Long-time dynamics for the energy-criticalHarmonic Map Heat Flow and Nonlinear HeatEquationbyDimitrios RoxanasA thesis submitted in partial fulfilment of the requirements forthe degree ofDoctor of PhilosophyinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)The University of British Columbia(Vancouver)May 2017c© Dimitrios Roxanas, 2017AbstractThe main focus of this thesis is on critical parabolic problems, in particular, theharmonic map heat flow into the 2-sphere~ut = ∆~u+ |∇~u|2~u, ~u : R2 → S2,and the focusing nonlinear heat equationut = ∆u+ |u| 4d−2u, u : Rd → R, d ≥ 3.The focus of this work has been on long-time dynamics, stability and singularityformation, and the investigation of the role of special, soliton-like solutions, to theasymptotic behaviour of solutions.Harmonic Map Heat Flow: We consider m-corotational solutions to the harmonicmap heat flow from R2 to S2. We first work in a class of maps with trivial topologyand energy of the initial data below two times the energy of the stationary harmonicmap solutions. We give a new proof of global existence and decay. The proof is basedon the “concentration-compactness plus rigidity” approach of Kenig and Merle andrelies on the dissipation of energy and a new profile decomposition.We also treat m-corotational maps (m ≥ 4) with non-trivial topology and energyof the initial data less than three times the energy of the stationary harmonic mapsolutions. Through a new stability argument we rule out finite-time blow-up andshow that the global solution asymptotically converges to a harmonic map.Nonlinear Heat Equation: We also study solutions of the focusing energy-criticaliinonlinear heat equation ut−∆u−|u|2u = 0 in R4. We show that solutions emanatingfrom initial data with energy and H˙1−norm below those of the stationary solutionW are global and decay to zero. We first show that global solutions dissipate to zero.The proof is based on a refined small data theory, L2−dissipation and an approx-imation argument. We then follow the “concentration-compactness plus rigidity”roadmap of Kenig and Merle (and in particular the approach taken by Kenig andKoch for Navier-Stokes) to exclude finite-time blow-up. Our proof extends to alldimensions d ≥ 3.iiiLay SummaryThis thesis focuses on the study of some evolution Partial Differential Equations.These are equations that govern the evolution in-time of quantities appearing in themodeling of phenomena in Science and Engineering, for example density or size ofpopulation. In particular, we study two nonlinear Heat Equations, the HarmonicMap Heat flow and a Heat Equation with a polynomial nonlinear term. In general,when working with an equation that models a real-life phenomenon, that a solutionceases to exist in finite-time signifies a possible flaw in the modeling step, and con-sequently the potential unsuitability of this equation, in its current form, as a goodapproximation of the natural phenomenon observed. In this work, we give criteriaunder which the solutions to the above equations, not only exist for all times, butalso enjoy specific desirable properties after, possibly, long time.ivPrefaceMuch of the original material in the following document is adapted from two ofthe author’s research preprints: [74] and [75] (the research is joint work with histhesis supervisor, Dr Stephen Gustafson). In particular, all of Chapter 2, whichevolves around the proof of Theorem 2.2, along with section 3.2, where the proofof Theorem 3.4 is presented, form the main content of [74], “Global regularity andasymptotic convergence for the higher-degree 2d corotational harmonic map heat flowto S2”. Chapter 4 is adapted from [75], “Global, decaying solutions below the groundstate for a critical heat equation”. The manuscripts have been written and will besubmitted for publication soon.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Landau-Lifshitz and Harmonic Map Heat Flow . . . . . . . . 41.1.2 Nonlinear Heat Equation . . . . . . . . . . . . . . . . . . . . . 162 Harmonic Map Heat Flow-Below Threshold . . . . . . . . . . . . . . 222.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2 Some analytical ingredients . . . . . . . . . . . . . . . . . . . . . . . 282.2.1 Energy properties of maps in E0 . . . . . . . . . . . . . . . . . 282.2.2 Local wellposedness for maps in E0 . . . . . . . . . . . . . . . 302.2.3 Stability under perturbations . . . . . . . . . . . . . . . . . . 352.2.4 Profile decomposition . . . . . . . . . . . . . . . . . . . . . . . 402.3 Minimal blow-up solution . . . . . . . . . . . . . . . . . . . . . . . . 502.4 Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56vi3 Harmonic Map Heat Flow-Above Threshold . . . . . . . . . . . . . 573.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.1.1 Global results for Landau-Lifshitz and Schro¨dinger maps . . . 573.2 Corotational maps in E1 . . . . . . . . . . . . . . . . . . . . . . . . . 603.2.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.2.2 Conclusions and future directions . . . . . . . . . . . . . . . . 744 Global solutions of a focusing energy-critical Heat Equation in R4 774.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 774.2 Some analytical ingredients . . . . . . . . . . . . . . . . . . . . . . . 824.2.1 Local theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.2.2 Profile decomposition . . . . . . . . . . . . . . . . . . . . . . . 854.2.3 Variational estimates . . . . . . . . . . . . . . . . . . . . . . . 864.3 Asymptotic decay of global solutions . . . . . . . . . . . . . . . . . . 874.4 Minimal blow-up solution . . . . . . . . . . . . . . . . . . . . . . . . 914.5 Rigidity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.6 Blow-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.7 Directions for future research . . . . . . . . . . . . . . . . . . . . . . 116Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118viiAcknowledgementsPrimarily, I would like to thank my supervisor Stephen Gustafson. No part of thisthesis would have been possible without his insight, feedback, and at times, criticalintervention. I am very grateful for his patience, time and advice, his shaping of mymathematical thinking, and also his support towards my career goals.I want to thank the members of my supervisory committee, Tai-Peng Tsai, Young-Heon Kim for their valuable feedback, support, and of course, their time. I also wantto take this opportunity to thank Ailana Fraser, Malabika Pramanik and Yaniv Planfor their support, and the great courses I took with them; they really made a dif-ference in my mathematical understanding. Moreover, I would like to thank myinstructors-in-charge Dan Coombs, Andrew Rechnitzer, Greg Martin and Fok-ShuenLeung, for all the experience and teaching advice they shared with me, as well as forthe time and effort they spent on organizing and troubleshooting, which resulted inmy having more time for my research.I would also like to acknowledge the abundant support I received from St John’sGraduate College, my moving there was definitely one of the best decisions I haveever made. In particular, I want to thank my many friends, especially Abhi, Derek,Edmundo, Felipe, Lena and Sheng for their friendship and encouragement, and alsothe Principals for their work and their support in difficult times. I am also pleasedto acknowledge the financial support from “St John’s College Reginald and AnnieVan Fellowship 2016”.viiiI wish to thank my parents, brothers and sister, and my extended family for theirlove and understanding, and all of my friends for their ample support all these years.A special mention goes to Dimitris and Pam, Maxime, Myrto, Samuli, Spyros andShaya.Last but definitely not least, I would like to thank Laura. I am incredibly gratefulfor the good times, the abundant support and all the encouragement, especially whenfacing very stressful situations and hard decisions. Thank you for everything!ixChapter 1Introduction1.1 General introductionNonlinear evolution equations (partial differential equations with a time variable)arise throughout the sciences, describing the dynamics of various systems. Classi-cal examples include nonlinear heat, wave and Schro¨dinger equations. Evolutionequations have also been used as an important tool in geometry; for example, theapplication of the Ricci flow to the Poincare´ conjecture, and generally the use of heatflows applied to prove results in global differential geometry.The main focus of this thesis is on critical parabolic problems, in particular, theharmonic map heat flow into the 2-sphere{~ut = ∆~u+ |∇~u|2~u, ~u : [0, T )× R2 → S2,~u(0, x) = ~u0(x)and the focusing nonlinear heat equation{ut = ∆u+ |u| 4d−2u, d ≥ 3, u : [0, T )× Rd → R,u(0, x) = u0(x)In order to place this work into a broader context, we first take a more gen-1eral look at the study of nonlinear evolution equations, many of which originate inphysics. From a mathematical perspective their study centers around the followingfundamental questions in connection with the Cauchy (initial data) problem: forgiven initial data, does there exist a local-in-time solution? Does this solution ex-ist for all time? If it ceases to exist after finite time (blowup), why? What is theasymptotic behavior of a global-in-time solution? Does it behave like a free linearsolution in the long run? Two basic features of a nonlinear heat equation play animportant role when one tries to answer these questions: the scaling invariance of theequation and the presence of a monotone quantity, which we will call “the energy”.The behaviour of the energy with respect to the scaling of the equation suggests aclassification of the equation as being either energy sub-critical, energy critical, orenergy super-critical.To both equations there is an associated energy functional of the general formE(u) =∫|∇u|2 +∫F (u), which is dissipated by solutions, and a scaling transfor-mation of the form uλ(t, x) = λβu(λ2t, λx), some β, which leaves each equation andits associated energy invariant. This invariance suggests that the dissipation of theLaplacian, ∆u, is balanced by the corresponding nonlinearity. In cases where thedissipation dominates, the solution should be global and decaying, behaving like thelinear heat equation ut = ∆u. If however the nonlinearity is stronger, the solutionmay exhibit nonlinear behaviour, such as soliton formation or blow-up by concentra-tion. As a result, the questions of global existence and decay of solutions become verydelicate and require special technology to answer. We aim to identify the thresholdbelow which the dissipation is dominating, resulting in global solutions that even-tually relax to equilibria, with the goal of writing criteria on the initial data whichwill guarantee either existence for all times and decay, or singularity formation. Inboth the cases of the Harmonic Map Heat Flow and the one of the Nonlinear HeatEquation, the threshold is defined in terms of the corresponding stationary solutions.Stationary solutions are global-in-time, but have no time decay, and they have beenthe objects on which blowing-up solutions have been built, e.g., ([114, 115, 118]).For both problems we work with functions whose (a priori) regularity matches2that of the dissipative energy, i.e., Sobolev spaces of one derivative, which we willcall the “energy space”. For several of the results obtained, the approach is largelymotivated by the recent developments in critical dispersive equations and in particu-lar, the “concentration-compactness plus rigidity” roadmap of Kenig and Merle [79].Unlike the nonlinear Schro¨dinger equation for example, heat equations don’t enjoyconservation of L2 (“mass”). This implies the need for a modified approach to thedispersive technology (as in [78]). For example, the irreversibility of the heat equa-tion together with the lack of mass conservation, require a more involved approachin the “rigidity” argument in Chapter 4. On the other hand, we gain a smoothingeffect and the monotonicity of the energy. To use it, especially for decay questions,we need to rule out the only obstruction to strict decrease of energy, namely thestationary solutions.All the results described below are on critical, in a scaling sense, problems. Whenworking on such questions, new difficulties (compared to subcritical problems) areencountered for both the local and the global theory. This is mostly because we arenow more or less forced to work exclusively with scale-invariant norms, which limitsthe tools available. For example, the time of existence given by the local theory willdepend on the profile of the data - not just on its norm. Because of this, even apriori bounds are not sufficient by themselves to upgrade local to global wellposed-ness: bounded energy norm does not exclude the possibility that the solution couldconcentrate at a point in finite time causing the lifespan of the local theory to shrinkto zero as we try to iterate the local result to extend the solution. In our framework,the energy norm does stay bounded, and blow-up can only happen because of con-centration. For global wellposedness and decay our task becomes to find a good wayof measuring concentration and show it cannot occur, at least below some naturalthreshold.Before we go more into the details of this work, let us briefly mention someother challenging aspects of these two problems, beyond their critical nature. Forthe harmonic map heat flow, the one challenge is the geometry and topology of thetarget. We are concerned with the most physically relevant case of maps with valuesin S2. It is known [47] that for manifolds with negative scalar curvature, there is3no obstruction to all solutions being globally smooth; while for S2 and other targetswith positive or sign-changing curvature the existence of non-trivial static solutions-harmonic maps- presents a possible obstruction [122]. From now on we will restrictthis discussion to the case of S2. We will be working within the class of corotationalsolutions (see Chapter 2 for details). Within this class, one is led to make a choiceof boundary conditions; these correspond to assignments for ~u(0), ~u(∞). Differentchoices of these conditions result in dramatically different results. Roughly speak-ing, there are combinations of topologies and energies that prohibit the existence ofobstructions to global regularity and decay, while others make it possible for singu-larities to occur. A second factor, whose importance is discussed in Chapters 2 and3, is the degree Z∗ 3 m := 14pi∫R2 (∂1~u ∧ ∂2~u) ·~u which measures the number of timesthe map “wraps” around the unit-sphere. We will see that the higher |m| is, themore favourable decay we can expect, in a sense to be quantified in later chapters.For the case of the nonlinear heat equation, the additional challenges stem fromthe focusing nature of the nonlinearity. The energy functional in this case, say,for d = 4, is given by E(u) =∫R4(12|∇u|2 − 14|u|4)dx. We refer to the term12∫R4|∇u|2dx as the kinetic energy and to the term 14∫R4|u|4dx as the potentialenergy. Notice that the potential energy is negative which reflects the focusing na-ture of the nonlinearity. The energy does not provide any a priori control on thecritical norm, ‖u‖H˙1 , which complicates matters significantly.1.1.1 Landau-Lifshitz and Harmonic Map Heat FlowJust as the harmonic map equation is a geometric analogue of the classsical Laplaceequation for harmonic functions, so the classical linear evolution PDEs, the Heat,the Wave and Schro¨dinger equations, have geometric “map” analogues: the Har-monic Map Heat Flow, Wave Map and Schro¨dinger Map equations. These equa-tions are nonlinear when the target space geometry is nontrivial. Quite remarkably,these equations are all of physical (as well as mathematical) interest, at least whenthe target space is the 2-sphere, arising in the study of ferromagnets (and anti-4ferromagnets), liquid crystals, and general relativity. Part of this dissertation isdevoted to our results for map evolution equations, focusing on the harmonic mapheat flow, a special case of the Landau-Lifshitz family of equations, which also in-cludes the Schro¨dinger map equation as a special case; our aim is to address thebasic global questions: singularity formation versus global regularity, and long-timeasymptotics.We will begin by giving a brief history, focusing mostly on recent developments.Let us begin with the harmonic map equation. From the outset, we fix a specificchoice of domain and target manifold for our maps:u : Rn → S2,mostly n = 2. We realize S2 as the unit sphere in R3, S2 := {u = (u1, u2, u3) ∈ R3 :|u| = 1} ⊂ R3 (Notation: vectors will be bold-faced throughout.) Harmonic mapsare critical points of the Dirichlet energy functionalE(u) :=12∫Rn|∇u|2dx = 12∫Rnn∑j=13∑k=1(∂uk∂xj)2dxand so (if smooth enough) solve the corresponding Euler-Lagrange equation0 = −E ′(u) = P u∆u = ∆u + |∇u|2u, (1.1)where Pu denotes the orthogonal projection from R3 onto the tangent planeTuS2 := {ξ ∈ R3 : ξ · u = 0}to S2 at u. Equation 1.1 is the equation for harmonic maps between Rn and S2. Itgeneralizes Laplace’s equation to maps.5Map evolution equations.Now we let our maps vary with time as well, so that for each time t ≥ 0,u(·, t) : Rn → S2,or equivalently u(x, t) = (u1(x, t), u2(x, t), u3(x, t)) ∈ R3 with the pointwise con-straint |u(x, t)| ≡ 1.Harmonic map heat-flow: Harmonic maps (between general Riemannian mani-folds) have for many years been of interest to differential geometers, and in orderto study them [47] introduced the gradient-flow