;R3)ˆ T1t0−(v˜,∂tζ )+(∇v˜,∇ζ )+(v˜,(v˜ ·∇)ζ )+(p˜,divζ )dt = (v˜,ζ )(t0) = (v,ζ )(t0),and for any ζ ∈C∞c ((0, t0]×R3;R3)ˆ t00−(v,∂tζ )+(∇v,∇ζ )+(v,(v ·∇)ζ )+(p,divζ )dt =−(v,ζ )(t0),from the weak continuity of v˜ at t0 from the right and that of v at t0, we can provethat (u, p) satisfies (NS) in the distribution sense: For any ζ ∈C∞c ((0,T1)×R3;R3)ˆ T10−(u,∂tζ )+(∇u,∇ζ )+(u,(u ·∇)ζ )+(q,divζ )dt = 0.Also, since we already have local L2-weak continuity of u on [0,T1]\{t0}, it isenough to check it at t0; for any ϕ ∈ L2(R3) with a compact support,limt→t−0(u,ϕ)(t) = limt→t−0(v,ϕ)(t) = (v,ϕ)(t0) = limt→t+0(v˜,ϕ)(t) = limt→t+0(u,ϕ)(t).Finally, we prove the local energy inequality (3.3.1). Indeed, for any t ∈ (0, t0],the inequality follows from the one of v. For t ∈ (t0,T1), we add the inequality of v133in [0, t0] to the one of v˜ in [t0, t] to get, for any non-negative ξ ∈C∞c ((0,T1)×R3),ˆ|u|2ξ (t)dx+2ˆ t0ˆ|∇u|2ξdxds=ˆ|v˜|2ξ (t)dx+2ˆ t00ˆ|∇v|2ξdxds+2ˆ tt0ˆ|∇v˜|2ξdxds≤ˆ t00ˆ|v|2(∂sξ +∆ξ )+(|v|2+2p)(v ·∇)ξdxds+ˆ tt0ˆ|v˜|2(∂sξ +∆ξ )+(|v˜|2+2 p˜)(v˜ ·∇)ξdxds=ˆ t0ˆ|u|2(∂sξ +∆ξ )+(|u|2+2q)(u ·∇)ξdxds.Therefore, (u,q) is a local energy solution on [0,T1) and is an extension of(v, p).Then, by Lemma 3.5.1 and the proof of Corollary 3.4.8, we can find t1 ∈(t0 + 78 S, t0 + S) such that u(t1)−V (t1) ∈ E4. Repeating the above argument withnew initial time t1, we can get a local energy solution in [0, t1 + S). Iterating thisprocess, we get a local energy solution global in time. Note that ‖V‖L∞([t1,∞)×R3) ≤‖V‖L∞([t0,∞)×R3) whenever t1 > t0, so that on each step, we can extend the timeinterval for the existence by at least 78 S.134Chapter 4ConclusionIn this Chapter, we summarize the works in previous chapters and provide openquestions related to the materials discussed in the dissertation.The work in Chapter 2 presents that the logarithmically regularized 2D Eulerequation (LE) for 0 < γ ≤ 12 is strongly ill-posed in the borderline Sobolev spaceH1(R2)∩ H˙−1(R2). More precisely, we first showed that for any given compactlysupported smooth initial data, a non-compactly supported perturbation always ex-ists such that it is arbitrary small in the borderline space but the unique solution forthe perturbed initial data in some other space leaves the borderline space instanta-neously. Then, we also construct a compactly supported perturbation with similarproperties, but it requires an additional condition on the given smooth initial data:it is odd at least in one variable.This result completely solves the well-posedness problem of (LE) in Sobolevspaces. Especially in the borderline space, it closes a gap between the local well-posedness for γ > 12 and the strong ill-posedness for γ = 0. Namely, the localwell-posedness result is sharp, while the strong ill-posedness still holds even undera slight regularization of the velocity in the Euler vorticity equation keeping thesame borderline space. Furthermore, it provides a deeper understanding of thelocal well-posedness of the 2D Euler equation in the borderline Sobolev spacewhich has attracted much attention in recent decades.In a broader perspective, clarifying the well-posedness of an evolutionary par-tial differential equation has a great significance. For example, in the physical135application, approximate solutions are mostly considered. For any smooth solu-tion, a small error in initial data still makes the approximate solution stay close tothe real one. However, in a borderline space, a solution can be very sensitive to asmall change in initial data. Indeed, in our analysis on (LE), the difference betweenthe real solution and the approximate one is infinity in the sense of the borderlinespace norm. In other words, it gives a warning sign in the usage of data in theborderline space in physical