equations for the energy E , the har-monic map heat flow equations ∂u∂t= −E ′(u), which in our setting reads∂u∂t= ∆u + |∇u|2u (1.2)The harmonic map heat flow generalizes the linear heat equation to maps.Landau-Lifshitz equations: Physically, equation (1.2) is the special case b = 0of the Landau-Lifshitz (sometimes Landau-Lifshitz-Gilbert) equations modeling thedynamics of the magnetization u in an isotropic ferromagnet ([89, 85]):∂u∂t= aPu∆u + bu× Pu∆u = a∆u + |∇u|2u + bu×∆u, (1.3)with a ≥ 0, b ∈ R, and where × denotes the usual cross-product in R3. In fact, equa-tion (1.3) itself is a special case of a more general equation incorporating additionalphysical effects such as anisotropy, and demagnetization.Schro¨dinger maps: The opposite limiting case (a = 0, i.e., no dissipation) of (1.3)can be written as∂u∂t= u×∆u = −JuE ′(u) (1.4)where the operatorJu := u× · : TuS2 → TuS2gives a rotation through pi/2 on the tangent plane TuS2, and so endows S2 with a6complex structure. Thus equation (1.4) can immediately be written for general mapsfrom Riemannian manifolds into Ka¨hler manifolds ([20, 48]). Since it generalizesthe linear Schro¨dinger equation to maps, (1.4) is known as the Schro¨dinger map(sometimes Schro¨dinger flow) equation.Wave maps: Finally, the wave map equation is Pu(∂2u∂t2−∆u) = 0, which in oursetting is (∂2∂t2−∆)u +(∣∣∣∣∂u∂t∣∣∣∣2 − |∇u|2)u = 0 (1.5)generalizes the linear wave equation to maps. The Wave Map problem is one ofthe most interesting and challenging nonlinear hyperbolic problems. It has a nat-ural formulation as the Euler-Lagrange system for a map between manifolds. Theparticular case of three dimensions is of special interest in high energy physics andsometimes called by physicists “the nonlinear sigma model”. The nuclei of atomsare held together by forces mediated by the pi mesons. These are a set of threeparticles whose masses are small compared to the nuclei themselves, so to a firstapproximation they can be considered to be massless, i.e., travelling at the speed oflight. If interactions among them are ignored, the pi mesons are described by a fieldsatisfying the wave equation. Interactions would add nonlinearities. A remarkablefact of physics is that the interactions among the pi mesons are described, to a goodapproximation ([99, 68, 69, 94]), by considering the target manifold to be the sphereS3 and replacing the wave equation by the corresponding wave map equation. Butits main interest, aside from the inherent mathematical one, is in general relativity,where it is studied as a (comparatively simple) model for understanding singularityformation (for some background, see, e.g., [119]). There have been several attemptsto modify this model to remove the possibility of finite-time blow up and to retaintopological solitons, the most famous modification is due to Skyrme. For an accountof recent works we refer to [31, 55, 90, 96].7The energy landscape and equivariant symmetry.The energy E(u) plays a central role in all of our analysis. We begin by observingthat the energy behaves well under the various dynamics introduced above.Energy identity. Formally taking the dot product of (1.3) withE ′(u) = −∆u− |∇u|2u ∈ TuS2and integrating in space and time yields the basic energy identityE(u(t)) + a∫ t0∫Rn|∆u + |∇u|2 u|2dxdt = E(u(0)). (1.6)For (1.4) (a = 0) this means energy conservation, while for a > 0 (including the (1.2)case (b = 0)), the energy is nonincreasing. A conserved Hamiltonian functional for(1.5) is obtained by adding12∫Rn∣∣∣∣∂u∂t∣∣∣∣2 dx to E(u).Two space dimensions is energy critical. The energy scales the following wayE(u(·)) = λ2−nE(u( ·λ))for λ > 0, which makes the space dimension n = 2 “energy critical”. This hasimportant consequences (see below) and in particular leads to the intuition thatn = 2 should be a borderline case for the formation of singularities for our mapdynamics. So n = 2 turns out to be particularly interesting mathematically (andn = 2 and n = 3 are physically the most interesting space dimensions). For thesereasons, we specialize to n = 2 from now on.Equivariant symmetry. Since the analysis of our flow is a challenging problem,a good starting point is to impose some symmetry. For now we will restrict ourattention to m-equivariant maps u : R2 → S2 ⊂ R3, of the formu(r, θ) = emθRv(r)8where (r, θ) are polar coordinates on R2,v : [0,∞)→ S2, R is the matrix generatingrotations around the u3−axis and m ∈ Z. We will also consider a subset of m-equivariant maps: the set of m-corotational maps; these are maps of the formu(r, θ) = (cos(mθ) sin(u(r, t)), sin(mθ) sin(u(r, t)), cos(u(r, t))),for u : [0,∞]→ R. Then, if u(r, t) solves (1.2), u(r, t) satisfies the equationut = urr +1rur −m2 sin 2u2r2. (1.7)This subclass is preserved by (1.2), as well as by the Wave Map, and is much usedin the corresponding literature, since the map equations reduce to a scalar PDE foru(r, t). This subclass is notably not preserved by (1.3) or (1.4), just as the wave andheat equations preserve real functions, while the Schro¨dinger equation (or a heat-Schro¨dinger mix) does not. We will work in the m-equivariant class for most of whatfollows in this introduction.Topological lower bound on the energy. There is a well-known energy lower boundE(u) ≥ 4pi|deg(u)|where Z∗ 3 m = deg(u) = 14pi∫R2(∂1u ∧ ∂2u) · u is the degree of the map u,considered (compactifying the domain R2 via stereographic projection) as a mapfrom S2 to itself. This bound is particularly easy to understand when u is an m-equivariant map, so thatE(u) = pi∫ ∞0(∣∣∣∣∂v∂r∣∣∣∣2 + m2r2 (v21 + v22))rdr.If E(u) < ∞, then v is continuous, and the limits limr→0v(r) and limr→∞v(r) exist (see[70]), and so we must have v(0),v(∞) = ±kˆ, where kˆ = (0, 0, 1). Without loss ofgenerality we fix v(0) = −kˆ. The two cases v(∞) = ±kˆ then correspond to differenttopological classes of maps. We denote by Σm the class of m-equivariant maps with9v(∞) = kˆ :Σm = {u : R2 → S2|u = emθRv(r), E(u) <∞,v(0) = −kˆ,v(∞) = kˆ}.For u ∈ Σm, the energy E(u) can be rewritten by “completing the square”:E(u) = pi∫ ∞0(∣∣∣∣∂v∂r∣∣∣∣2 + m2r2 |JvRv|2)rdr = pi∫ ∞0∣∣∣∣∂v∂r − |m|r JvRv∣∣∣∣2 rdr + Emin,withEmin = 2pi∫ ∞0vr · |m|rJvRvrdr = 2pi|m|∫ ∞0v3rdr = 4pi|m|.Thus we arrive at the conclusion thatu ∈ Σm ⇒ E(u) ≥ 4pi|m|Harmonic maps: This topological lower bound is clearly saturated if and only if∂v∂r=|m|rJvRvand the minimal energy is attained precisely an explicit two-parameter family ofharmonic maps:Om := {e(mθ+α)Rh(rs) : s > 0, α ∈ R}whereh(r) = h1(r)0h3(r) h1(r) = 2r|m| + r−|m|, h3(r) =r|m| − r−|m|r|m| + r−|m|(1.8)The rotation parameter α is determined only up to shifts of 2pi (i.e., really α ∈ S1).Note that Om is just the orbit of the harmonic map emθRh(r) under the symmetriesof the energy E which preserve equivariance: scaling and rotation. Of course, theseharmonic maps are static solutions of all of the map evolution equations introduced10above.Recent History:here we describe some of the important results for the various map dynamics de-scribed above, continuing to focus on maps from R2 to S2.Harmonic map heat-flow: Of the map evolution problems we are considering,(1.2) has been studied the longest, and is certainly the best understood. The ques-tion of singularity formation and characterization of possible blow-up has attracteda lot of attention in the last 30 years or so. On a compact manifold domain, Struwe[122] showed that if blow-up is to occur, it can only happen in an energy concen-tration scenario, and the concentration of energy results in the bubbling off of anon-trivial harmonic map at a finite number of pointsu(tn, an + λ(tn)x)tn↗T−−−→ Q, λ(tn)→ 0,locally in space; but otherwise, weak solutions to (1.2) with finite energy data, existglobally and are smooth: if at time T a singularity occurs the flow develops a bubblewhich then separates, and restarts from the weak limit (as t ↗ T ). If anothersingularity occurs at a later time, the flow goes through the same process. At leastthe full energy of a harmonic map is lost every time a singularity occurs, and nobubbling can occur when there is energy less than the lowest energy of a non-trivialharmonic map.Later work by Qing [110], Ding and Tian [38], Qing and Tian [111], Topping[125, 126] (see also the book [97]) showed that the convergence to a “body map”(what remains after the bubbles are removed) is strong away from the singular points,and also that near the bubble points all the energy is accounted for by the body mapand the bubbles - statements of the same flavour were also made for t → ∞. Weagain refer to the above works for more details.Working in the subclass of the corotational solutions with m = 1, and on a disk,[22] showed that, indeed, finite time blow-up does occur in some situations, themethods relying on the maximum principle and sub(super)-solutions.11Until recently, not much information was available on the admissible rates ofblow-up, as well as the relation between the possibility of singularity formation andthe degree m or a good description of the blow-up profile. A formal analysis by Vande Berg, Hulshof and King [9], and a rigorous construction verifying their resultsby Raphael and Schweyer [114], [115] shows that (if T is the blow-up time), for1-corotational maps, initial data u0, L ∈ N∗u(t, r)−Q( rλ(t))→ u∗, as t→ T, in H˙1,λ(t) = c(u0)(1 + ot→T (1))(T − t)L| log(T − t)| 2L2L−1, c(u0) > 0,with the case L = 1 providing the generic blow-up rate.On the other hand, Grotowski and Shatah ([65]), using maximum principle meth-ods as well, and assuming particular bounds on the initial data, showed that on theunit disc in R2 blow-up will not occur in finite-time for degrees m ≥ 2. One of ourgoals is to extend this result to all of R2 and give a maximum principle-free proof.Landau-Lifshitz equation. Once the Schro¨dinger-type term (b 6= 0) is includedin (1.3), our understanding diminishes considerably. Though the problem is stilldissipative, maximum principle-type arguments are not available, and even partialregularity results become more difficult and weaker (see, e.g., [84] and the referencestherein). Singularity formation is an open question, partly because the corotationalclass is no longer preserved. Indeed, the (1.2) blow-up may not provide a reliableguide for the (1.3) problem. In the recent papers [70, 71, 66, 72], for equivariantmaps from R2 to S2, if m ≥ 3, near-minimal energy solutions are shown not to formsingularities and to converge to harmonic maps as t goes to infinity. When m = 2this asymptotic stability may fail: in the case of heat-flow with a further symmetryrestriction, it is shown that more exotic asymptotics are possible, including infinite-time concentration (blow-up).Schro¨dinger maps. In the absence of dissipation (a = 0), the analysis be-comes still more difficult. Even the local theory is just beginning to be understood([5, 48, 76, 103, 120]; see also [77, 108] for the “modified Schro¨dinger map” case).12For the class of data we consider in the next subsection, m-equivariant solutions withenergy near the minimal energy 4pi|m|, an energy-space local wellposedness result isgiven in [71]. It is worth remarking that the existence time provided by this theoremdepends not on the energy (reflecting the energy-space critical nature of the equationin dimension n = 2), but rather on more refined information about the initial data:the “length scale” of the H˙1-nearest harmonic map (see [71] for details).Very few global results are known, and these only the equivariant case. The globalresults of [70, 71] we describe in the next section, showing asymptotic stability ofharmonic maps for m ≥ 3, can be thought of as above threshold analogues of [20]for large energy, where the problem is considerably enriched by the presence of theharmonic map family. The case of energy below the energy of a harmonic map hasbeen treated ([7, 8]), for maps with values either in S2 or H2, and also the casem = 0 in [73]. We also mention the recent important paper [106] which establishedfinite-time blow-up for the case m = 1.Wave maps: Wave maps have received more attention for a longer time than haveSchro¨dinger maps. There is a large literature, especially concerning local questions,which we will not attempt to summarize here (see, e.g., [119] for some background).Because of the close connection with the problem we are focusing on, we mentiononly that the possibility of finite-time blow-up for the energy-space critical (n = 2)wave maps was established only quite recently, first in [116] for higher degree equiv-ariant maps, and then in [87] for degree m = 1. In a series of works, Duyckaerts,Kenig and Merle ([41], [42],[43],[44]) have established soliton resolution for all ener-gies in the case of the radial wave equation in 3+1 dimensions, completely describingglobal, type I and type II blow-up solutions. Their results build on the Kenig-Merle“concentration-compactness plus rigidity” roadmap ([79], [80]) and the newly devised“energy channel method” for the wave equation. In the spirit of this work, similar re-sults have been obtained for the corotational Wave Map equation in 2+1 dimensions(to the 2-sphere) by Coˆte-Kenig-Merle [32], Coˆte-Kenig-Lawrie-Schlag [33], [34], [35],and Coˆte [30]; for other targets and geometries we refer to [86, 91, 92, 93]. Mostof these results rely on the presence of an underlying wave structure: they makefrequent use of the finite-speed of propagation as well as the energy channel method.13Note however, that the second order (in time) nature of the above equations is notstrong enough to prevent finite-time blow-up; blow-up solutions have been exhibitedin [87, 88], [113], [116] (even for high m).Main ResultsIn this thesis we specialize to the harmonic map heat flow for corotational maps ofdegree m :u(·, t) : (r, θ)→ (cos(mθ) sin(u(r, t)), sin(mθ) sin(u(r, t)), cos(u(r, t))) (1.9)Then, u(r, t) satisfies {ut = urr +1rur −m2 sin 2u2r2u(0, r) = u0(r)(1.10)The energy is given byE(u(t, r)) := pi∫ ∞0(u2r +m2 sin2(u)r2)rdr.To streamline the presentation of our results we make the following definitionsE0 := {u0 : E(u0) < 2E(Q);u(0) = 0, limr→∞u(r) = 0},which contains no non-trivial harmonic maps; maps in E0 have degree 0.E1 := {u0 : E(Q) ≤ E(u0) < 3E(Q);u(0) = pi, limr→∞u(r) = 0},Maps in E1 have degree m; the harmonic maps are also contained in this class.The harmonic maps are explicitly given byQs(r) = pi − 2 arctan((r/s)m), s > 0.These are the maps which minimize the energy within E1.14In Chapter 2 we first address the below threshold case of E0. The purpose ofthis work is twofold: to extend the classical analysis to non-compact domains, andmore importantly to revisit Struwe’s theory [122] in terms of the “concentration-compactness plus rigidity approach” of Kenig and Merle [79] with the aim of even-tually obtaining more detailed results. We prove the following theorem (Chapter2):Theorem 1.1. In the class E0, solutions to (1.10) with m ≥ 2, exist globally in timeand are smooth; furthermore, u(t)t→∞−−−→ 0 in energy.We first establish local well-posedness for maps in E0. Decay of global solutionsfollows from an approximation by an appropriately chosen linear solution. We thenadapt the “concentration-compactness plus rigidity” approach to show global exis-tence and decay. The key tool here is a profile decomposition for a bounded sequencein the energy space, applied on an appropriate sequence of initial data. There wasno readily available profile decomposition directly applicable to our setting; the diffi-culty stems mainly from the absence of some Sobolev embeddings in dimension two,so we take an indirect approach by first establishing estimates on the linear evolutionin higher dimensions which then connect back to our problem through a reductiontrick.Given the previous result, it is natural to ask what happens in the presence ofnon-trivial topologies and higher energies. In the equivariant setting, global regu-larity and asymptotic stability were shown ([72]) for m−equivariant solutions withm ≥ 3, for data u0 with E(u0) = E(Q) + δ, δ 1.However, our class E1 includes harmonic maps and maps with energy much largerthan the energy of the harmonic map, hence the approach taken in the above works(using “coordinate systems” around the family of harmonic maps, reflecting thenear-minimality of the energy) is hard to implement. We restrict ourselves to thecorotational setting and take a different approach to prove (Chapter 3)Theorem 1.2. The solution to (1.10) with m ≥ 4, and u0 ∈ E1 is global and smoothwith u(r, t)→ Qs∞(r), as t→∞ for some s∞ > 0.15As in [122], if the solution blows-up in finite time, it does so by bubbling off anon-trivial harmonic map. Following [110], if u(t) ∈ E1 is a solution blowing up attime T , with E(u) < 3E(Q), there exists a sequence of times tn → T , a sequenceof scales λn and a map w0 ∈ E0, such that u(tn, r) = Q( rλn ) + w0(r) + ξ(tn), withξ(tn)→ 0 in X2 as n→∞.We will start with data u(tn), where tn is very close to the blow-up time T andthen use a perturbative approach,based on modulation decomposing the solution uas u(t, r) = Q( rs(t)) +w(t, r) + ξ(t, r), where w is the solution corresponding to initialdata w0(r). We can eventually show that concentration is not possible in this case,hence the solution extends past T, providing a contradiction.1.1.2 Nonlinear Heat EquationFor the rest of this section we will focus on the nonlinear heat equationut −∆u− |u|p−1u = 0. (1.11)Our study of the heat equation was not motivated by a particular physical or ge-ometric application. We were however intrigued by the relative lack of results in theliterature concerning critical problems, and because of the similarities with the Non-linear Schro¨dinger equation on an algebraic level, we decided to investigate the useof the dispersive roadmap provided by the “concentration-compactness plus rigidity”approach of Kenig and Merle [79]. The resulting methodology is very different fromthe classical works on the nonlinear heat equation: we make no use of maximum andcomparison principles or use techniques such as the “intersection number”, nor do werely directly on a rescaling/blowing-up argument like Struwe’s [122] for the HarmonicMap Heat Flow to characterize singularities. Our approach combines elements fromthe analysis of the energy-critical NLS ([79, 82]), as well as ideas from the literatureon the Navier-Stokes system ([56, 78]), with new insights and deviations from knownarguments required at several places.The study of nonlinear heat equations has been a subject of intense work and the16literature is vast; it would be impossible to provide a complete list, so we content our-selves with a brief review focused on the case of domain Rd−case and refer the readerto the recent book [112] for a comprehensive review of the literature. For treatmentsof the Cauchy problem in Lp and Sobolev spaces under various assumptions on thenonlinearity and the initial data we refer to the works [128, 129, 13]. Most of thework on semilinear heat equations has been on subcritical problems p < 2dd−2 . For thesemilinear heat equation, no matter how weak is the initial regularity (let’s say in Lpspaces), blow-up occurs always in L∞ due to the regularizing effect, see [128]. Theseminal papers [62],[63],[64] of Giga and Kohn introduced the study of heat equationsthrough similarity variables and characterized blow-up solutions. In continuation ofthese works, Merle [104] gave a first construction of a solution with arbitrarily givenblow-up points and together with Zaag, they provided detailed uniform estimatesfor the blow-up rate, descriptions of the blow-up set, and stability results for theblow-up profile; we refer to [107] and the references therein. We remark that theblow-up in the subcritical case for L∞−solutions is known to be of Type I (in thesense that lim supt→Tmax(Tmax − t)1p−1‖u(·, t)‖L∞ < +∞) and Type I blow-up solutions areknown to behave like self-similar solutions near the blow-up point. More precisely,at any point a ∈ Rd where |u(a, t)| t→Tmax−−−−→∞, one can find a bounded solution ψ(y)of∆ψ − 12y · ∇ψ − 1p− 1ψ + |ψ|p−1ψ = 0, y ∈ Rd (1.12)such that u behaves like a self-similar solution (i.e., of the form u(x, t) = (T −t)−1p−1ψ( x−a√T−t)) in a certain “local” sense:u(a+√Tmax − ty, t) ∼ (Tmax − t)−1p−1ψ(y), as t→ Tmax.For supercritical problems, we refer to the series of works by Matano and Merle[100, 101, 102]. It is shown that there is no Type II blow-up for 3 ≤ d ≤ 10, whilefor d ≥ 11 it is possible for algebraic non-linearities with a large enough exponent(p > pJL := 1 +4d−4−2√d−1 , the Joseph-Lundgren exponent). It is also shown thata Type I blowing up solution behaves like a self-similar solution, while a Type II17converges (in some sense) to a stationary solution. We also refer to the recent results[30] (d ≥ 11, bounded domain) and [26] and to the recent preprint [10] for results inMorrey spaces.For the critical case, there are a few blow-up constructions [52, 118] in the caseof domain Rd, which inspired our project. For a bounded domain, we point to therecent construction [28] (d ≥ 5) of a solution blowing up by bubbling in infinite time,and on R3, the infinite-time blow-up construction [36]. We also refer to the work[53] dealing with the continuation problem for reaction-diffusion equations undervarious assumptions and range of exponents. We finally mention the recent result ofCollot, Merle and Raphael [27], where a complete classification of solutions near thestationary solution W for d ≥ 7 is provided. In particular, they show that Type IIblow-up is ruled out in d ≥ 7 “near” W.For a set of criteria (of a different nature to those we provide below) for globalexistence/blow-up in terms of the initial data we refer the reader to [18] where theauthors prove that given 0 < α < 2d, there exists a function ψ with the followingproperties: the solution of the equation ut = ∆u + |u|au, x ∈ Rd with the initialcondition u0 = ψ is global. On the other hand, the solution with the initial conditionu0 = λψ blows up in finite time if λ > 0 is either sufficiently small or sufficientlylarge.Finally, for results on the relation between the regularity of the nonlinear termand the regularity of the corresponding solutions we refer to the work [19].Main ResultsIn what follows, we consider the focusing energy-critical nonlinear heat equationin four space dimensions: ut −∆u− |u|2u = 0. We are interested in H˙1 solutions ofthe Cauchy Problemut = ∆u+ |u|2uu(t0, x) = u0(x) ∈ H˙1(R4)(1.13)18The energy for u ∈ H˙1 is defined byE(u) :=∫R412|∇u|2 − 14|u|4dx.We refer to the gradient term in E as the kinetic energy, and the second term asthe potential energy. Note that the potential energy is negative, which expresses thefocusing nature of the nonlinearity. Equation (1.13) is the L2−gradient flow for E.The energy is dissipatedddtE(u(t)) = −∫R4u2tdx ≤ 0and that the scaling uλ(t, x) = λu(λ2t, λx) leaves both the equation and the energyinvariant reflect the energy-critical nature of the problem.The stationary equation ∆W + |W |2W = 0 is known to admit the solutionW = W (x) =11 + |x|28which (along with its rescalings and translations) plays an important role in ouranalysis. By the work of Aubin-Talenti [2, 123], W is known to saturate the Sobolevinequality:∀u ∈ H˙1, ‖u‖L4 ≤[ ‖W‖L4‖∇W‖L2]‖∇u‖L2 , (1.14)One motivation for this project was the following result :Theorem 1.3. (Schweyer [118]) Let W be the Talenti-Aubin solution. Then forany a∗ > 0, there is a radially symmetric initial datum u0 ∈ H1(R4), with E(W ) <E(u0) < E(W ) + a∗, so that the corresponding solution to (1.13) blows-up in finitetime T = T (u0).We observe that the solution u(t, x) = W (x) = 1(1+|x|28)∈ H˙1(R4) is a stationarysolution to the equation which, while global, doesn’t decay. On the other hand, thelocal wellposedness theory shows that all maximal-lifespan solutions with sufficiently19small kinetic energy are global and decay to zero. The above results and observationssuggest that the static solution W will play an important role in determining whichinitial data lead to globally smooth and decaying solutions and which ones lead tosingularity formation. In particular, we show in Chapter 4 that below the thresholdset by the static solution all solutions are global and that, in addition, infinite-timeblow-up is not possible and global solutions have to decay to zero in H˙1.Theorem 1.4. Let u0 ∈ H˙1(R4) satisfy‖∇u0‖L2 < ‖∇W‖L2 and E(u0) < E(W ). (1.15)Then, the corresponding solution u to (1.13) is global and limt→∞‖u(t)‖H˙1 = 0.We give the proof for the case d = 4, but the result can be easily transferredto solutions in any d ≥ 3. Our proof makes no use of parabolic comparison princi-ples and can be used verbatim for complex-valued solutions. Assuming ‖∇u0‖L2 <‖∇W‖L2 and E(u0) < E(W ) we first show that global solutions decay to zero. Thisis done through a reduction to a refined small-data theory and arguments from thelocal-in-time theory. The result is new in the framework of heat equations andof independent interest. To rule out blow-up, our approach is again based on the“concentration-compactness plus rigidity” roadmap of Kenig and Merle [79] as imple-mented for the Navier-Stokes in [78]. Ruling out finite-time blow-up is different fromthe NLS case: we use compactness and the asymptotic vanishing of the L2−normon balls as in [79]; however there is no mass conservation in our case and the ir-reversibility of the equation requires a more involved backwards uniqueness step asin [78]. Another technical challenge is in precluding the scenario where the centerof concentration is moving off to infinity. Infinite-time blow-up can be ruled out byappealing to our decay result.We have also proved a blow-up criterion by a mild variant of a classical argumentin [95], which, modulo a decay assumption on the initial data, shows that our resultsare sharp20Proposition 1.5. Let u0 ∈ H1(R4) and δ0 > 0 such thatE(u0) ≤ (1− δ0)E(W ) and ‖∇u0‖L2 ≥ ‖∇W‖L2 (1.16)Then the corresponding solution u blows-up in finite-time.Whether removing the L2 assumption is possible is an interesting question forfuture consideration.21Chapter 2Harmonic Map Heat Flow-BelowThreshold2.1 IntroductionThis chapter is devoted to the presentation of our investigations on the globalregularity problem for the harmonic map heat flow. In particular, we will presentglobal wellposedness and decay results below a certain threshold.The harmonic map heat flow is given by the equationut = ∆u + |∇u|2u, u(x, 0) = u0(x) (2.1)where u(·, t) : R2 → S2, and S2 is the unit 2-sphereS2 := {u = (u1, u2, u3) : |u| = 1} ⊂ R3,∆ denotes the Laplace operator in R2 and |∇u|2 = ∑2j=1∑3i=1( ∂ui∂xj )2Equation (2.1) is the L2−gradient flow of the energy functionalE(u) =12∫R2|∇u|2dx22Taking, at least formally, the scalar product with ut = ∆u + |∇u|2u and integratingover R2 × [0, t), we obtain:E(u(t)) +∫ t0∫R2|ut|2 = E(u(0))which implies that the energy is non-increasing.A more geometric way to write (2.1) isut =2∑j=1Dj∂ju = Pu∆u,where Pu denotes the orthogonal projection from R3 onto the tangent plane to S2at u:TuS2 := {ξ ∈ R3 : ξ · u = 0},∂j =∂∂xjis the usual partial derivative and Dj the covariant derivative (affine con-nection) acting on vector fields ξ(x) ∈ Tu(x)S2 :Djξ := Pu∂jξ = ∂jξ − (∂jξ · u)u = ∂jξ + (∂ju · ξ)u.Equation (2.1) is a particular case of the harmonic map heat flow between Rieman-nian manifolds, introduced by Eells and Sampson [47]. Static solutions of (2.1) areharmonic maps from R2 to S2.The reason why we chose to work in two dimensions, beyond the physical rele-vance, is that this is when the energy E(u) is invariant under scaling,E(u(·)) = E(u( ·λ))which is interesting with respect to the questions of singularity formation and globalexistence.The question of singularity formation and characterization of possible blow-uphas attracted a lot of attention in the last 30 years or so. On a compact manifold23domain, Struwe [122] showed that if blow-up is to occur, it can only happen in anenergy concentration scenario, and the concentration of energy results in the bubblingoff of a non-trivial harmonic map at a finite number of pointsu(tn, an + λ(tn)x)→ Q, λ(tn)→ 0locally in space; but otherwise, weak solutions to (1.2) with finite energy data, existglobally and are smooth. Later work by Qing [110], Ding and Tian [38], Qing andTian [111], Topping [125, 126] (see also the book [97]) showed that the convergenceto a “body map” (what remains after the bubbles are removed) is strong away fromthe singular points, and also that near the bubble points all the energy is accountedfor by the body map and the bubbles - statements of the same flavour were alsomade for t→∞. We again refer to the above works for more details.Working in the subclass of the corotational solutions with m = 1, and on adisk, [22] showed that, indeed, finite time blow-up does occur in some situations,the methods relying on the maximum principle and sub(super)-solutions. A formalanalysis by Van de Berg, Hulshof and King [9], and a rigorous construction verifyingtheir results by Raphael and Schweyer [114], [115] shows that (if T is the blow-uptime), for 1-corotational maps, initial data u0, L ∈ N∗u(t, r)−Q( rλ(t))→ u∗, as t→ T, in H˙1,λ(t) = c(u0)(1 + ot→T (1))(T − t)L| log(T − t)| 2L2L−1, c(u0) > 0,with the case L = 1 providing the generic blow-up rate.On the other hand, Grotowski and Shatah ([65]), using maximum principle meth-ods as well, and assuming particular bounds on the initial data, showed that on theunit disc in R2 blow-up will not occur in finite-time for degrees m ≥ 2. One of ourgoals is to extend this result to the case of domain R2 and give a maximum principle-free proof.In this work we will specialize to maps with some symmetry, namely co-rotational24maps of degree m, i.e., maps of the formu(r, θ) = (cos(mθ) sin(u(r, t)), sin(mθ) sin(u(r, t)), cos(u(r, t))),Since |∇u|2 = u2r +m2r2sin2(u) (easy to check), and writing the Laplacian in polarcoordinates, plugging the above ansatz into the equation givescos(mθ) cos(u(t, r))ut = cos(mθ) sin(u(t, r))u2r + cos(mθ) sin(u(t, r))m2r2sin2(u(t, r))+ cos(mθ)(− sin(u(t, r))u2r) + cos(mθ) cos(u(t, r))urr+ cos(mθ) cos(u(t, r))urr− cos(mθ)m2r2sin(u(t, r))orcos(u(t, r))ut = cos(u(t, r))urr + cos(u(t, r))urr+ sin(u(t, r))m2r2sin2(u(t, r))− m2r2sin(u(t, r))= cos(u(t, r))urr + cos(u(t, r))urr− sin(u(t, r))m2r2cos2(u(t, r))= cos(u(t, r))urr + cos(u(t, r))urr− m22r2sin(2(u(t, r)) cos(u(t, r)),which simplifies to the following equation for the angle u :ut = urr +1rur −m2 sin 2u2r2(2.2)Moreover, if (2.2) holds, then the other components of ~u are also easily seen to satisfythe heat-flow equation. We make the following definitions:∆ru = urr +1rur, the radial Laplacian in R2,∆mu = (∆r − m2r2)u,(2.3)25thus, we can rewrite the equation asut = (∆r − m2r2)u+m2r2(u− sin 2u2),and symbolically as ut = ∆mu + F (u), denoting the new nonlinear term by F (u) =m2r2(u− sin 2u2).The energy for these maps is given byE(u(t, r)) := pi∫ ∞0(u2r +m2 sin2(u)r2)rdr.The stationary solutions are Q(r) = pi − 2 arctan(rm), and for any s > 0, Q( rs) isalso a solution. These maps minimize the energy within their topology class, andthey are very important objects when addressing the questions of global existenceand singularity formation, in that they are the objects which provide the naturalthresholds for global wellposedness and decay (in a sense to become precise later).We make the following definitionsE0 := {u : E(u) < 2E(Q);u(0) = 0, limr→∞u(r) = 0},E1 := {u : E(Q) ≤ E(u0) < 3E(Q);u(0) = pi, limr→∞u(r) = 0},Remark 2.1. The assumption of finite energy can be easily shown to be sufficientto guarantee the existence of the above limits.The main goal of this chapter is the proof of the following theorem:Theorem 2.2. Assuming u0 ∈ E0, and m ≥ 2 the corresponding solution u(r, t) to(2.2) exists globally in time and is smooth; furthermore, E(u(t))t→∞−−−→ 0.We give a proof which is suited to non-compact domains which makes use of the“concentration-compactness plus rigidity approach” of Kenig and Merle [79], orig-inally developed for dispersive equations. We will establish both local and global26wellposedness for maps in E0. Our results rest upon the extended literature on criti-cal dispersive equations concerning global existence and scattering, and also providealternative ways to look at parabolic flows. We won’t try to provide a complete list ofreferences for all the recent work on the issues of global existence and scattering fordispersive equations, which has become a very active area of research following thebreakthrough ideas of Bourgain [12] and of Colliander-Keel-Staffilani-Takaoka-Tao[24] (the “induction on the energy” method), and in particular Kenig and Merle [79](the “concentration-compactness plus rigidity” approach). We refer the reader tothe excellent notes [82] for more background information.We will now present an outline of the approach we take in this chapter, with moredetails to follow in the subsequent sections.First, we establish local wellposedness for maps in E0. Despite the presence ofsome results in the literature, they are written within the classical parabolic frame-work so we prefer to redo the theory specializing to the co-rotational class and in away that can be compared to the dispersive literature for related problems. In par-ticular, this is mostly apparent in our blow-up alternative which, in this formulation,bonds well with the Kenig-Merle approach.A “concentration-compactness plus rigidity” procedure excludes the possibilityof finite-time blow-up. The key tools here are a profile decomposition for a boundedsequence in an H˙1-like space, applied on an appropriate sequence of initial data, anda stability-under-small-perturbations result. As a result, we establish the existenceof a putative minimal counterexample; the dissipation of energy contradicts its exis-tence implying global existence and decay to zero.There was no readily available profile decomposition directly applicable to oursetting, so this was one of the main intermediate tasks to achieve. The difficultystems mainly from the absence of some Sobolev embeddings in dimension two, sowe take an indirect approach by first establishing estimates on the linear evolutionin higher dimensions which then connect back to our problem through a reductiontrick.272.2 Some analytical ingredients2.2.1 Energy properties of maps in E0We will first show that maps in the class E0, the energy space is naturally endowedwith the X2-norm :‖u‖2X2 =∫ ∞0(|ur|2 +m2|ur|2)rdr.This equivalence only holds for the class E0.By finiteness of the energy, the limits at zero and infinity exist.Lemma 2.3. If u ∈ E0, with E(u) ≤ 2E(Q) − δ1, for some δ1 > 0, there is aδ2 = δ2(δ1) > 0 such that|u(r)| ≤ pi − δ2, (2.4)Proof. As in [119] we defineG(u) := pi∫ u0m| sin(s)|dsandEr2r1 (u) := pi∫ r2r1(u2r +m2 sin2(u)r2)rdr.Then for all r1, r2 ∈ [0,∞), by the Fundamental Theorem of Calculus and Young’sinequality:|G(u(r2))−G(u(r1))| =∣∣∣∣∫ r2r1∂∂rG(u(r))dr∣∣∣∣ = pi ∣∣∣∣∫ r2r1m| sinu|urdr∣∣∣∣≤ pi2∫ r2r1(m2 sin2(u)r2+ u2r)rdr ≤12Er2r1 (u)(2.5)In this case G(u(∞)) = G(0) = 0, G(u(0)) = G(pi) = 