computing.As an extension of this work, a possible direction is working on other spaces.For instance, one can consider Besov spaces of logarithmic smoothness, which ismore sophisticated than Sobolev spaces. This will provide a sharper result on thewell-posedness of (LE). On the other hand, the Lipschitz type spaces as in [5] canbe another option. Along these lines, we can deepen our understanding of the 2DEuler dynamics in integer Cm spaces in [5].The strong ill-posedness scheme can be applied to other problems. One ofthe most challenging but interesting problems is the well-posedness of the inviscidsurface quasi-geostrophic equation∂tθ +∇⊥(−∆)12θ ·∇θ = 0θ |t=0 = θ0.This equation in the two-dimensional whole domain has special importance amongincompressible fluid equations because it has strong analogies with the 3D Eulerequation. In this case, its borderline Sobolev space is also changed to H2(R2). Thescheme perfectly works in the construction of local solutions having the criticalSobolev norm inflation. However, in order to apply the patching argument, weneed to guarantee the uniform lifespan of the local solutions. It can be one of thedefects in the scheme that we need to overcome.In summary, our result in Chapter 2 improves the understanding of the behav-ior of the solution to the logarithmically regularized 2D Euler equations. Also, itstrengthens the scheme and shows the potential of its applications to many otherfluid equations.In Chapter 3, we suggest a construction scheme for a global-in-time local en-ergy solution of the Navier-Stokes with non-decaying initial data. More precisely,136this scheme works for any initial data whose non-decaying part is in L3uloc(R3) anddecaying part is in E2σ (R3), having slow oscillation decay. Here, we do not restrictthe rate of the decay. Furthermore, the analysis in the global existence explainshow the solution behaves in any positive time. Indeed, the solution to the heatequation with the non-decaying part of given initial data governs the non-decayingpart of the constructed solution.In practice, non-decaying flows are widely known — for example, constantflows and periodic flows. Most research, however, has studied decaying flowsbecause of difficulties arising from the pressure. In this context, our scheme notonly has practical usages but also enrich the analysis of Navier-Stokes flows withnon-decaying initial data. Indeed, it makes a vast improvement from most recentwork on the global existence with non-decaying initial data in [38].Despite the significant improvement, one of the defects of the scheme is therequirement on the local regularity of non-decaying part of the initial data. Thisassumption follows from a technical difficulty of the scheme. On the other hand,this suggests a possible direction of future work, weakening the assumption on thislocal regularity of non-decaying part.In summary, the scheme introduced in Chapter 3 says that even for a rough ini-tial datum with no decay at spatial infinity, a Navier-Stokes flow can exist globallyin time. Our result has rich applications to the reality and also deepens the un-derstanding regarding a solution to the Navier-Stokes equation with non-decayinginitial data.137Bibliography[1] Barraza, O., Self-similar solutions in weak Lp-spaces of the Navier-Stokesequations. Rev. Mat. Iberoamericana 12 (1996), 411-439. → page 73[2] Bahouri, H., Chemin, J.-Y., and Danchin, R., Fourier analysis and nonlinearpartial differential equations. Grundlehren der MathematischenWissenschaften [Fundamental Principles of Mathematical Sciences], 343.Springer, Heidelberg, 2011. → page 73[3] Beale, J.T., Kato, T., Majda, A.: Remarks on the breakdown of smoothsolutions for the 3D Euler equations. Comm. 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