0. From (2.5) for any r > 0:|G(u(r))| = |G(u(r))−G(u(0))| ≤ 12Er0(u)28and|G(u(r))| = |G(u(∞))−G(u(r))| ≤ 12E∞r (u).Thus2|G(u(r))| ≤ 12E(u) ≤ E(Q)− δ12,hence for every r:|G(u(r))| ≤ 12E(Q)− δ14.Note that G is odd, increasing on [−pi, pi] and, G(pi) = 2mpi = E(Q)2and sinceG−1(−E(Q)2) < u(r, t) < G−1(E(Q)2), there is a δ2 > 0|u(r)| ≤ pi − δ2as claimed.This information is enough in order to prove the desired equivalence between theenergy and the X2-norm. In particular, we prove:Proposition 2.4. The class E0 is naturally endowed with the norm ‖ · ‖X2 , in thesense that, given δ1 > 0, there exist C1, C2 > 0 such that C2‖u‖2X2 ≤ E(u) ≤C1‖u‖2X2 , for all u ∈ E0 with E0 ≤ 2E(Q)− δ1.Proof. By elementary calculus; the previous lemma granting the uniform constants.We remark that in E0, there are no nontrivial harmonic maps as a result of theboundary conditions and the monotonicity of the stationary solutions (see [29]).292.2.2 Local wellposedness for maps in E0From now on, unless otherwise specified, all the norms will be considered withthe measure rdr. We define rLp := {measurable z : z = rf, f ∈ Lp}, and the spacesXp equipped with the norm‖u‖pXp :=∫ ∞0(|ur|p +m2|ur|p)rdr.We say that a function u : I × R→ R, I = [0, T ) is a solution to the problem{ut = (∆r − m2r2 )u+ F (u),u(0) = u0 ∈ X2,(2.6)where F (u) = m2r2(u− sin 2u2), if it lies in CtX2r ∩L4t rL4r(K), for every compact K ⊂ I,and obeys the following Duhamel formula for every t ∈ I :u(t) = et∆mu0 +∫ t0e(t−s)∆mF (u)ds. (2.7)We can summarize the local theory in the following theorem:Theorem 2.5. (Local wellposedness)1. (Local Existence) Let u0 ∈ X2. There exists an > 0 such that, if I = [0, T ),and ‖et∆mu0‖L4t (I;rL4) < , then there exists a unique solution to (2.6) withu ∈ C(I;X2), ‖u‖L4t (I;rL4) ≤ 2. To each initial datum u0 we can associate amaximal interval of existence I = [0, Tmax(u0)), where Tmax(u0) can be +∞.2. (Blow-up Criterion) Tmax(u0) < +∞⇒ ‖u‖L4t ([0,Tmax(u0)];rL4r) = +∞.3. (Dissipation) If Tmax(u0) = +∞ and ‖u‖L4t ([0,∞];rL4r) < +∞, then‖u(t)‖X2 → 0, as t→ +∞.4. (Small data implies global existence and dissipation) If ‖u0‖X2 is sufficientlysmall, the solution is global and decays to zero, in the above sense.305. (Continuous Dependence on Initial Data) The solution depends continuouslyon the initial data. Furthermore, Tmax is a lower semi-continuous function ofthe initial data.The proof relies on the space-time estimates established in [66]. From now on, wewill be referring to them as “the space-time estimates”:For H = −∆r + m2r2 , d = 2,m ≥ 2 :(i)‖e−tHφ‖Lpr . t−(1/a−1/p)‖φ‖La , 1 ≤ a ≤ p ≤ ∞ (2.8)(ii)‖e−tHφ‖LqtLpr ≤ C‖φ‖La ,1q=1a− 1p, 1 < a ≤ q (2.9)(iii)‖∫ t0e−(t−s)Hf(s)ds‖LqtLpr . ‖f‖Lq˜′t Lp˜′r (2.10)for admissible pairs (q,p),(q˜, p˜) and the Ho¨lder-dual pair (q˜′, p˜′).(iv)‖e−tHφ‖L∞t X2∩L2tX∞+‖∫ t0e−(t−s)Hf(s)ds‖L∞t X2∩L2tX∞ . ‖φ‖X2 +‖f‖L1tX2+L2tX1(2.11)In dimension d = 2, an admissible pair (q, p) satisfies : 1q+ 1p= 12, including theendpoint (2, 2,∞).The proof of the local wellposedness is a mild variant of the classical work ofCazenave-Weissler [21] for the critical NLS, and is based on a fixed point theorem.The strategy is standard but we will give the details of the proof for completeness.Proof. We will show local existence and uniqueness, give a blow-up alternative andthen also prove decay for global solutions.31Local Existence: Define the solution operatorΦu0(v) := et∆mu0 +∫ t0e(t−s)∆mF (v)ds.We will use Banach’s Fixed Point Theorem by showing Φu0(·) is a contractionmapping on Ba,b := {v on I × [0,∞) : ‖v‖L4(I;rL4) ≤ a, ‖v‖L∞(I;X2(R2)) ≤ b}, a,b tobe determined. In what follows, standard pointwise estimates based on the Taylorexpansions of trigonometric functions and the Mean Value Theorem are employed.We will first show that Φu0(·) maps Ba,b to itself (for a particular choice of a, b).Set M := ‖u0‖X2 . Using Duhamel’s formula, the space-time estimates, Ho¨lder’sinequality and the assumption on the evolution of the initial data:‖Φ(u)‖L∞X2 .M + ‖Fr‖L2L1 + ‖Fr‖L2L1 . ‖u3r3‖L2L1 + ‖uru2r2‖L2L1.M + ‖u‖L∞X2 · ‖u‖2L4rL4 .M + b · a2⇒ ‖Φ(u)‖L4rL4 < + ‖F (u)‖L4/3t rL4/3r (I) . + ‖u‖3L4tL4r. + a3Denote by C the largest of all the implied constants, and make the following choices:pick b = 2CM and choose a such that C · a2 ≤ 12. Then pick = a2. Under thesechoices, Φ maps Ba,b to itself.Note that the intersection space can be equipped with the metric d(u, v) = ‖u−v‖L4rL4 + ‖u− v‖L∞X2 which makes the space L∞X2 ∩ L4rL4 complete. It remainsto be shown that the mapping above is a contraction:‖Φ(u)− Φ(v)‖L∞X2 .∥∥∥∥F (u)− F (v)r∥∥∥∥L4/3L4/3+ ‖(F (u)− F (v))r‖L4/3L4/3.∥∥∥∥u− vr (|u|+ |v|)2r2∥∥∥∥L4/3L4/3+∥∥∥∥|vr| |u− v|r (|u|+ |v|)r∥∥∥∥L4/3L4/3. ‖u− v‖L4rL4‖u+ v‖2L4rL4 + ‖vr‖L4rL4‖u− v‖L4rL4‖|u|+ |v|‖L4rL4≤ C(‖u‖L4rL4 , ‖v‖L4rL4 , ‖v‖L∞X2) ‖u− v‖L4rL4 ,32and ‖Φ(u) − Φ(v)‖L4rL4 is treated the same way. The constant C above, dependsin a multiplicative way on ‖u‖L4rL4 , ‖v‖L4rL4 ; hence, by further shrinking we canarrange for Φ to be a contraction. Continuity in time follows again directly from thespace-time estimates.Maximal Interval of ExistenceIf u(1), u(2) are solutions of (2.6) on the interval I = [t0, T ) with the same initialdata at t0, then u(1) ≡ u(2) on I. To see that, for any τ we can partition [0, τ)into finitely many subintervals Ij, such that ‖u(i)‖L4rL4(Ij) ≤ 2. Then, if t0 ∈ Ij0for some j0, the uniqueness of the fixed point in the proof above, yields an in-terval Iˆ 3 t0, where u(1) ≡ u(2). A continuation argument proves uniqueness inall of I. This allows us to associate a maximal interval to given initial data, i.e.,I(u0) = [0, 0 + Tmax(u0)), Tmax(u0) > 0, whereTmax(u0) = sup{T (u0) > 0 : there is a solution of (2.6) on [t0, t0 + T (u0)]}Blow-Up Criterion: If Tmax(u0) < +∞, then ‖u‖L4t ([0,Tmax(u0));rL4r) = +∞.Assume, for contradiction, that Tmax(u0) < +∞ but ‖u‖L4rL4([0,Tmax(u0)) < +∞. Forsome fixed ˜, subdivide the interval [0, Tmax(u0)) into a finite number of intervalsIj so that ‖u‖L4rL4(Ij) ≤ ˜. Picking a time τ close to Tmax(u0) (how close is tobe determined) and solving the equation on [τ, Tmax(u0)) using Duhamel’s formula:u(t) = e(t−τ)∆mu(τ) +∫ tτe(t−s)∆mF (u(s))ds, and solve for e(t−τ)∆mu(τ). To estimate‖e(t−τ)∆mu(τ)‖L4rL4([τ,Tmax(u0))) : arguing as before‖e(t−τ)∆mu(τ)‖L4rL4([τ,Tmax(u0))) ≤ ‖u‖L4rL4([τ,Tmax(u0))) + C‖u‖3L4rL4([τ,Tmax(u0))),where the constant C is independent of τ, Tmax. Arranging for τ to be such thatC‖u‖3L4rL4([τ,Tmax(u0))) ≤ ‖u‖L4rL4([τ,Tmax(u0))) < ˜, with ˜ ≤ 4 (where is in the exis-tence proof) we get: ‖e(t−τ)∆mu(τ)‖L4rL4([τ,Tmax(u0))) ≤ 2 . Thus, by continuity, we canfind 0 such that‖e(t−τ)∆mu(τ)‖L4rL4([τ,Tmax(u0)+0))) < ,and hence, the solution extends past Tmax(u0), which contradicts the assumed max-33imality of this lifespan.Proof of the dissipation result:We will compare with the linear evolution: to begin, observe that by assumption,for any > 0, there is a T > 0 : ‖u‖3L4rL4([T,+∞)) < ‖u‖L4rL4([T,+∞)) < 2 .We first show that for such a T, ‖u(t) − e(t−T )∆mu(T )‖X2 → 0, as t → +∞ whichwould imply the statement since‖u(t)‖X2 ≤ ‖u(t)− e(t−T )∆mu(T )‖X2 + ‖e(t−T )∆mu(T )‖X2and the last term goes to zero (for all T > 0) by Young’s inequality using the explicitkernel for the linear evolution and the density of L1∩L2 in L2. In particular, for every, there is N > 0 such that, for all n > N : ‖φn−u(T )‖X2 < 4 , where φn ∈ X1∩X2.However for data φn ∈ X1, putting the kernel in X2 in the application of Young’sinequality, we get ‖e(t−T )∆mφn‖X2 < 4 , t > T˜ , for all n, some T˜ corresponding to this, while for all times t, we get ‖e(t−T )∆m(u(T )− φn)‖X2 < 4 , by (2.8) and density.Definew(t) = u(t)− e(t−T )∆mu(T ).Next, rearranging and using w(T ) = 0 we getu(t)− w(t) = e(t−T )∆m [u(T )− w(T )]and thus, w satisfieswt = ∆mw + F (u)(since u is a solution). Writing Duhamel’s formula with initial time t0 = T and againusing w(T ) = 0, and (2.10) :w(t) =∫ tTe(t−s)∆mF (u)ds.34‖u(t)− e(t−T )∆mu(T )‖X2 = ‖w(t)‖X2 ≤ ‖w‖L∞X2 ≤ ‖F (u)‖L4/3rL4/3. ‖u‖3L4rL4([T,t]) . completing the proof, since was arbitrary.Notice that our local theory combined with the previous section on the equivalenceof the X2 and the energy topology implies that if u0 ∈ E0 the boundary conditionspersist in time, i.e. u(t, ·) ∈ E0 throughout its lifespan.2.2.3 Stability under perturbationsAn important consequence of the local existence proof, is the following Perturba-tion Theorem on which much of this work is relying. We are interested in developinga stability theory, in the sense that we would like to prove the existence of a solutionto (2.6), given an approximate one. This will be done in two steps, the first one(Short-time perturbations) assuming some “smallness”, and the second one (Long-time perturbations), without the smallness assumption, iterating the argument ofthe first step. To the best of our knowledge, theorems of this type were first provedby Colliander-Keel-Staffilani-Tao-Takaoka [23] (and later improved and simplified byTao-Visan, cf. [124].) In what follows, define z0 := z(t = 0). Moreover, we will beusing the standard notation ‖z‖S(I) for admissible, and ‖z‖N(I) for the dual norms.Theorem 2.6. (Short-time perturbations)Let I ⊂ R be a time interval of the form [0, T ). Let u˜ be defined on I × [0,∞) andsatisfyu˜t −∆mu˜+ F (u˜) = e,‖u˜‖L∞(I;X2) < +∞, (2.12)35and let u0 ∈ X2. Assume also the smallness conditions‖u˜‖L4(I;rL4) ≤ 0 (2.13)‖et∆m(u0 − u˜0)‖L4(I;rL4) ≤ (2.14)‖e‖L4/3(I;rL4/3) ≤ (2.15)for some 0 < ≤ 0. Then, there exists a solution of (2.6) on I×[0,∞) with u(0) = u0satisfying‖u− u˜‖L4(I;rL4) . . (2.16)Remark 2.7. By (2.9), the assumption ‖et∆m(u0 − u˜0)‖L4rL4(I) ≤ is redundant if‖u0 − u˜0‖X2 is small.Proof. By the wellposedness theory reviewed above, it suffices to prove (2.16) as an apriori estimate. Let w = u− u˜. Then, w satisfies the following initial value problem{wt −∆mw = F (u˜+ w)− F (u˜)− e,w0 = u0 − u˜0For t ∈ I, define A(t) := ‖F (u˜+ w)− F (u˜)‖L4/3rL4/3([0,t]).We have F (u˜+ w)− F (u˜) = m2r2w + m22r2[sin(2u˜)− sin[2(u˜+ w)]].Using once more the pointwise bounds coming from the Taylor expansion:‖F (u˜+ w)− F (u˜)‖L4/3rL4/3 . ‖(|w|u˜2 + |u˜|w2 + |w|3)r3‖L4/3L4/3. ‖w‖3L4rL4 + ‖u˜2wr3‖L4/3L4/3 + ‖u˜w2r3‖L4/3L4/3and this by Ho¨lder’s:A(t) . ‖w‖3L4rL4 + ‖w‖2L4rL4‖u˜‖L4rL4 + ‖w‖L4rL4‖u˜‖2L4rL4 . (2.17)Also, from the integral formula for w, the space-time estimates and the assumptions36(2.14) and (2.15),‖w‖L4rL4(I) . + A(t). (2.18)Combining (2.17) and (2.18), by a continuity argument, picking 0 small enough, wecan deduceA(t) . which finishes the proof of 2.16.Subdividing an arbitrary interval I so that the smallness assumption (2.13) ap-plies, and by verifying the assumptions of the previous theorem, we can keep iteratingthe process to cover the whole interval, having only to adjust a finite number oftimes; in particular, we can prove the following:Theorem 2.8. (Long-time perturbations) Let I = [0, T ), a time interval.Let u˜ be defined on I × [0,∞) and satisfyu˜t −∆mu˜+ F (u˜) = e, in I× R2‖u˜‖L∞(I;X2) ≤M (2.19)‖u˜‖L4(I;rL4) ≤ L (2.20)Let u0 be such that‖u0 − u˜0‖X2 ≤M ′ (2.21)for some constants M,M ′, L > 0. Assume also the smallness conditions‖e(t−t0)∆m(u0 − u˜0)‖L4(I;rL4) ≤ (2.22)‖e‖L4/3(I;rL4/3) ≤ (2.23)for some 0 < ≤ 1 = 1(M,M ′, L). Then, there exists a solution of (2.6) withu(0) = u0 satisfying‖u− u˜‖L4rL4(I) ≤ C(M ′,M,L)M ′ (2.24)Proof. Subdivide I into J = J(0, L) subintervals Ij = [tj, tj+1], 0 ≤ j < J such that37‖u˜‖L4rL4(Ij) ≤ 0, where 0 = 0 is as in the previous theorem.Consider the first subinterval I1 = [0, t1] (assume for simplicity t0 = 0. By theprevious theorem, for small enough (depending only on M ′) we have‖u− u˜‖L4rL4(I1) ≤ C(M ′)M ′ (2.25a)‖F (u)− F (u˜)‖L4/3rL4/3(I1) ≤ C(M ′) (2.25b)Now, to proceed with I2 : in order to apply the Short-time Perturbations Theorem2.6, we need to verify the stated assumptions.To estimate ‖u(t1)−u˜(t1)‖X2 : writing Duhamel’s formula for w = u−u˜ on I1 = [0, t1],for t = t1 we get:‖w(t1)‖X2 = ‖u(t1)− u˜(t1)‖X2 ≤ ‖et1∆mw(0)‖X2+‖∫ t10e(t1−s)∆m [F (u)− F (u˜)]ds‖X2 + ‖∫ t10e(t1−s)∆meds‖X2 .To estimate ‖∫ t10e(t1−s)∆m [F (u)− F (u˜)]ds‖X2 , we use (2.10) to get‖F (u)− F (u˜)‖L4/3rL4/3([0,t1]) ≤ C1(M ′) by the result for the first interval.By (2.10): ‖∫ t10e(t1−s)∆meds‖X2 ≤ ‖e‖L4/3rL4/3([0,t1]) ≤ ‖e‖L4/3rL4/3(I) ≤ .For ‖et1∆mw(0)‖X2 ≤ ‖et∆mw(0)‖L∞X2([0,t1]) ≤ ‖w(0)‖X2 = ‖u(0) − u˜(0)‖X2 ≤ M ′,by assumption. Combining all the above estimates‖w(t1)‖X2 ≤ C1(M ′)+M ′ + .Trivially by the assumption, ‖e‖L4/3rL4/3(I2) ≤ , so it remains to estimate‖e(t−t1)∆mw(t1)‖L4rL4(I2).38To this end, using Duhamel’s formula on [0, t1], for t = t1, we get:‖e(t−t1)∆mw(t1)‖L4rL4(I2) = ‖e(t−t1)∆m [et1∆mw(0) +∫ t10e(t1−s)∆m [F (u)− F (u˜)]ds+∫ t10e(t1−s)∆meds]‖L4rL4(I2)≤ ‖e(t−t1)∆met1∆mw(0)‖L4rL4(I2) +∥∥∥∥e(t−t1)∆m ∫ t10e(t1−s)∆m [F (u)− F (u˜)]ds∥∥∥∥L4rL4(I2)+∥∥∥∥e(t−t1)∆m ∫ t10e(t1−s)∆meds∥∥∥∥L4rL4(I2).We can see that∫ t10e(t1−s)∆m [F (u)− F (u˜)]ds,∫ t10e(t1−s)∆m eds are in rL2. Thus, by (2.8)≤ ‖et∆mw(t0)‖L4rL4(I2) + ‖∫ t10e(t1−s)∆m [F (u)− F (u˜)]ds‖rL2 + ‖∫ t10e(t1−s)∆meds‖rL2≤ ‖et∆mw(t0)‖L4rL4(I2) + ‖F (u)− F (u˜)‖L4/3rL4/3(I1)+‖e‖L4/3rL4/3(I1) ≤ 2+ C1(M ′)employing (2.10) (on I1) and the same arguments as before. Hence, we can applythe Short-time Perturbations Theorem 2.6 to conclude the result for I2 as well.Obviously, we can keep iterating the process to cover the whole interval having onlyto adjust a finite number of times and thus, complete the proof.Using the last theorem we can easily deduce continuous dependence on initialdata, by the following argument: choose u0 = u0,n, u = un, u0,nX2−→ u˜0, for n thatare large enough for‖u0,n − u˜0‖X2 ≤ ≤ 1(M,L),where 1 is the one required by the previous theorem.Also, applying (2.9): ‖e(t−t0)∆m(u0,n − u˜0)‖L4rL4(I) . ; since un are solutions of39(2.6), e ≡ 0. So the above theorem affirms that the solutions un exist on I, and‖un − u˜‖L4rL4(I) n→∞−−−→ 0.2.2.4 Profile decompositionThe main goal of this paragraph is to prove the following proposition whichis the main tool (along with the Perturbation Theorem) used to establish globalexistence and decay. Profile Decompositions have become one of the main tools in thetreatment of global behaviour for critical equations. The idea is to characterize theloss of compactness in some embedding, and in some way recover some compactness.It can be traced back to ideas in [98], [13], [121], [117] and their modern “evolution”counterparts [3], [79] and [80].Proposition 2.9. Let {un}n be a bounded sequence in X2. Then, after possiblypassing to a subsequence (in which case, we suppress notation and rename it unagain), there exist a family of radial functions {φj}∞j=1 ⊂ X2 and scales λjn > 0 suchthat:un(x) =J∑j=1φj(xλjn) + wJn(x), (2.26)wJn ∈ X2 is such that:limJ→∞lim supn‖et∆mwJn‖L4t rL4r = 0, (2.27)wJn(λjnx) ⇀ 0, in X2, ∀j ≤ J. (2.28)Moreover, the scales are asymptotically orthogonal, in the sense thatλjnλin+λinλjn→ +∞, ∀i 6= j (2.29)40Furthermore, for all J ≥ 1 we have the following decoupling properties:‖un‖2X2 =J∑j=1‖φj‖2X2 + ‖wJn‖2X2 + on(1) (2.30)E(un) =J∑j=1E(φj) + E(wJn) + on(1). (2.31)The procedure through which one establishes such a decomposition has becomestandard by now, for example see [3],[81], thus we will only present the equation-specific parts of the argument.There are two general roadmaps to follow in establishing such a decomposition.To get the convergence of the error wJn in the appropriate space-time norm, one caneither use directly a refinement of space-time estimates on the linear propagator, ora refinement on a Sobolev inequality through which the refinement of the space-timeestimates will follow, making use of interpolation arguments. The first approach isa more modern one, yet it would require more work in our case. Modification of ar-guments used in the Schro¨dinger case cannot directly be made due to the lack of ananalogue of the restriction theorems used. For the second approach, a remark is thatthe case of dimension two is very special due to the lack of the usual embeddings.Our strategy evolves around a refinement of a Sobolev inequality in which thisdimension issue is directly addressed. We first establish (2.27) for the homogeneouslinear heat equation for radial functions in higher dimensions. We make use of arefined Sobolev inequality, first proved in [3] for d = 3 and later generalized to d > 3in [14]. Then, through an isomorphism between H˙1 and X2, we connect the aboveestimate to our spaces for the 2d problem, and use interpolation again to obtain thedesired convergence in the norm in which the blow-up criterion was stated.Definition 2.10. We call a pair (q, p) L2-admissible in dimension d, if the followingrelation is satisfied2q+dp=d2, (2.32)41and H˙1-admissible if2q+dp=d− 22. (2.33)We define the following Besov norm on L2 :Ik(f) :=(∫2k≤|ξ|≤2k+1|fˆ(ξ)|2dξ)1/2, ‖f‖B := supk∈ZIk(f).The following refinement of the Sobolev inequality was proved in [14] (Lemma 3.1)Lemma 2.11. (Refined Sobolev) For d ≥ 3 there is a constant C = C(d) > 0 suchthat for every u ∈ H˙1(Rd), we have‖u‖Lp ≤ C‖∇u‖2pL2‖∇u‖1− 2pB , (2.34)where p = 2dd−2 (sometimes denoted by 2∗).The next result, proved originally in [59], provides a decomposition of bounded se-quences in L2(Rd) (for a different, but equivalent Besov norm). Here, we specializeto radial functions.Proposition 2.12. Let {fn}n be a bounded sequence of radially symmetric functionsin L2(Rd), d ≥ 3. Then, there exist a subsequence (still denoted by {fn}n), a sequenceof scales {λjn}n ⊂ (0,∞) satisfying (2.29), and bounded radial {gj}j, {rJn}n ⊂ L2(Rd),such that for every J ≥ 1, x ∈ Rdfn(x) =J∑j=11(λjn)d/2gj(xλjn) + rJn(x), (2.35a)‖fn‖2L2 =J∑j=1‖ 1(λjn)d/2gj(xλjn)‖2L2 + ‖rJn‖2L2 + on(1), (2.35b)limJ→∞lim supn→∞‖rJn‖B = 0. (2.35c)42Applying the above result to a bounded radially symmetric sequence {vn} ⊂ H˙1(Rd),we conclude that there is a subsequence (again denoted by {vn}), a family of scalesλjn as before (satisfying the orthogonality property (2.29)), and a family of (boundedin H˙1) radial functions ψj such thatvn(x) =J∑j=1ψjn(x) + w˜Jn(x),∀J ≥ 1, , (2.36)where ψjn(x) :=1(λjn)d2−1ψj( xλjn),‖vn‖2H˙1 =J∑j=1‖ψj‖2H˙1+ ‖w˜Jn‖2H˙1 + on(1)andlim supn‖∇w˜Jn‖B J→∞−−−→ 0. (2.37)Let us consider the homogeneous heat equation on Rdvt = ∆v (2.38)and denote the linear propagator by S(t) := et∆, i.e. the solution with given initialdata v0, is v(t) = S(t)v0. Evolving (2.36) by the linear propagator we get:S(t)vn =J∑i=1S(t)ψjn + S(t)w˜Jn .Our first goal is to estimate S(t)w˜Jn in an appropriate space-time norm. If σ is afunction on Rd, we define σ(D) byσ̂(D)f(ξ) := σ(ξ)fˆ(ξ).Also define σk(ξ) := χ2k≤|ξ|≤2k+1(ξ), k ∈ Z. Then if z a solution to (2.38), by commu-tation of Fourier multipliers with derivatives, σk(ξ)z is also a solution to (2.38). By43dissipation we have, for any k, that‖∇(σk(ξ)z(t))‖L2 ≤ ‖∇(σk(ξ)z0)‖L2 .Using Plancherel’s identity and properties of the Fourier transform and the definitionof multipliers:‖∇(σkz)‖L2 = ‖∇̂(σkz)‖L2 = ‖|ξ|σkzˆ‖L2 = Ik(∇z)Hence, by taking supremum in k,‖∇z(t)‖B ≤ ‖∇z0‖B.Applying this general observation to the evolution starting with initial data w˜Jn :‖∇(S(t)w˜Jn)‖L∞t Bx ≤ ‖∇w˜Jn‖B.Then, due to (2.37) we concludelimJ→∞lim supn→∞‖∇(S(t)w˜Jn)‖L∞t Bx = 0. (2.39)For every t > 0, (2.34) gives:‖S(t)w˜Jn‖L2dd−2x. ‖∇(S(t)w˜Jn)‖2(d−2)2dL2 · ‖∇(S(t)w˜Jn)‖1− 2(d−2)2dB. ‖∇w˜Jn‖2(d−2)2dL2 · ‖∇(S(t)w˜Jn)‖1− 2(d−2)2dB⇒ ‖S(t)w˜Jn‖L∞t L2dd−2x. ‖∇w˜Jn‖2(d−2)2dL2 · ‖∇(S(t)w˜Jn)‖1− 2(d−2)2dL∞t Bx.Since, by assumption, ‖∇w˜Jn‖L2 is uniformly bounded, using (2.39):limJ→∞lim supn→∞‖S(t)w˜Jn‖L∞t L2dd−2x= 0. (2.40)44The radial Laplacian in dimension d = 2m+2 is ∆m = ∂rr+2m+1r∂r =1r2m+1∂r(r2m+1∂r).Letting v = urm, if u is radial and ut = ∆mu in R2, then v solves vt = ∆v. It isstraightforward to verify, using the Hardy inequality in dimension d = 2m+ 2, thatthis map is an isomorphism between our space X2 and H˙1rad(R2m+2) (e.g., see Lemma4 in [32]).The reason we are using this transform is so that we can make use of (2.2.4) byconnecting the spaces used in our LWP theory in two space dimensions with theones involved in the preceding argument. First, we make the following observation:suppose (r, p) is an L2-admissible pair in d = 2, i.e., 1r+ 1p= 12. Then‖ur‖rLrt (I;Lpr(rdr)) =∫I(∫ ∞0r1−pr2m+2|u|pr2m+1dr)r/pUsing the above transformation u = v · rm we get:∫I(∫ ∞0r1−pr2m+2rmp|v|pr2m+1dr)r/p=∫I(∫ ∞0rp(m−1)−2m|v|pr2m+1dr)r/p.Letting p = 2mm−1 , and thus r = 2m, we observe that‖ur‖Lrt (I;Lpr(rdr)) = ‖v‖Lrt (I;Lpr(R2m+2,r2m+1dr) (2.41)for this choice of (r,p) (which is an H˙1−admissible pair in dimension 2m+ 2). Thisobservation is the connecting link between the two-dimensional problem and thehigher-dimensional estimates. So for un bounded in X2, vn =unrmis bounded inH˙1(Rd) and we have (2.36). First, one has to show thatlimJ→∞lim supn→∞‖S(t)w˜Jn‖L2m(I;L 2mm−1 (R2m+2,r2m+1dr)) = 0. (2.42)For this we use interpolation and the previous result (2.40) for d = 2m+ 2. We have‖S(t)w˜Jn‖L2mt L 2mm−1 ≤ ‖S(t)w˜Jn‖m−1mL∞t L2m+2mr· ‖S(t)w˜Jn‖1mL2tL2(m+1)m−1rTaking limJ→∞lim supn→∞, and noting that the second term is uniformly bounded (by the45standard space-time estimates for the heat equation, see, e.g., [60]), the claim (2.42)follows. Undoing the transformation un = rmvn in (2.36) yieldsun(x) =J∑j=1φj(rλjn) + wJn , wJn = rmw˜Jn .Now, again by interpolation‖ur‖L4tL4r ≤ ‖ur‖m2(m−1)L2mt L2mm−1r· ‖ur‖m−22(m−1)L2tL∞r.Invoking (2.42), we get (2.27), i.e.,limJ→∞lim supn‖et∆mwJn‖L4t rL4r = 0.The rest of the proof of the profile decomposition follows the same arguments as inthe references cited at the beginning of this section and is thus omitted. We stillhave to show the asymptotic energy splitting with this choice of a space, i.e.(2.31):Proof. From now on, we will be systematically dropping the pi factor in the definitionof the energy. Expanding using the definition,E(un) =∫ ∞0[J∑j=1(1λjn)2(φjr(rλjn))2 +J∑j=1,i<j1λjnλinφjr(rλjn)φir(rλin) + (wJn,r(r))2+ 2J∑j=1wJn,r(r)1λjnφjr(rλjn) +m2r2sin2(J∑j=1φj(rλjn) + wJn,r(r))]rdr,We want to show that E(un)−J∑j=1E(φj)− E(wJn) is on(1).46Expanding this out∫ ∞0J∑j=1,i<j1λjnλinφjr(rλjn)φir(rλin)rdr + 2∫ ∞0[J∑j=1wJn,r(r)1λjnφjr(rλjn)]rdr+∫ ∞0m2r2[sin2(J∑j=1φj(rλjn) + wJn(r))−J∑j=1sin2(φj(rλjn))− sin2(wJn(r))]rdrFor the first two sums it suffices to look at single pairs and show they all are on(1).Using an approximation argument, we can assume every function involved is in C∞cand that all the supports lie in some ball B(0, R). The argument is standard so weonly give a sketch: for the first sum, we just change variables, assuming withoutloss of generality si,jn :=λjnλingoes to zero. Then, by Ho¨lder, each term in the sum isbounded by ‖φir‖X2∫ si,jn R0(φjr(r))2rdr = on(1). For the second one, change variablesagain and employ the weak convergence of wJn(λjnr) to zero, for all j ≤ J, i.e., (2.28).For the rest we will use of the trigonometric identitysin2(a+ b)− sin2(a)− sin2(b) = 12sin(2a) sin(2b)− 2 sin2(a) sin2(b)and the derived from it inequality| sin2(a+ b)− sin2(a)− sin2(b)| ≤ C|a||b| (2.43)for some C > 0.We want to show that∣∣∣∣∣∫ ∞0m2r2[sin2(J∑j=1φj(rλjn) + wJn(r))−J∑j=1sin2(φj(rλjn))− sin2(wJn(r))]rdr∣∣∣∣∣ = on(1)Using (2.43) J−1 times, this can be reduced to showing the following two estimates:47∫ ∞0|φj( rλjn)||φi( rλin)|r2rdr = on(1), i 6= j (2.44)∫ ∞0|wJn(r)|φj( rλjn )|r2rdr = on(1), for any j ≤ J (2.45)The proof of (2.44) follows the same rescaling argument as before. For (2.45), sincethe only information we have for the scales is their relation to each other and sincewe cannot assume convergence to zero, infinity or even existence of the limit at all, achange of variables will not be enough. Observe however, that the obvious change ofvariables makes the term |wJn(λjnr′)| appear, which suggests that somehow we haveto use the weak convergence to zero of the term which appears in the absolute value.Because of the absolute values we can’t directly obtain the result, yet the situationis not hopeless: a change of variables gives∫ ∞0|wJn(λjnr′)||φj(r′)|r′dr′(r′)2. Observe thatby Ho¨lder’s inequality and the definition of the X2-norm, the integrand is in L1(rdr),independent of n, since the sequence wJn is uniformly bounded in X2.Then, again by the fact that ‖φjr‖L2(rdr) < +∞, for every > 0 we can find anR = R() (which, since we will be working with very small , can be picked biggerthan 1) such that, (∫r≥R|φj(r)r|2rdr)1/2<3M,(∫r≤ 1R|φj(r)r|2rdr)1/2<3M,whereM := supn‖wJn‖X2 < +∞.Now, we split the integral at hand in three regions, using ( for all > 0) thisR = R().The first and the last, by Ho¨lder’s and the choice of R, are < 3, while for the one inthe middle we have by the obvious bounds for r and the fact that 1R< 1 by choice,48that ∫1R≤r≤R|wJn(λjnr)||φj(r)|rdr(r)2<∫1R≤r≤R|wJn(λjnr)||φj(r)|dr.But, on a bounded domain Ω = [a, b], a > 0, it is very easy to see that a u ∈ X2([a, b])is a function in H1([a, b]).Indeed,‖u‖2H1([a,b]) =∫a≤r≤b|ur|2dr +∫a≤r≤b|u|2dr =∫a≤r≤b|ur|2rrdr + |ur|2r2dr≤ 1a∫a≤r≤b|ur|2rdr + b∫a≤r≤b|ur|2rdr≤ C(∫a≤r≤b|ur|2rdr +∫a≤r≤b|ur|2rdr), C = max{1a, b};but this is exactly C‖u‖X2([a,b]).Now, we can invoke the continuous embedding of X2([a, b]) in H1([a, b]) we justproved, and the compact one of H1([a, b]) in L2([a, b]), to get that wJn(λjnr) goes to0 in L2([a, b]).With all this in hand, going back to the integral in question, we have that it issmaller by23+(∫1R≤r≤R|wJn(λjnr)|2dr)1/2·(∫1R≤r≤R|φj(r)|2dr)1/2,which in turn is below23+(∫1R≤r≤R|wJn(λjnr)|2dr)1/2· ‖φj‖L2 . Taking n largeenough, because of the strong convergence to zero that we proved earlier:(∫1R≤r≤R|wJn(λjnr|2dr)1/2<3‖φj‖L2 ,which completes the proof.492.3 Minimal blow-up solutionFor u0 ∈ E0 we defineEc = inf{E(u0) : u solves (2.6) with u(0) = u0, ‖u‖L4rL4([0,Tmax)) = +∞};i.e., we consider the infimum of the energies of initial data such that the correspondingsolutions fail to be global or decay to zero, in the sense discussed in the local theory.Note that Tmax can be finite (blow-up), or infinite (global but not decaying solution).Observe that Theorem 2.2 is equivalent to Ec ≥ 2E(Q). Also, a priori, Ec > 0. To seethat, note that we have shown that for maps in this class the energy is comparableto the X2-norm so small energy implies small (in X2− norm) initial data, and bythe space-time estimates ‖et∆mu0‖L4rL4(R+) is small, which in turn implies that u(t)is global and decays to zero.We will follow the contradiction approach of Kenig and Merle; we will assumeEc < 2E(Q) and proceed in two main steps: first, we show that a unique criticalelement exists. A critical element is initial data with energy Ec giving rise to asolution that that either fails to exist globally or decay to zero. The proof is basedon the profile decomposition (2.9) and the long-time perturbation theorem 2.8 (whichallows us to construct true solutions from approximate ones, thus obtaining profiledecompositions for the solution of the full non-linear equation), and the variationalinformation coming from the energy. The second step, is the so called “rigidity”part. In this part, we show that such a critical element cannot exist, thus reachinga contradiction. This is the content of the next section. Unlike the results in the“dispersive” literature, where a lot of work is required, our rigidity part is trivialbecause of the dissipation of energy.We now turn out attention to the existence of a critical element. In particular,we prove the following proposition, which is the main goal of this section:Proposition 2.13. Assume Ec < 2E(Q). There exists a function u0,c in X2 withE(u0,c) = Ec < 2E(Q), such that if uc(t, r) is the solution of (2.6) with initial datau0,c and maximal interval of existence I = [0, Tmax(u0,c)), then ‖uc‖L4rL4(I) = +∞.50Proof. Let {u0,n}n be a sequence in X2 such that E(u0,n) → Ec, n → ∞ fromabove, and the corresponding solutions un of (2.6) with maximal intervals of existenceIn = [0, Tmax(u0,n)) satisfy ‖un‖L4rL4(In) = +∞. By the comparability of the energyand the X2−norm the sequence {u0,n}n is bounded in X2, just by the convergence ofE(u0,n). Thus, passing to a subsequence, if necessary, we have the following profiledecompositionu0,n(r) =J∑j=1φj(rλjn) + wJn(r)with the stated properties in Proposition 2.9.For future reference, define a nonlinear profile vj : Ij × [0,∞)→ R associated toφj, to be the maximal-lifespan solution to (2.2) with initial data φj.For each j, n ≥ 1, define vjn : Ijn × [0,∞) → R, by vjn(r, t) = vj( t(λjn)2 ,rλjn), wherewe define Ijn := {t ∈ R+ :t(λjn)2∈ Ij}. I.e., each vjn is a solution to (2.2) with initialdata vjn(0) = φj( rλjn).We also have the energy decoupling in the limitE(u0,n) =J∑j=1E(φj) + E(wJn) + on(1), ∀J.Taking limn , we getEc =J∑j=1E(φj) + limnE(wJn)which by the positivity of every term impliesJ∑j=1E(φj) ≤ Ec, for any J. This, inturn, for the same reason givessupjE(φj) ≤ Ec.We eventually want to show that φj = 0, j ≥ 2 and E(φ1) = Ec. We consider thefollowing possibilities:51Case 1 : there is an > 0 such thatsupjE(φj) ≤ Ec − ,from which we will derive a contradiction. Since supjE(φj) < Ec, by the definitionof the Ec, all the φj ’s give rise to global and decaying to 0 solutions. Now, definethe approximate solution (to un(t) of the nonlinear equation with initial data u0,n)uJn(r, t) =J∑j=1vjn(r, t) + et∆mwJn(r).What we want to show is that uJn is a good approximate solution to un (for n, Jsufficiently large) in the sense of the Long-time Perturbations Theorem 2.8. Thiswould imply that un(t) is global, which is a contradiction.First, to see that ‖uJn‖L4rL4(R+) < +∞ : for any > 0, (2.27) provides a J such thatlimn‖uJn‖L4rL4(R+) ≤ limn‖J∑j=1vjn‖L4rL4(R+) + limn‖et∆mwJn‖L4rL4(R+)≤J∑j=1‖vj‖L4rL4(R+) + .To conclude the claim, we will show that the latter norms are bounded uniformly inJ . We can split the sum into two parts (for every fixed J); one over 1 ≤ j ≤ J0,and the rest. Let 0 be such that Theorem 2.5 affirms that if ‖u0‖X2 ≤ 0, then thecorresponding solution u is global and decays to zero, and pick J0 such that∑j≥J0E(φj) ≤ 0,which we can do again by the summability of the series; this shows that for j ≥ J0,the corresponding L4rL4-norms are uniformly (in J) bounded by some M(0), and52then choose the maximum bound among this M and the J0−1 bounds for the others.Hence,‖uJn‖L4rL4(R+) ≤ C,(for some C and n large enough).We need to show that the assumptions in the Long-time Perturbations Theorem2.8 hold. From the profile decomposition, ‖uJn(0) − un(0)‖X2 = 0,∀J, n and also,‖et∆m(uJn(0)− un(0))‖L4rL4(R+) = 0, ∀J, n. The perturbed PDE for uJn(t) is∂tuJn −∆muJn =J∑j=1F (vjn),hence the error is given byeJn = F (uJn)−J∑j=1F (vjn),where F is the nonlinear term F (u) = m2r2(u− sin 2u2). We will show that the error issmall in the dual norm ‖ · ‖L4/3t rL4/3r, again for sufficiently large n and J.By the explicit formula for F, the error iseJn =m22r2(2uJn − sin(2uJn)−J∑j=1(2vjn − sin(2vjn)).For simplicity, define W Jn (r, t) := et∆mwJn(r). We will make use of the followingtrigonometric relations:| sin(2u) + sin(2v)− sin(2u+ 2v)| = |2 sin(2u) sin2(v) + 2 sin(2v) sin2(u)|. |u||v|2 + |v||u|2.Using the above estimate:53|J∑j=1sin(vjn)+ sin(WJn )− sin(J∑j=1vjn +WJn )± sin(J∑j=1vjn)|. |J∑j=1sin(vjn)− sin(J∑j=1vjn)|+ |W Jn ||J∑j=1vjn|2 + |W Jn |2|J∑j=1vjn|Define A := |W Jn | |J∑j=1vjn|2 + |W Jn |2 |J∑j=1vjn|, B := |J∑j=1sin(vjn)− sin(J∑j=1vjn)|.By Ho¨lder’s:‖A‖L4/3rL4/3 ≤ ‖W Jn ‖L4rL4 ‖J∑j=1vjn‖2L4rL4 + ‖W Jn ‖2L4rL4 ‖J∑j=1vjn‖L4rL4≤ ‖W Jn ‖L4rL4(J∑j=1‖vjn‖L4rL4)2 + ‖W Jn ‖2L4rL4(J∑j=1‖vjn‖L4rL4).But, by (2.27)limJ→∞lim supn→∞‖W Jn ‖L4rL4 = 0,and hence, by the scaling invariance of the L4rL4−norm,limJ→∞lim supn→∞‖A‖L4/3rL4/3 = 0.For term B, again by adding and subtracting sin(J−1∑j=1vjn) we get, using the trigono-metric inequality:|J−1∑j=1sin(vjn)− sin(J−1∑j=1vjn)|+ |vJn ||J−1∑j=1vjn|2 + |vJn |2|J−1∑j=1vjn|We will show how to treat the second term, and after that the procedure can beeasily iterated. It consists of terms of the form |vjn||vjn|2 and |vjn|2|vjn| (employingYoung’s inequality for the product terms |vjn||vin|). We treat terms of the first type,54namely ‖ |vjn|2r2|vjn|r‖L4/3L4/3 (and the others follow in the same way).As before, we can employ an approximation argument and therefore assumeeverything is smooth and compactly supported in space-time, say on [0, T ]× [0, R].Without loss of generality, assume sJ,jn :=λJnλjn→ 0. Changing variables (in space andtime) and Ho¨lder’s inequality we get that the above is controlled by‖vj‖2L4rL4 ‖vj‖L4rL4([0,(sJ,jn )2T ]×[0,(sJ,jn )R])n→∞−−−→ 0.Everything else can be treated the same way, provinglimJ→∞lim supn→∞‖eJn‖L4/3rL4/3 = 0.Thus, we have shown that uJn(t) is a good approximate solution and hence, by theLong-time Perturbations Theorem 2.8, un(t) has to be global for sufficiently largen and J. As noted before, this contradicts the way the sequence of initial data hasbeen picked. So, Case 1 led to a contradiction and the only remaining possibility isthe alternativeCase 2 :supjE(φj) = EcThis immediately implies (possibly after a relabeling) that φj = 0, j ≥ 2 and theprofile decomposition simplifies tou0,n(r) = φ1(rλ1n) + w1n(r).By the energy splitting and the fact that E(u0,n)→ Ec, we getlimnE(w1n) = 0. (2.46)By (2.46) and the comparability of the energy and the X2-norm, defining u˜0,n(r) :=u0,n(λ1nr) we get u˜0,n → φ1, strongly in X2. As the proof shows, E(φ1) = Ec, andit turns out that φ1 gives rise to a blowing-up or non-decaying solution. To see55that, assume it doesn’t and again employ the Long-time perturbation Theorem 2.8to reach a contradiction. Thus, φ1 is the critical element we were looking for.2.4 RigidityIn this short section, we will show that the critical element found in the previoussection cannot possibly exist, hence completing the contradiction argument and theproof of Theorem 2.2.Proposition 2.14. The critical element found in Proposition 2.13 cannot exist.Proof. For a solution u(t) emanating from initial data u0, for all times t in themaximal interval of existenceE(u(t)) ≤ E(u0),with equality if and only if u is a stationary solution.So, we consider the following two cases for initial data φ1, for which we knowE(φ) = Ec ∈ (0, 2E(Q)):Scenario 1 : φ1 is a harmonic map, in which case u(t) is just a stationary solution.However, in this topology class and range of energies, there is no non-trivial harmonicmap (e.g. by [29], Proposition 1), hence the solution is global.Scenario 2 : φ1 is not a harmonic map. Then, the energy is strictly decreasing andimmediately drops below Ec for any t1 > 0. Employing uniqueness of solutions andthe definition of Ec, the new solution starting at t1 is global and decays, contradictingthe properties of φ. We have thus reached a contradiction which concludes the proofof the theorem.Having treated the below threshold case, the next goal becomes the investigationof higher energies and non-trivial topologies. This is the content of the next chapter.56Chapter 3Harmonic Map Heat Flow-AboveThreshold3.1 IntroductionThis chapter is devoted to our treatment of an above threshold case in the coro-tational harmonic map heat flow. We show that the solutions in this scenario areglobally smooth and asymptotically relax to a stationary solution.3.1.1 Global results for Landau-Lifshitz and Schro¨dinger mapsTo put our results into context (and because we will make use of some of therelated arguments), we will first state some recent results concerning the questionof global regularity vs singularity formation for the Landau-Lifshitz (LL) family ofequations∂u∂t= aPu∆u + bu× Pu∆u = a∆u + |∇u|2u + b×∆u, (3.1)with a ≥ 0, b ∈ R, which of course includes as special cases the harmonic map heat-flow (a = 1, b = 0) and the Schro¨dinger map (a = 0, b = 1). This is mostly work fromthe papers [70, 71] which address the Schro¨dinger case, and the paper [66] which57addresses the heat-flow case. This introduction concerns m-equivariant maps withenergy near the minimal energy Emin = 4pi|m| (the harmonic map energy), and sothe standing assumption on the initial data, unless otherwise specified, will beu0 ∈ Σm, E(u0) = 4pi|m|+ δ20, δ0 1Letu(t) ∈ C([0, T ); Σm)(Σm is topologized with the energy (H˙1 norm) be the solution of 3.1 correspondingto the initial data u0 (which is a priori just a local-in-time solution-see [71] for localwellposedness for this class of data).The next result shows that when the degree is sufficiently high, singularities willnot form, and moreover, solutions converge to specific harmonic maps as t→∞.Theorem 3.1. ([66, 70, 71] global regularity and asymptotic stability for high de-gree). For 3.1 with a > 0, assume |m| ≥ 3. For (a = 0), assume |m| ≥ 4. Thenumber δ0 is sufficiently small. Then1. there is no finite-time blow-up: the solution can be extended to u ∈ C([0,∞); Σm).2. For any r ∈ (2,∞], p ∈ [2,∞) with 1r+ 1p= 12, we have‖∇[u(x, t)− e(mθ+α(t))Rh( rs(t))]‖LrtLpx((R2×[0,∞)) . Cpδ0(if a > 0 we may include (r, p) = (2,∞))3. furthermore, there exist s+ > 0 and α+ withs(t)→ s+, α(t)→ α+, as t→∞.We remark that if T < ∞, then T is the maximal existence time (u(t) doesn’t58extend past T as a solution continuous into Σm) if and only iflim inft↗T−s(t) = 0.This theorem can be viewed, on one hand, as an orbital stability result for the familyOm of harmonic maps (at least up to the possible blow-up time), and on the otherhand as a characterization of blow-up for energy near Emin : solutions blow-up if andonly if the “length-scale” s(t) goes to zero. The space-time estimates above implyasymptotic convergence of the solutions to the family of harmonic maps in a space-time norm (“dispersive”) sense, which is the best we can expect for the Schro¨dingercase a = 0).The analysis of [70, 71, 66] in the equivariant setting fails to extend to m ≤ 3and a new approach was required. Handling the case m = 3 was one of the mainresults of [72]:Theorem 3.2. Let m ≥ 3, a = a + ib ∈ C/{0}, and a ≥ 0. Then there exists δ > 0such that for any u(0, x) ∈ Σm with E(u(0)) ≤ 4pim + δ2, there is a unique globalsolution u ∈ C([0,∞); Σm) of 3.1, satisfying ∇u ∈ L2t,loc([0,∞);L∞x ). Moreover, forsome µ ∈ C we have‖u(t)− emθRh[µ]‖L∞x + aE(u(t)− emθRh[µ])→ 0, as t→∞, (3.2)where emθRh(r/s) describes all the harmonic maps in Σm.Every solution with energy close to the minimum converges to one of the har-monic maps uniformly in x as t→∞. Even for the higher degrees m ≥ 4 this resultis stronger than the previous ones ([70, 71, 66]), where the convergence was givenonly in time average. Moreover, note that in the dissipative case (a > 0), solutionsconverge to a harmonic map also in the energy norm, while this is impossible for theconservative Schro¨dinger flow (b = 0).The analysis for the case m = 2 is trickier and the results weaker: in the harmonicmap heat flow case the strong asymptotic stability result of the previous theorem for59m ≥ 3 is no longer valid; instead, more exotic asymptotics are possible, includinginfinite-time concentration (blow-up).Concerning blow-up:Theorem 3.3. ([66] harmonic map heat flow blow-up for m = 1). Let m = 1. Forany δ > 0, there exists u0 ∈ Σ1 with 0 < E(u0)−4pi ≤ δ2 such that the correspondingsolution of the harmonic map heat flow blows up in finite time, in the sense that, forexample, ‖∇u(·, t)‖L∞x →∞.This result is an adaptation of the blow-up proof of [22] for a disk domain, tothe case of R2. Explicit blowing-up solutions (for m = 1) were provided recentlyby Raphael and Schewyer [114, 115], showing that (if T is the blow-up time), for1-corotational maps, initial data u0, L ∈ N∗u(t,r)−Q( rλ(t))→ u∗, as t→ T, in H˙1,λ(t) = c(u0)(1 + ot→T (1))(T − t)L| log(T − t)| 2L2L−1, c(u0) > 0.For a recent construction of a blowing-up solution on a bounded domain we refer to[37].3.2 Corotational maps in E1Moving to higher energies and non-trivial topologies one also encounters, otherthan harmonic maps, objects with energy much larger than the energy of the har-monic map; the approach taken in the above works (using “coordinate systems”about the family of harmonic maps, reflecting the near-minimality of the energy)cannot be implemented. In what follows all maps are in the class E1 (unless other-wise stated),E1 := {u : 2E(Q) ≤ E(u0) < 3E(Q);u(0) = pi, limr→∞u(r) = 0}60In this section we restrict ourselves to the corotational setting and take a differentapproach to proveTheorem 3.4. The solution to the m-corotational harmonic map heat flow in theclass E1 with m ≥ 4, is global and smooth with E(u(·, t) − Q( ·s∞ )) → 0, as t → ∞for some s∞ > 0.As before, we consider solutions u(r, t) ofut = urr +1rur +m22r2sin(2u) (3.3)with finite energyE(u) :=12∫ ∞0(u2r +m2r2sin2(u))r dr <∞and boundary conditionsu(0, t) = pi, u(∞, t) = 0. (3.4)Denote the static solution with these boundary conditions asQ(r) = pi − 2 tan−1(rm),and define the following quantitiesh(r) := sin(Q(r)) =2rm1 + r2m, hˆ(r) := cos(Q(r)) =r2m − 1r2m + 1For later use, we also record the easy computations hr = −mrhhˆ and hˆr =mrh2.Denote rescalings asQs(r) := Q(r/s), hs(r) = h(r/s), etc., s > 0.61Recall that the energy space (for maps with trivial topology) is:X2 = {w : [0,∞) 7→ R |∫ ∞0(w2r +w2r2)r dr <∞}.The results in this section are valid in the range m ≥ 4. It is an open questionwhether this behaviour is also expected of maps with m = 2, 3.3.2.1 Main resultsAs we have discussed earlier, the mechanism of singularity formation is that thesolution blows-up because of energy concentration, by bubbling off a harmonic map.The first important step is to show that concentration cannot happen at infinity(note that because of the corotational symmetry a singularity can only occur at theorigin or at infinity). This is the content of the next lemma:Lemma 3.5. Let u be a finite energy smooth solution on (3.3) on (0, T ). No energyconcentration at spatial infinity is possible:limR→∞lim supt→TE(u(t);BcR) = 0Before we present the proof of the lemma, some remarks are in order: amongother things, this result grants us direct access to many of the classical results inthe literature that were concerned with the case of a compact domain, for examplethe local theory irrespective of the topology assumptions, which also enables theuse of the same classical arguments to characterize blowing-up solutions (which onlyrequires local analysis).Proof. The energy dissipation relationE(u(t2))− E(u(t1)) =∫ t2t1ddtE(u(s))ds =∫ t2t1−‖ut‖2L2ds (3.5)for t2 > t1 will be of use.62We will first choose a radial smooth cut-off function ψ such thatψ(r) ={0 if r ≤ 11 if r ≥ 2and define ψR(r) := ψ(rR).If there was energy concentration at spatial infinity at time t = T :lim supt↗TE(u(t);BcR) ≥ δ > 0,∀R,for some δ > 0. So we could find sequences of radii Rn ↗∞ and times tn ↗ T suchthat limnE(u(tn), BcRn) ≥ δ > 0. We also define the “exterior” energy in terms of theabove cut-off:EˆR(t) :=∫ ∞0ψR(r)(u2r +m2r2sin2(u))rdrBy the finiteness of the energy, for any t0 < T there is an R0 > 1, such thatEˆR0(t0) ≤ δ4 . By assumption, there is T > t1 > t0 : EˆR0(t1) ≥ δ2 .Now, by direct calculation:ddtEˆR0(t) = −∫ ∞0ψR0u2t rdr −∫ ∞0urutdψR0drrdr.Putting everything together and usingddrψR(r) =1Rψ′(rR):δ4≤∫ t1t0ddtE(u(t);BcR0)dt = −∫ t1t0∫ ∞0ψR0u2t rdr −∫ t1t0∫ ∞0urutψ′R0rdr.≤(∫ t1t0∫ ∞0u2t rdr)1/2·(∫ t1t0∫ ∞0u2r(ψ′R0)2)1/2. 1R0(t1 − t0)1/2E1/2(u0),(because of 3.5) which yields a contradiction taking t0 ↗ T.Following [122] (or [6] directly in the corotational setting) and using Lemma 3.5 :63Lemma 3.6. If the solution blows-up in finite time, say T,u(tj, s(tj)r)→ Q, s(tj)→ 0in X2loc along a sequence {tj}n ↗ T.However working locally removes any knowledge of the topology of the map,which is determined by the behavior of the map at spatial infinity. We will improvethe above result in the corotational setting by working globally in space in the energytopology. Here we are forced to account for the topological restrictions of non-trivialdegree maps, and in fact we shall use these restrictions, along with our degree zerotheory (see the previous chapter), to our advantage.The improved convergence is as given in the following Proposition (a “no neck”proposition), which is a direct adaptation of Theorem 1.1 in [110]. The only possibil-ity not excluded in that work, and the only potential hindrance to a global-in-spaceconvergence, was the loss of energy at infinity; but this is already ruled out by Lemma3.5.Proposition 3.7. Let u0 ∈ E1 and u(t) the corresponding solution to (3.3) blowingup at time t = T withE(u0) < 3E(Q). (3.6)Then there exists a sequence of times tj ↗ T, a sequence of scales sj = o(√T − tj),a map w0 ∈ E0, and a decompositionu(r, tj) = Q(rsj) + w0(r) + ξ(r, tj) (3.7)such that ξ(tj) : ξ(tj, 0) = limr→∞ξ(tj, r) = 0 and ξ(tj)→ 0, in X2 as j →∞.In other words, if an m-corotational heat-flow with E(u(·, 0)) < 3E(Q) forms afirst singularity at time t = T < ∞, we can conclude that there are tj → T− and0 < sj → 0 such that E(u(tj)−Qsj−w0)→ 0 for some w0 ∈ X2 with E(w0) < 2E(Q).Note that the combination of the energy bound (3.6) and the boundary conditions64(3.4) prohibit the formation of more than one bubble.The rest of the section is devoted to proving that finite-time blow-up cannot happen;in particular, since the only possibility is blow-up by energy-concentration, it sufficesto show the following proposition:Proposition 3.8. Assume m ≥ 4. Suppose u(r, t) is a smooth solution of (3.3) on[0, T ) such that along some sequence tj → T−, there are sj > 0 such thatu(·, tj)−Qsj → w0 in X2. (3.8)for some w0 ∈ X2 with E(w0) < 2E(Q). Then sj 6→ 0.The argument has two main themes. First, it is a variant of the kind of localwellposedness-“stability” argument which is typical for nonlinear dispersive PDE,except that here we are in a topologically non-trivial setting, in which the staticsolution family is present. Second, because of this, we need to linearize about atime-dependent rescaled static solution (modulation theory), and specifically to ex-tend the [72] proof of asymptotic stability of equivariant harmonic maps to solutionsat higher energies, which are far from the static ones.Its proof is based on two main ingredients. For the first of these, we introduce thesolution w(r, t) of (3.3) with initial data at t = tj given by w0:wt − wrr − 1rwr − m22r2sin(2w) = 0w(r, tj) = w0(r) ∈ X2, E(w0) < 2E(Q). (3.9)By the results of Chapter 2, we know that w is a global, smooth solution with‖w‖L∞t X2∩L2t (X∞∩rX2)([tj ,∞) <∞. (3.10)For later use, we record one consequence of the higher regularity gained after the65initial time:wr2∈ L2tX2([t∗,∞)) for every t∗ > tj. (3.11)This follows from the observations that by standard parabolic regularity estimates(for example by performing energy-type estimates on the differentiated PDE), thefunction v(x, t) = w(r, t)eimθ satisfies D2v ∈ L2tL2x([t∗,∞)) , and that w/r2 and wr/rare controlled pointwise by |D2v| (for any m ≥ 2).For fixed s > 0, Qs is also a (static) solution of (3.3). Since the PDE is nonlinear,of course the sum Qs + w is not a solution:(∂t − ∂2r −1r∂r − m22r2sin(2 · ))(Qs + w)=m22r2(sin(2Qs) + sin(2w)− sin(2Qs + 2w))=m22r2(sin(2Qs)(1− cos(2w)) + sin(2w)(1− cos(2Qs)) =: Eqn(Qs + w).However, Qs(t) + w is a good approximate solution over short time intervals in thesense:Lemma 3.9. ‖Eqn(Qs(t) +w)‖L2tX1 . ‖w‖L2tX∞ + ‖w‖2L4tX4 and therefore by (3.10),‖Eqn(Qs(t) + w)‖L2tX1[tj ,T ) → 0 as tj → T − . (3.12)Remark: We do not need it here, but if 0 < s(t) 1, then Qs(t) + w is a goodapproximate solution globally, in the sense that ‖Eqn(Qs(t) + w)‖L2tX1[tj ,∞) → 0 assupt∈[tj ,∞)s(t)→ 0.Proof. This is an easy consequence of the elementary pointwise estimates|Eqn(Qs + w)| . 1r2(hsw2 + (hs)2w)|∂rEqn(Qs + w)| . 1r3(hsw2 + (hs)2w)+1r2(hs|w||wr|+ (hs)2|wr|).66Then using ‖hsr‖L2 . 1, ‖hsr2 ‖L1 . 1, and Ho¨lder, the Lemma follows.Lemma 3.10. (Linear Estimates) Assume m ≥ 4. Let ξ(·, t) ∈ X2 be a solution ofthe inhomogeneous linearized equation about Qs, where s = s(t) > 0 is a differentiablefunction of time,{∂tξ +Hsξ = f(r, t)ξ(r, 0) = ξ0(r)}Hs := −∂2r −1r∂r − m2r2cos(2Qs),which also satisfies the orthogonality condition(ξ(·, t), hs(t))L2rdr≡ 0. (3.13)Then we have the estimates‖ξ‖L∞t X2∩L2tX∞ . ‖ξ0‖X2 + ‖f‖L1tX2+L2tX1 + ‖s˙‖L2t . (3.14)Proof. The idea comes from [72] where it appeared as a linearization of a generalizedHasimoto transformation, while here we apply it directly at the linear level: exploitthe factorized form of the linearized operatorHs = (Ls)∗Ls, Ls = ∂r +mrcos(Qs) = hs∂r(hs)−1,the fact that the reverse factorization is positive,(Ls)(Ls)∗ = −∂2r−1r∂r+1r2(1 +m2 − 2m cos(Qs)) ≥ −∂2r− 1r∂r+ (m− 1)2r2 . (3.15)Applying Ls to the linearized equation produces∂tη + Ls(Ls)∗η = Lsf + (∂tLs)η η := Lsξ, ∂tLs =m2r1s(hs)2s˙Multiplying this equation by η, integrating over space and time, and using (3.15)67gives‖η‖2L∞t L2 + ‖η‖2L2tX2 . ‖Lsξ0‖2L2 + ‖(Lsf)η‖L1tL1 + ‖1s(hs)2ηr‖L2tL1‖s˙‖L2t .Using Ho¨lder on the right, then Young’s inequality, as well as ‖1s(hs)2‖L2 . 1, yields‖η‖L∞t L2∩L2tX2 . ‖ξ0‖X2 + ‖f‖L1tX2+L2tX1 + ‖s˙‖L2t .Finally, in [72] it was shown that we can invert Ls under the orthogonality condi-tion (3.13) to bound ξ:m ≥ 4, (ξ, hs)L2 = 0 =⇒ ‖ξ‖Xp . ‖Lsξ‖Lp , 2 ≤ p ≤ ∞.Together with the standard embedding ‖η‖L∞ . ‖η‖X2 this completes the proof.Proof. (of the Proposition 3.8) Let w(r, t) be as in (3.9). For t ∈ [tj, T ), the idea isto write the solution u(r, t) in the formu(r, t) = Qs(t)(r) + w(r, t) + ξ(r, t), (3.16)where s(t) > 0 is chosen so that the orthogonality condition (3.13) holds. The factthat we can make such a choice follows from a standard implicit function theoremargument:Lemma 3.11. There is 0 > 0 such that for any s0 > 0 and any ξ ∈ X2 with‖ξ‖X2 ≤ 0, there is 0 < s = s(ξ, s0) such thatQs0 + ξ = Qs + ξ˜ with(ξ˜, hs)L2rdr= 0,∣∣∣∣ ss0 − 1∣∣∣∣+ ‖ξ˜‖X2 . ‖ξ‖X2 ≤ 0.Proof. First take s0 = 1. For s > 0 and ξ ∈ X2 defineg(s; ξ) := (Q−Qs + ξ, hs)L2rdr ,a smooth function of s and ξ because the spatial decay of h(r) implies ‖rh(r)‖L2rdr <68∞ (provided m > 2). We observe g(1; 0) = 0, and∂sg(1; 0) =((−mshs, hs) + (Q−Qs + ξ, ∂shs))|s=1,ξ=0 = −m‖h‖2L2rdr 6= 0,so by the Implicit Function Theorem there is 0 > 0 such that for all ξ with ‖ξ‖X2 ≤0, there is s = s(ξ) with |s − 1| . ‖ξ‖X2 such that g(s; ξ) = 0. Then also ξ˜ :=ξ+Q−Qs =⇒ ‖ξ˜‖X2 . ‖ξ‖X2 + |s−1| . ‖ξ‖X2 . The case of general s0 > 0 followsfrom simple rescaling, and the scale invariance of the X2-norm.This lemma shows that as long asinfs>0‖u(·, t)− w(·, t)−Qs‖X2 < 0, (3.17)we may write u in the form (3.16), with orthogonality (3.13) holding.In particular, (3.8) implies that for any 0 < δ0 < 0, by taking j large enough,and therefore ‖u(·, tj)−Qs(tj) − w0‖X2 small enough, we may writeu(tj, r) = Qs(0) + w0(r) + ξ(r, 0),(ξ(·, 0), hs(0)) = 0, ‖ξ(·, 0)‖X2 ≤ δ0. (3.18)So by continuity, (3.17) holds on some non-empty time interval I = [tj, τ), tj < τ ≤T , on which we may write u(r, t) as in (3.16) with orthogonality (3.13).Moreover by regularity of u(r, t), by shrinking τ even more if needed, we mayalso assume‖ξ‖L∞t X2∩L2tX∞([tj ,τ)) ≤ δ230 , (3.19)which in particular implies (3.17) for δ0 sufficiently small.We will use a standard “continuity argument”. That is, we will carry out all ourestimates over the time interval I = [tj, τ) under the assumption (3.19), and thenconclude that we may take τ = T provided δ0 is chosen sufficiently small.Inserting (3.16) into the PDE and using standard trigonometric identities yieldsthe following equation for ξ:(∂t +Hs)ξ = −ms˙shs + Eqn(Qs + w) +m22r2(V sin(2ξ) +N) , (3.20)69whereV = cos(2Qs)(cos(2w)− 1))− sin(2Qs) sin(2w),and N contains only terms super-linear in ξ coming from the termscos(2Qs)[2(w + ξ)− sin(2(w + ξ))] andsin(2Qs)[1− cos(2(w + ξ))].Rather than write out all the terms of N explicitly, we just record the elementaryestimates|N | . (hs + |w|)ξ2 + |ξ|3|Nr| . (1 + |w|)(|wr|+ hsr)ξ2 +1r|ξ|3 + (hs + |w|)|ξ||ξr|+ ξ2|ξr|.(3.21)Our goal is to estimate all the terms on the right side of (3.20) in appropriate space-time norms, so that we may apply the linear estimates (3.14).For the first term, using ‖1shs‖X1 . 1, we have‖ −ms˙shs‖L2tX1 . ‖s˙‖L2t . (3.22)The main estimates for V are|V | . w2 + hs|w| =⇒ ‖ 1r2V ‖L2 . ‖wr‖2L4 + ‖wr‖L∞ ,using ‖hsr‖L2 . 1, and|Vr| . |w||wr|+ 1r(hs)2w2 + hs|wr|+ 1rhs|w|=⇒ ‖1rVr‖L2 . ‖wr‖L4‖wr‖L4 + ‖wr‖2L4 + ‖wr‖L∞ + ‖wr‖L∞ ,using ‖hs‖L∞ . 1 and ‖hsr ‖L2 . 1. Combining these, we obtain a spatial-norm70estimate on the linear term on the right side of (3.20),‖m22r2V sin(2ξ)‖X1 .(‖w‖2X4 + ‖w‖X∞) ‖ξ‖X2 ,and from there a space-time estimate:‖m22r2V sin(2ξ)‖L2tX1 .(‖w‖2L4tX4 + ‖w‖L2tX∞)‖ξ‖L∞t X2 , (3.23)where, recall, the time interval over which these norms are taken is I = [tj, τ).Finally, from (3.21), we estimate the nonlinear terms:‖ 1r3N‖L1 .(‖hsr‖L2 + ‖wr‖L2)‖ξr‖2L4 + ‖ξr‖3L3 . ‖ξr‖2L4 + ‖ξr‖3L3 ,and using ‖w‖L∞ . ‖w‖X2 . 1, and ‖hs‖X2 . 1,‖ 1r2Nr‖L1 . ‖ξr‖2L4 + ‖ξr‖3L3 + ‖ξr‖L4‖ξr‖L4 + ‖ξr‖2L4‖ξr‖L2 .These last two give‖ 1r2N‖X1 . ‖ξ‖2X4 + ‖ξ‖3X3 + ‖ξ‖2X4‖ξ‖X2 ,and then the spacetime estimate:‖m22r2N‖L2tX1 . ‖ξ‖2L4tX4(1 + ‖ξ‖L∞t X2) + ‖ξ‖3L6tX3 . (3.24)Now applying the linear estimates (3.14) to (3.20), using (3.18), (3.22), (3.12)(taking j larger as needed), (3.23), and (3.24), as well as (3.10), we get‖ξ‖L∞t X2∩L2tX∞ ≤ C(δ0 + ‖s˙‖L2t +(‖w‖2L4tX4 + ‖w‖L2tX∞)‖ξ‖L∞t X2+‖ξ‖2L4tX4(1 + ‖ξ‖L∞t X2) + ‖ξ‖3L6tX3).(3.25)By (3.10), by choosing j larger still if needed we can assure that on the interval71[tj, T ) ⊃ I,C(‖w‖2L4tX4([tj ,T ) + ‖w‖L2tX∞([tj ,T ))<12, (3.26)so that the estimate (3.25) becomes‖ξ‖L∞t X2∩L2tX∞ . δ0 + ‖s˙‖L2t + ‖ξ‖2L∞t X2∩L2tX∞ + ‖ξ‖3L∞t X2∩L2tX∞ ,and then by using (3.19),‖ξ‖L∞t X2∩L2tX∞ . δ0 + ‖s˙‖L2t . (3.27)It remains to estimate s˙. For this, we differentiate the orthogonality relation (3.13),rewritten as (ξ,1shs)L2rdrfor convenience of calculation, with respect to t, and use the equation (3.20) for ξ:0 =(ξ, −s˙ 1r2(r(r2h)′)s)+(−ms˙shs + Eqn(Qs + w) +m22r2(V sin(2ξ) +N) ,1shs)where we used Hshs = 0. The first term is bounded by|s˙|‖ξr‖L2‖1r(r(r2h)′)s‖L2 . |s˙|‖ξ‖X2 ,while (−ms˙shs,1shs)= −ms˙‖1r(rh)s‖2L2 = −ms˙‖h‖2L2 ,so(m‖h‖2L2 +O(‖ξ‖X2))s˙ =(Eqn(Qs + w) +m22r2(V sin(2ξ) +N) ,1shs). (3.28)Then by (3.19) and ‖r 1shs‖L∞ = ‖rh‖L∞ . 1,|s˙| . ‖Eqn(Qs + w)‖X1 + ‖ 1r2V sin(2ξ)‖X1 + ‖ 1r2N‖X1 ,72and so by (3.12), (3.23), (3.10), (3.21) and (3.19):‖s˙‖L2t . δ0 +(‖w‖2L4tX4 + ‖w‖L2tX∞)‖ξ‖L∞t X2 .Using (3.27) then shows‖s˙‖L2t ≤ C(δ0 +(‖w‖2L4tX4 + ‖w‖L2tX∞)‖s˙‖L2t).As above, by taking j larger if needed we can ensure (3.26) and so (using again (3.19))‖s˙‖L2t + ‖ξ‖L∞t X2∩L2t∩X∞ . δ0. (3.29)This now shows that in our bootstrap assumption (3.19), since we take δ0 δ2/30 ,we may indeed take τ = T , and all of our previous estimates hold on the full timeinterval [tj, T ).It remains to show that s(t) stays bounded away from zero. Recall the pointwisebounds used above|V | . w2 + hs|w|, |N | . (hs + |w|)ξ2 + |ξ|3, Eqn(Qs + w) . 1r2(hsw2 + (hs)2w).We isolate the term in the equation (3.28) for s coming from the part of Eqn(Qs+w)which behaves linearly in w, and write:s˙s= v1 + v2 + v3where|v1| . ‖wr3‖L2‖1s(rh3)s‖L2 . ‖|w|r2‖X2 ∈ L2t ,|v2| . ‖w2r2‖L∞‖ 1s2(h2)s‖L1 . ‖w‖2X∞ ∈ L1t73and|v3| . ‖w2r2‖L2‖ξr‖L∞‖1s(rh)s‖L2 + ‖wr‖L∞‖ξr‖L∞‖ 1s2hs‖L1+ ‖ξ2r2‖L∞‖ 1s2(hs)3‖L1 + ‖ξ2r2‖L2(‖wr‖L∞ + ‖ξr‖L∞)‖1s(rh)s‖L2. ‖w‖2X4‖ξ‖X∞ + ‖w‖X∞‖ξ‖X∞ + ‖ξ‖2X∞ + ‖ξ‖2X4(‖w‖X∞ + ‖ξ‖X∞)∈ L1t .So s˙s∈ L1t + L2t over [t∗, T ), and by the Fundamental Theorem of Calculus, andCauchy-Schwarz,supt∗≤t<T∣∣∣∣log( s(t)s(t∗))∣∣∣∣ ≤ ‖ s˙s‖L1t ([t∗,T )) +√t− t∗‖ s˙s‖L2t ([t∗,T )) <∞,so that s(t) remains bounded away from zero, as required.Now notice that Proposition 3.8 directly prohibits concentration, and so showsthat an m ≥ 4-equivariant heat-flow cannot form a finite-time singularity. Hencesuch a heat-flow is global. Moreover, it cannot form a singularity at infinite timet = ∞, since such this would produce a sequence tj → ∞, with 0 < sj → 0 or ∞,along which u(·, tj) − Qsj → v0 with X2 3 v0 a static solution, hence v0 ≡ 0. Thisis however prohibited by the asymptotic stability result of [72]. Hence we must haveE(u(·, t)−Qs∞)→ 0 for some s∞ > 0.3.2.2 Conclusions and future directionsThe map equations discussed in this introduction are of both physical and geo-metric interest, and yet it is only very recently that the global behavior of solutionsis starting to be understood, and that only in very limited settings. Much workremains to be done.74In continuation of this work, we plan to investigate the general case and in par-ticular address the following problem:Problem: Characterize blowing-up and global solutions to the corotational har-monic map heat flow (from R2 → S2) with arbitrary initial energy and any choice oftopology.Despite some technical difficulties, we believe this characterization to be within thereach of our techniques. Results of this flavour have already been established in thecase of corotational Wave Maps (WM) (e.g., [30, 33, 34]).A more ambitious next step is to address the general problem in the equivariant class:Problem: Characterize blowing-up and global solutions to the equivariant harmonicmap heat flow/Landau-Lifshitz (from R2 → S2) with arbitrary initial energy and anychoice of topology, and in particular, investigate the case of large data, not just closeto the family of stationary solutions, which has been addressed in [71, 72].The very interesting case of Schro¨dinger Maps is quite challenging and seems torequire the development of new techniques.Problem: The construction of either blowing-up or global solutions, which is stilllargely unexplored. For the full Landau-Lifshitz equation, once the Schro¨dinger-type term (b 6= 0) is included, our understanding diminishes considerably. Thoughthe problem is still dissipative, maximum principle-type arguments are not readilyapplicable, and even partial regularity results become more difficult and weaker. Sin-gularity formation is an open question, partly because the corotational class is nolonger preserved. Indeed, the harmonic map heat flow blow-up may not provide areliable guide for the Landau-Lifshitz problem. We do however venture the conjec-ture:Conjecture: The Landau-Lifshitz equation from R2 → S2, is globally well-posed,75at least when m ≥ 4.We have access to potentially useful linear estimates, but some work is required ona technical level: a choice of gauge must be made to write the equation in a formamenable to estimates and the profile decomposition should take into account thecompatibility condition imposed by this choice of gauge. Nevertheless, these are is-sues that have been addressed before for the case of the Schro¨dinger Map Equation[7] and we expect these results to provide a guide.The full physical model. The Landau-Lifshitz equation described above is a fairlysimple model, in that it incorporates only the “exchange energy”, and (for a > 0)dissipation. The question of how physically important effects such as anisotropy, anddemagnetization (a nonlocal term) affect solutions has barely been addressed.76Chapter 4Global solutions of a focusingenergy-critical Heat Equation in R44.1 IntroductionThis chapter is devoted to identifying the natural threshold below which all solu-tions of a nonlinear heat equation are global and decay.In what follows, we will mostly consider the focusing energy-critical nonlinearheat equation in four space dimensions:in particular, we consider the Cauchy problem{ut = ∆u+ |u|2uu(0, x) = u0(x) ∈ H˙1(R4)(4.1)for u(x, t) ∈ C with initial data in the energy spaceH˙1(R4) = {u ∈ L4(R4;C) ∣∣ ‖u‖2H˙1=∫R4|∇u(x)|2 dx <∞}.77This is the L2 gradient-flow equation for an energy, defined for u ∈ H˙1 asE(u) =∫R4(12|∇u|2 − 14|u|4)dx,and so in particular the energy is (formally) dissipated along solutions of (4.1):ddtE(u(t)) = −∫R4|ut|2 dx ≤ 0. (4.2)We refer to the gradient term in E as the kinetic energy, and the second term asthe potential energy. The fact that the potential energy is negative expresses thefocusing nature of the nonlinearity. Problem (4.1) is energy-critical in the sense thatthe scalinguλ(t, x) = λu(λ2t, λx), λ > 0 (4.3)leaves invariant the equation, the potential energy, and in particular the kineticenergy, which is the square of the energy norm ‖ · ‖H˙1 .Static solutions of (4.1), which play a key role here, solve the elliptic equation∆W + |W |2W = 0. (4.4)The functionW = W (x) =1(1 + |x|28)∈ H˙1(R4), 6∈ L2(R4)is a well-known solution. Its scalings by (4.3), and spatial translations of these areagain static solutions, and multiples of these are well-known [2, 123] to be the uniqueextremizers of the Sobolev inequality∀u ∈ H˙1, ‖u‖L4 ≤ C‖∇u‖L2 , C = ‖W‖L4‖∇W‖L2 =1‖∇W‖22the best constant. (4.5)As for time-dependent solutions, a suitable local existence theory – see Theo-rem 4.6 for details – ensures the existence of a unique smooth solution u ∈ C(I; H˙1(R4))on a maximal time interval I = [0, Tmax(u0)). The main result of this chapter states78that initial data lying ‘below’ W gives rise to global smooth solutions of (4.1) whichdecay to zero:Theorem 4.1. Let u0 ∈ H˙1(R4) satisfyE(u0) ≤ E(W ), ‖∇u0‖L2 < ‖∇W‖L2 . (4.6)Then the solution u of (4.1) is global (Tmax(u0) =∞) and satisfieslimt→∞‖u(t)‖H˙1 = 0. (4.7)The conditions (4.6) define a non-empty set, since by the Sobolev inequality (4.5)it includes all initial data of sufficiently small kinetic energy. Moreover, condi-tions (4.6) are sharp for global existence and decay in several senses. Firstly, if thekinetic energy inequality is replaced by equality, W itself provides a non-decaying(though still global) solution. Secondly, if the kinetic energy inequality is reversed,and under the additional assumption u0 ∈ L2(R4), by a slight variant of a classicalargument [95] we find that the solution blows up in finite time:Theorem 4.2. Let u0 ∈ H1(R4) withE(u0) < E(W ), ‖∇u0‖L2 ≥ ‖∇W‖L2 .Then the solution u of (4.1) has finite maximal lifespan: Tmax(u0) <∞.Thirdly, for any a∗ > 0, [118] constructed finite-time blow-up solutions withinitial data u0 ∈ H1(R4) satisfying E(W ) < E(u0) < E(W ) + a∗. See also [52] forformal constructions of blow-up solutions close to W .It follows from classical variational bounds – see Lemma 4.9 – and energy dis-sipation (4.2), that any solution u on a time interval I = [0, T ) whose initial datasatisfies (4.6), necessarily satisfiessupt∈I‖∇u‖L2 < ‖∇W‖L2 . (4.8)79So it will suffice to show that the conclusions of Theorem 4.1 hold for any solutionsatisfying (4.8). Indeed, we will prove:1. If I = [0,∞) and (4.8) holds, then limt→∞‖∇u(t)‖L2 = 0. This is given asTheorem 4.10.2. For any solution satisfying (4.8), Tmax(u(0)) = ∞. This is given as Corol-lary 4.18.That static solutions provide the natural threshold for global existence and decay,as in (4.8), is a classical phenomenon (eg. [122]) for critical equations, particularlywell-studied in the setting of parabolic problems, mostly on compact domains, (e.g.,[40, 61, 97, 125]) via ‘blow-up’-type arguments: first, failure of a solution to extendsmoothly is shown, by a local regularity estimate, to imply (kinetic) energy con-centration; then, near a point of concentration, rescaled subsequences are shown toconverge locally to a non-trivial static solution; finally, elliptic/variational consider-ations prohibit non-trivial static solutions below the threshold.The main purpose of our work is twofold: first, to establish the global-regularity-below-threshold result Theorem 4.1 on the full space R4; second, to do so not byway of the classical strategy sketched above, but instead via Kenig-Merle’s [79, 80]“concentration-compactness plus rigidity” approach to critical dispersive equations,similar to Kenig-Koch’s [78] implementation for the Navier-Stokes equations.The argument is structured as follows. First, in Section 4.3, we prove the energy-norm decay of global solutions which satisfy (4.8), Theorem 4.10. The strategy is thatemployed for the Navier-Stokes equations in [56]: reduce the problem to establishingthe decay of small solutions (which is a refinement of the local theory) by exploitingthe L2−dissipation relation, using a solution-splitting argument to overcome the factthat the solution fails to lie in L2. Second, in Section 4.4, we prove the existenceand compactness (modulo symmetries) of a “critical” element – a counterexampleto global existence and decay, which is minimal with respect to supt‖∇u(t)‖L2 , fol-lowing closely the work [82]. See Theorem 4.13. The technical tools are a profiledecomposition compatible with the heat equation (described in Section 4.2.2) and a80perturbation result for the linear heat equation, based on the local theory (Propo-sition 4.7). Finally, in Section 4.5, we exclude the possibility of a compact solutionwith finite maximal existence time in Theorem 4.17. This part is based on classicalparabolic tools. We first show that the centre of compactness remains bounded, byexploiting energy dissipation. Then a local small-energy regularity criterion, togetherwith backwards uniqueness and unique continuation theorems as in [78], imply thetriviality of the critical element.There is a vast literature on the semilinear heat equation ut = ∆u+ |u|p−1u. Wecontent ourselves here with a brief review focused on the case of domain Rd, and referthe reader to the recent book [112] for a more comprehensive review of the literature(until 2007). For treatments of the Cauchy problem in Lp and Sobolev spaces undervarious assumptions on the nonlinearity and the initial data, see [128, 129, 13].Much of the work concerns (energy) subcritical (p < d+2d−2) problems. The seminalpapers [62, 63, 64] introduced the study of heat equations through similarity variablesand characterized blow-up solutions. In continuation of these works, [104] gave afirst construction of a solution with arbitrarily given blow-up points, and see [107](and references therein) for estimates of the blow-up rate, descriptions of the blow-up set, and stability results for the blow-up profile. We remark that blow-up inthe subcritical case for L∞−solutions is known to be of Type I, in the sense thatlim supt→Tmax(Tmax − t)1p−1‖u(·, t)‖L∞ < +∞, and Type I blow-up solutions are knownto behave like self-similar solutions near the blow-up point. For a different set ofcriteria for global existence/blow-up in terms of the initial data we refer the readerto [18]. For results on the relation between the regularity of the nonlinear term andthe regularity of the corresponding solutions, see [19].For supercritical problems, [100, 101, 102] show that there is no Type II blow-upfor 3 ≤ d ≤ 10, while for d ≥ 11 it is possible if p is large enough. It is also shownthat a Type I blow-up solution behaves like a self-similar solution, while a Type IIconverges (in some sense) to a stationary solution. We also refer to the recent results[30] (d ≥ 11, bounded domain), and [26] and to the preprint [10] for results in Morreyspaces.For the critical case, we have already mentioned the finite-time blow-up construc-81tions [52, 118], and we point recent contructions of infinite-time blowup (bubbling)on bounded domains (d ≥ 5) [28] , and on R3 [36]. The work [53] deals with thecontinuation problem for reaction-diffusion equations. We finally mention the re-cent result [27], where a complete classification of solutions sufficiently close to thestationary solution W is provided for d ≥ 7: such solutions either exhibit Type-Iblow-up; dissipate to zero; or converge to (a slightly rescaled, translated) W . Inparticular, Type II blow-up is ruled out in d ≥ 7 near W.Remark 4.3. Theorem 4.1 extends to the energy critical problem for the nonlinearheat equation in general dimension d ≥ 3:ut = ∆u+ |u| 4d−2uu(t0, x) = u0(x) ∈ H˙1(Rd)(4.9)For simplicity of presentation, we will give the proof only for the case d = 4. As willbe apparent from the proof, the result can be easily transferred to solutions of (4.9)(and we will note the specific parts of the proof that are not dimension-independentand remark on the modifications that are required).Remark 4.4. Our proof makes no use of any parabolic comparison principles, andso applies to complex-valued solutions. That said, for ease of writing some estimateswe will sometimes replace the nonlinearity |u|2u with u3, though the estimates remaintrue in the C-valued case.4.2 Some analytical ingredients4.2.1 Local theoryWe first make precise what we mean by a solution in the energy space:Definition 4.5. A function u : I × R4 → C on a time interval I = [0, T ) (0 < T ≤∞) is a solution of (4.1) if u ∈ (CtH˙1x ∩ L6t,x)([0, t] × R4); ∇u ∈ L3x,t([0, t] × R4);82D2u, ut ∈ L2tL2x([0, t]× R4) for all t ∈ I; and the Duhamel formulau(t) = et∆u0 +∫ t0e(t−s)∆F (u(s))ds, (4.10)is satisfied for all t ∈ I, where F (u) = |u|2u. We refer to the interval I as the lifespanof u. We say that u is a maximal-lifespan solution if the solution cannot be extendedto any strictly larger interval. We say that u is a global solution if I = R+ := [0,+∞).We will often measure the space-time size of solutions on a time interval I in L6x,t,denotingSI(u) :=∫I∫R4|u(t, x)|6dxdt, ‖u‖S(I) := SI(u) 16 =(∫I∫R4|u(t, x)|6dxdt) 16.A local wellposedness theory in the energy space H˙1(R4), analogous to that forthe corresponding critical nonlinear Schro¨dinger equation (see e.g., [21]), is easilyconstructed, based on the Sobolev inequality and space-time estimates for the heatequation on R4 ([60]),‖et∆φ‖Lpx(R4) . t−2(1/a−1/p)‖φ‖La , 1 ≤ a ≤ p ≤ ∞‖et∆φ‖LqtLpx(R+×R4) . ‖φ‖La ,1q+2p=2a, 1 < a ≤ q‖∫ t0e(t−s)∆f(s)ds‖LqtLpx(R+×R4) . ‖f‖Lq˜′t Lp˜′x (R+×R4),1q+2p=1q˜+2p˜= 1,1q+1q′=1p+1p′= 1,(4.11)and (q˜′, p˜′) the dual to any admissible pair (q˜, p˜).We also refer the reader to [11, 128] for a treatment of the Cauchy problem in thecritical Lebesque space L2dd−2 ; the arguments directly adapt to show wellposednessin H˙1. One can use a fixed-point argument to construct local-in-time solutions forarbitrary initial data in H˙1(R4); however, as usual when working in critical scaling83spaces, the time of existence depends on the profile of the initial data, not merely onits H˙1-norm. We summarize:Theorem 4.6. (Local well-posedness) Assume u0 ∈ H˙1(R4).1. (Local existence) There exists a unique, maximal-lifespan solution to the CauchyProblem (4.1) in I × R4, I = [0, Tmax(u0))2. (Continuous dependence) The solution depends continuously on the initial data(in both the H˙1 and the SI-induced topologies). Furthermore, Tmax is a lower-semicontinuous function of the initial data.3. (Blow-up criterion) If Tmax(u0) < +∞, then ‖u‖S([0,Tmax(u0))) = +∞4. (Energy dissipation) the energy E(u(t)) is a non-increasing function in time.More precisely, for 0 < t < Tmax,E(u(t)) +∫ t0∫R4|ut|2 dx dt = E(u0). (4.12)5. (Small data global existence) There is 0 > 0 such that if ‖et∆u0‖S(R+) ≤ 0,the solution u is global, Tmax(u0) =∞, and moreover‖u‖S(R+) + ‖∇u‖(L∞t L2x∩L3x,t)(R+×R4) + ‖D2u‖L2x,t(R+×R4) . 0. (4.13)This occurs in particular when ‖u0‖H˙1(R4) is sufficiently small.An extension of the proof of the local existence theorem implies the followingstability result (see, e.g., [83]):Proposition 4.7. (Perturbation result)For every E,L > 0 and > 0 there exists δ > 0 with the following property: assumeu˜ : I × R4 → R, I = [0, T ), is an approximate solution to (4.1) in the sense that‖∇e‖L32t,x(I×R4)≤ δ, e := u˜t −∆u˜− |u˜|2u˜,84and also‖u˜‖L∞t H˙1x(I×R4) ≤ E and ‖u˜‖S(I) ≤ L,then if u0 ∈ H˙1x(R4) is such that‖u0 − u˜(0)‖H˙1x(R4) ≤ δ,there exists a solution u : I × R4 → R of (4.1) with u(0) = u0, and such that‖u− u˜‖L∞t H˙1x(I×R4) + ‖u− u˜‖S(I) ≤ 4.2.2 Profile decompositionThe following proposition is the main tool (along with the Perturbation Proposi-tion 4.7) used to establish the existence of a critical element. The idea is to char-acterize the loss of compactness in some critical embedding; it can be traced backto ideas in [98], [13], [121], [117] and their modern “evolution” counterparts [3], [79]and [80].Proposition 4.8. (Profile Decomposition)Let {un}n be a bounded sequence of functions in H˙1(R4). Then, after possibly passingto a subsequence (in which case, we rename it un), there exists a family of functions{φj}∞j=1 ⊂ H˙1, scales λjn > 0 and centers xjn ∈ R4 such that:un(x) =J∑j=11λjnφj(x− xjnλjn) + wJn(x),wJn ∈ H˙1(R4) is such that:limJ→∞lim supn‖et∆wJn‖L6t,x(R+×R4) = 0, (4.14)λjnwJn(λjnx+ xjn) ⇀ 0, in H˙1(R4), ∀j ≤ J. (4.15)85Moreover, the scales are asymptotically orthogonal, in the sense thatλjnλin+λinλjn+|xin − xjn|2λjnλin→ +∞, ∀i 6= j (4.16)Furthermore, for all J ≥ 1 we have the following decoupling properties:‖un‖2H˙1 =J∑j=1‖φj‖2H˙1+ ‖wJn‖2H˙1 + on(1) (4.17)andE(un) =J∑j=1E(φj) + E(wJn) + on(1) (4.18)The proof follows exactly the same steps with the proof presented in Chapter 2- before the transformation that was connecting the higher-dimensional to the two-dimensional estimates- and it can be easily finished from (2.40) through the standardheat space-times estimates and the same interpolation procedure as before.4.2.3 Variational estimatesThe elementary variational inequalities we use are summarized here:Lemma 4.9. (Variational Estimates)1. If‖∇u0‖2L2 ≤ ‖∇W‖2L2 , E(u0) ≤ (1− δ0)E(W ), δ0 > 0,then there exists δ¯ = δ¯(δ0) > 0 such that for all t ∈ [0, Tmax(u0)), the solutionof (4.1) satisfies ∫|∇u(t)|2 ≤ (1− δ¯)∫|∇W |2. (4.19)2. If (4.19) holds, then∫(|∇u(t)|2 − |u(t)|4)dx ≥ δ¯∫|∇u(t)|2 (4.20)86and moreover E(u(t)) ≥ 0.Proof. The second statements are an immediate consequence of the sharp Sobolevinequality (4.5):∫(|∇u(t)|2 − |u(t)|4)dx ≥[1−(‖∇u(t)‖L2‖∇W‖L2)2]‖∇u‖2L2 & ‖∇u‖2L2while the first follows easily from Sobolev and energy dissipation (4.12); see, e.g.,Lemma 3.4/Theorem 3.9 in [79].4.3 Asymptotic decay of global solutionsIn this section we prove the following theorem:Theorem 4.10. If u ∈ C([0,∞); H˙1(R4)) is a solution to equation (4.1) whichmoreover satisfiessupt≥0‖∇u(t)‖L2 < ‖∇W‖L2 , (4.21)thenSR+(u) <∞ and limt→∞‖u(t)‖H˙1 = 0.Proof. The general strategy, drawn from the techniques of [56] for the Navier-Stokesequations, is as follows. We first show that global solutions for which SR+(u) <∞ –which includes small solutions by the small data theory (4.13) – decay to zero in theH˙1−norm. Second, we impose the extra assumption of H1− data, so that we mayexploit the L2−dissipation relation to show finiteness of ‖∇u‖L2x,t , which in turnsallows us to reduce matters to the case of small H˙1 data. Finally, to remove thisextra assumption, we split the initial data in frequency, and estimate a perturbedequation.Proposition 4.11. If u is a global solution of (4.1) with SR+(u) <∞, thenlimt→∞‖∇u(t, ·)‖L2 = 0. (4.22)87Proof. Let u ∈ (CtH˙1x ∩ L6t,x)(R+ × R4) be a global solution to (4.1). Just as oneproves the blow-up criterion for the local theory Theorem 4.6, we first show:Claim 4.12. ‖∇u‖L3t,x(R+×R4) <∞Proof. Since u ∈ L6t,x(R+ × R4), given η > 0, we may subdivide R+ = [0,∞) intoa finite number of subintervals Ij = [aj, aj+1), j = 0, 1, . . . , J , 0 = a0 < a1 < · · · <aJ = ∞, on which ‖u‖L6t,x(Ij) ≤ η. Taking ∇ in the Duhamel formula (4.10) andusing (4.11):‖∇u‖(L3t,x∩L∞t L2x)(I0) ≤ C‖et∆∇u0‖L2 + C‖∫ t0S(t− s)∇(u3)ds‖L3t,x(I0)≤ C‖u0‖H˙1 + C‖u2∇u‖L3/2t,x (I0)≤ C‖u0‖H˙1 + C‖u‖2L6t,x(I0)‖∇u‖L3t,x(I0),so by choosing η < 1√2Cwe ensure‖∇u‖(L3t,x∩L∞t L2x)(I0) ≤ 2C‖u0‖H˙1 .In particular ‖u(a1)‖H˙1 ≤ 2C‖u0‖H˙1 , and so we may repeat this argument onthe next interval I1 to find ‖∇u‖(L3t,x∩L∞t L2x)(I1) ≤ (2C)2‖u0‖H˙1 , and, continuing,‖∇u‖(L3t,x∩L∞t L2x)(Ij) ≤ (2C)j+1‖u0‖H˙1 , for j = 0, 1, . . . , J . The claim follows.Now denote the linear evolution by S(t) = et∆, so the solution in Duhamel formis writtenu(t) = S(t)u0 +∫ t0S(t− s)u3(s)dsLetI := S(t)u0, II :=∫ τ0S(t− s)u3(s)ds, III :=∫ tτS(t− s)u3(s)ds,for some τ to be determined later.For term I we will take advantage of the decay of the heat propagator. By density,88we can approximate ∇u0 by v ∈ L1 ∩ L2 and use a standard heat estimate:‖I‖H˙1 = ‖S(t)∇u0‖L2 ≤ ‖S(t)(∇u0 − v)‖L2 + ‖S(t)v‖L2≤ ‖∇u0 − v‖L2 + ‖S(t)v‖L2 .The first term can be made arbitrary small by the choice of v, while for the second,by (4.11), ‖S(t)v‖L2 → 0 as t→∞, hence‖I‖H˙1 → 0 as t→∞.We now treat term III, which will allow us to fix τ. By the claim, for any > 0, wecan find τ such that ‖u‖L6t,x([τ,∞)×R4), ‖∇u‖L3t,x([τ,∞)×R4) ≤ . Since we are consideringthe limit t → ∞, we may assume t > τ 1, and so by the same estimate of thenonlinear term as in the proof of the claim,‖III‖H˙1 . ‖u‖2L6t,x([τ,t)×R4)‖∇u‖L3t,x([τ,t)×R4) . 3.Having fixed τ in this manner, we turn to term II. First notice thatII =∫ τ0S(t− s)u3(s)ds = S(t− τ)∫ τ0S(τ − s)u3(s)ds.Since∫ τ0S(τ − s)u3(s)ds ∈ H˙1 (by u ∈ L6x,t and (4.11)), the same approximationargument used for term I shows‖II‖H˙1 = ‖S(t− τ)∫ τ0S(τ − s)u3(s)ds‖H˙1 t→∞−−−→ 0.Since was arbitrary, (4.22) follows.Now if we assume u0 ∈ H1(R4), multiplying (4.1) by u and integrating overspace-time yields the L2 dissipation relation‖u(t)‖2L2 = ‖u0‖2L2 + 2∫ t0∫R4[u4 − |∇u|2]dxds. (4.23)89Because of (4.21), we have the variational estimate (4.20) and so for some δ¯ > 0,supt≥0‖u(t)‖2L2 + 2δ¯‖∇u‖2L2t,x(R+×R4) ≤ ‖u0‖2L2This estimate immediately implies that for any 0 > 0, there is some time t0 suchthat ‖u(t0)‖H˙1 ≤ 0, and we can directly apply the small data result (4.13) (withinitial time t = t0) to conclude that SR+(u) < ∞, and so by Proposition 4.11,limt→∞‖u(t)‖H˙1 = 0, as required.To remove the extra assumption u0 ∈ L2, splitu0 = w0 + v0, ‖w0‖H˙1 1, v0 ∈ H1.Define w(t) to be the solution to (4.1) with initial data w0:wt = ∆w + w3w(0, x) = w0(x) ∈ H˙1(Rd)From the small data theory (4.13), w ∈ CtH˙1x(R+ × R4) is global, with‖w‖L6t,x(R+×R4) + ‖∇w‖(L∞t L2x∩L3t,x)(R+×R4) . ‖∇w0‖L2 1 (4.24)and by Proposition 4.11, ‖w(t)‖H˙1 t→∞−−−→ 0.Defining v by v := u− w, it will be a solution of the perturbed equationvt −∆v = v3 + 3w2v + 3wv2.Just as in the derivation of the L2-dissipation relation (4.23), multiply by v andintegrate in space-time:‖v(t)‖2L2 − ‖v0‖2L2 + 2∫ t0‖∇v‖2L2 = 2∫ t0‖v‖4L4 + 6∫ t0∫R4w2v2 + 6∫ t0∫R4wv3.By (4.24), picking ‖∇w0‖L2 small enough, ensures that condition (4.21) holds also90for v: supt≥0‖∇v(t)‖L2 < ‖∇W‖L2 . Hence by (4.20), for some δ¯ > 0,‖v(t)‖2L2 + δ¯∫ t0‖∇v‖2L2 . ‖v0‖2L2 +∫ t0∫R4w2v2 + 6∫ t0∫R4wv3,and so by Ho¨lder and Sobolev,‖v(t)‖2L2 + δ¯‖∇v‖2L2L2 . ‖v0‖2L2 + ‖w‖2L∞L4‖v‖2L2L4 + ‖w‖L∞L4‖v‖L∞L4‖v‖L2L4. ‖v0‖2L2 + ‖∇w‖2L∞L2‖∇v‖2L2L2 + ‖∇w‖L∞L2‖∇v‖L∞L2‖∇v‖L2L2 .So by (4.24), choosing ‖∇w0‖L2 small enough yields∫ ∞0‖∇v‖2L2dt <∞, and hencethere is T > 0 for which ‖∇v(T )‖L2 < ‖w0‖H˙1 and so ‖∇u(T )‖L2 ≤ 2‖∇w0‖L2 .Choosing ‖∇w0‖L2 smaller still, if necessary, we are able to apply the small dataresult (4.13) to conclude SR+(u) <∞, and moreover by Proposition 4.11,limt→∞‖u(t)‖H˙1 = 0,concluding the proof of the theorem.4.4 Minimal blow-up solutionFor any 0 ≤ E0 ≤ ‖∇W‖22, we defineL(E0) := sup{SI(u) | u a solution of (4.1) on I with supt∈I‖∇u(t)‖22 ≤ E0},where I = [0, T ) denotes the existence interval of the solution in question. L :[0, ‖∇W‖22]→ [0,∞] is a continuous (this follows from Proposition 4.7), non-decreasingfunction with L(‖∇W‖22) =∞. Moreover, from the small-data theory (4.13),L(E0) . E30 for E0 ≤ 0.91Thus, there exists a unique critical kinetic energy Ec ∈ (0, ‖∇W‖22] such thatL(E0) <∞ for E0 < Ec, L(E0) =∞ for E0 ≥ Ec.In particular, if u : I × R4 → R is a maximal-lifespan solution, thensupt∈I‖∇u(t)‖22 < Ec =⇒ u is global, and ‖u‖S(R+) ≤ L(supt∈I‖∇u(t)‖22) <∞.The goal of this section is the proof of the following theorem:Theorem 4.13. There is a maximal-lifespan solution uc : I ×R4 → R to (4.1) suchthat supt∈I‖∇uc(t)‖2L2 = Ec, ‖uc‖S(I) = +∞. Moreover, there are x(t) ∈ R4, λ(t) ∈ R+,such thatK ={1λ(t)uc(t,x− x(t)λ(t)) ∣∣ t ∈ I} (4.25)is precompact in H˙1.For the proof of this theorem we closely follow the arguments in [82] . Theextraction of this minimal blow-up solution (and its compactness up to scaling andtranslation) will be a consequence of the following proposition:Proposition 4.14. Let un : In × R4 be a sequence of solutions to (4.1) such thatlim supnsupt∈In‖∇un‖22 = Ec and limn→∞‖un‖S(In) = +∞. (4.26)where In are of the form [0, Tn). Denote the initial data by un(x, 0) = un,0(x). Thenthe sequence {un,0}n converges, modulo scaling and translations, in H˙1 (up to anextraction of a subsequence).Proof. The sequence {un,0}n is bounded in H˙1 by (4.26) so applying the profiledecomposition (up to a further subsequence) we getun,0(x) =J∑j=11λjnφj(x− xjnλjn) + wJn(x)92with the properties listed in Proposition 4.8.Define the nonlinear profiles vj : Ij × R4 → R, Ij = [0, T jmax), associated toφj by setting them to be the maximal-lifespan solutions of (4.1) with initial datavj(0) = φj.Also, for each j, n ≥ 1 we introduce vjn : Ijn × R4 → R byvjn(t) =1λjnvj(t(λjn)2,x− xjnλjn), Ijn := {t ∈ R :t(λjn)2∈ Ij}.Each vjn is a solution with vjn(0) =1λjnφ(x−xjnλjn) and maximal lifespan Ijn = [0, Tn,jmax),T n,jmax = (λjn)2T jmax.For large n, by the asymptotic decoupling of the kinetic energy (property (4.17)),there is a J0 ≥ 1 such that ‖∇φj‖22 ≤ 0 for all j ≥ J0, where 0 is as in Theorem 4.6,4. Hence, for j ≥ J0, the solutions vjn are global and decaying to zero, and moreoversupt∈R+‖∇vjn‖22 + ‖vjn‖2S(R+×R4) . ‖∇φj‖22 (4.27)by the small data theory (4.13).Claim 4.15. (There is at least one bad profile). There exists 1 ≤ j0 < J0 such that‖vj0‖S(Ij0 ) =∞.For contradiction, assume that for all 1 ≤ j < J0‖vj‖S(Ij) <∞ (4.28)which by the local theory implies Ij = Ijn = [0,∞) for all such j and for all n. Thegoal is to deduce a bound on ‖un‖S(In)for sufficiently large n. To do so, we will useProposition 4.7, for which we first need to introduce a good approximate solution.DefineuJn(t) =J∑j=1vjn(t) + et∆wJn . (4.29)We will show that for n and J large enough this is a good approximate solution93(in the sense of Proposition 4.7) and that ‖uJn‖S([0,+∞)) is uniformly bounded. Thevalidity of both points implies that the true solutions un should not satisfy (4.26),reaching a contradiction.First observe∑j≥1‖vjn‖2S([0,∞)) =J0−1∑j=1‖vjn‖2S([0,∞)) +∑j≥J0‖vjn‖2S([0,∞)) (4.30). 1 +∑j≥J0‖∇φj‖22 . 1 + Ec (4.31)where we have used (4.28), property (4.17) and (4.26).Now, using the above and (4.14) in Proposition 4.8:limJ→∞limn‖uJn‖S([0,+∞)) . 1 + Ec. (4.32)For convenience, denote‖u‖S˜(I) := ‖∇u‖L3x,t(I×R4).Under the assumption (4.28), we can also obtain‖vj‖S˜(Ij) <∞,and so similarly we havelimJ→∞limn‖uJn‖S˜([0,+∞)) <∞.To apply Proposition 4.7, it suffices to show that uJn asymptotically solves (4.1) inthe sense thatlimJ→∞limn‖∇[(∂t −∆)uJn − F (uJn)]‖L32t,x([0,+∞)×R4)= 0which reduces (adding and subtracting the term F (∑Jj=1 vjn) and using the triangle94inequality) to provinglimJ→∞limn‖∇[J∑j=1F (vjn)− F (J∑j=1vjn)]‖L32t,x([0,+∞)×R4)= 0 (4.33)andlimn‖∇[F (uJn − et∆wJn)− F (uJn)]‖L32t,x([0,+∞)×R4)= 0. (4.34)The following easy pointwise estimate will be of use:|∇[(J∑j=1F (vj)− F (J∑j=1vj)]| .J∑i 6=j|∇vj||vi|2. (4.35)We have shown that for all j ≥ 1 and n large enough vjn ∈ S˜([0,∞)), so usingproperty (4.16)limn‖|vjn|2∇vin‖L32t,x([0,∞)×R4)= 0for all i 6= j; thuslimn‖∇[(J∑j=1F (vj)− F (J∑j=1vj)]‖L32t,x.J limn→∞limn∑i 6=j‖∇vjn|vin|2‖L32t,x= 0settling (4.33).‖∇[F (uJn − et∆wJn)− F (uJn)]‖L32t,x. ‖∇et∆wJn‖L3t,x‖et∆wJn‖2L6t,x + ‖|uJn|2∇et∆wJn‖L32t,x+‖∇uJn‖L3t,x‖et∆wJn‖2L6t,x + ‖∇uJn‖L3t,x‖et∆wJn‖L6t,x‖uJn‖L6t,xThe first, third and fourth terms are easy to be seen to converge to zero (using thespace-time estimates, the fact that wJn is bounded in H˙1 and (4.14)), so (4.34) isreduced to showinglimJ→∞limn‖|uJn|2∇et∆wJn‖L32t,x= 0.95By Ho¨lder and the space-time estimates,‖|uJn|2∇et∆wJn‖L32t,x. ‖uJn‖32L6t,x‖∇et∆wJn‖12L3t,x‖uJn∇et∆wJn‖12L2t,x. ‖(J∑j=1vjn)∇et∆wJn‖12L2t,x+ ‖et∆wJn‖12L6t,x‖∇et∆wJn‖12L3t,x. ‖(J∑j=1vjn)∇et∆wJn‖12L2t,x+ ‖et∆wJn‖12L6t,x.Again due to (4.14) it suffices to provelimJ→∞limn‖(J∑j=1vjn)∇et∆wJn‖L2t,x = 0.For any η > 0 by summability, we see that there exists J ′ = J ′(η) ≥ 1 such that∑j≥J ′‖vjn‖S([0,∞)) ≤ η. For this J ′,limn‖(J∑j=J ′vjn)∇et∆wJn‖6L2t,x . limn(∑j≥J ′‖vjn‖S([0,∞)))‖∇et∆wJn‖6L3t,x . η.As η > 0 is arbitrary, it suffices to showlimJ→∞limn‖vjn∇et∆wJn‖L2t,x = 0, 1 ≤ j ≤ J ′.Changing variables and assuming (by density) vj ∈ C∞c (R+ × R4), by Ho¨lder andthe scale-invariance of the norms, proving (4.34) reduces to provinglimJ→∞limn‖∇et∆wJn‖L2t,x(K) = 0,for any compact K ∈ R+×R4. The proof is a direct modification of the proof of theLemma 2.5 in [83].We have verified all the requirements of the stability proposition (4.7), hence we96conclude that‖un‖S([0,∞)) . 1 + Eccontradicting (4.26).The problem now is that the kinetic energy is not conserved. The difficulty arisesfrom the possibility that the S-norm of several profiles is large over short times, whiletheir kinetic energy does not achieve the critical value until later. To finish the proofof proposition we have to prove that only one profile is responsible for the blow-up.We can now (after possibly rearranging the indices) assume there exists 1 ≤ J1 <J0 such that‖vj‖S(Ij) =∞, 1 ≤ j ≤ J1 and ‖vj‖S([0,∞)) <∞, j > J1Again, we follow the combinatorial argument of [83]: for each integer m,n ≥ 1, definean integer j = j(m,n) ∈ {1, ..., J1} and an interval Kmn of the form [0, τ ] bysup1≤j≤J1‖vjn‖S(Kmn ) = ‖vj(m,n)n ‖S(Kmn ) = m (4.36)By the pigeonhole principle, there is a 1 ≤ j ≤ J1 such that for infinitely many mone has j(m,n) = j1 for infinitely many n. Reordering the indices, if necessary, wemay assume j1 = 1. By the definition of the critical kinetic energylim supm→∞lim supn→∞supt∈Kmn‖∇v1n(t)‖22 ≥ Ec (4.37)By (4.36), all vjn have finite S-norms on Kmn for each m ≥ 1. In the same wayas before, we check again that the assumptions of Proposition 4.7 are satisfied toconclude that for J and n large enough, uJn is a good approximation to un on Kmn .In particular we have for each m ≥ 1,limJ→∞lim supn→∞‖uJn − un‖L∞t H˙1x(Kmn ×R4) = 0 (4.38)97Lemma 4.16. (Kinetic energy decoupling for later times). For all J ≥ 1 and m ≥ 1,lim supn→∞supt∈Kmn|‖∇uJn(t)‖22 −J∑j=1‖∇vjn(t)‖22 − ‖∇wJn‖22| = 0 (4.39)Proof. Fix J ≥ 1 and m ≥ 1. Then, for all t ∈ Kmn ,‖∇uJn(t)‖22 =< ∇uJn(t),∇uJn(t) >=J∑j=1‖∇vjn(t)‖22 + ‖∇wJn‖22+∑j 6=j′< ∇vjn(t),∇vj′n (t) > +2J∑j=1< ∇et∆wJn ,∇vjn(t) >It suffices to prove (for all sequences tn ∈ Kmn ) that< ∇vjn(tn),∇vj′n (tn) >n→∞−−−→ 0 (4.40)and< ∇etn∆wJn ,∇vjn(tn) > n→∞−−−→ 0. (4.41)Since tn ∈ Kmn ⊂ [0, T n,jmax), for all 1 ≤ j ≤ J1, we have tn,j := tn(λjn)2 ∈ Ij for allj ≥ 1. For j > J1 the lifespan is R+. By refining the sequence using the standarddiagonalization argument, we can assume that tn,j converges (+∞ is also possible)for every j.We deal with (4.40) first. If both tn,j, tn,j′ →∞, necessarily j, j′ > J1 and vj, vj′are global solutions satisfying the kinetic energy bound (4.21), so by Theorem (4.10)‖vj‖H˙1 , ‖vj′‖H˙1 t→∞−−−→ 0. Employing Ho¨lder’s inequality and the scaling invariance ofthe H˙1-norm, we get (4.40) for this case. When tn,j →∞ but tn,j′ → τj′ : using thecontinuity of the flow in H˙1 we can be, for the limit, replacing∇{ 1λj′nvj′(tn,j′ ,x− xjnλj′n)}with ∇{ 1λj′nvj′(τj′ ,x− xjnλj′n)}. By an L2- approximation, we can also assume we areworking with smooth, compactly supported functions. In this case, we can bound< ∇vjn(tn),∇vj′n (tn) > by ‖vj(tn,j)‖H˙1‖vj′(τj′)‖H˙1 → 0, as n → ∞. The remaining98case is when both tn,j and tn,j′ converge to finite τj, τj′ in the interior of Ij, Ij′respectively. We can replace as above tn,j, tn,j′ by τj, τj′ respectively, and perform achange of variables:< ∇vjn(tn),∇vj′n (tn) >=∫(λjnλj′n)2∇vj(τj, x),∇vj′(τj′ , λjnλj′nx+xjn − xj′nλj′n)dxwhich is going to zero assuming, without loss of generality thatλjnλj′n→ 0 and thefunctions in the integrand have been assumed to be compactly supported, thus con-cluding the case (4.40).For the case (4.41), performing a change of variable:< ∇etn∆wJn ,∇vjn(tn) >=< ∇etn,j∆[λjnwJn(λjnx+ xjn)],∇vj(tn,j) > .When tn,j → ∞, using Ho¨lder, the L2 − L2 heat estimate (and the boundedness ofwJn in H˙1 coming from the profile decomposition) and Theorem 4.10 as before, weget to the result. For the case tn,j → τj < +∞, we can, as before, replace tn,j by itslimit τj in the integral∫∇etn,j∆[λjnwJn(λjnx+xjn)] ·∇vj(τj, x)dx. . One can show (seeLemma 2.10 in [82], by an easy modification of Lemma 3.63 in [105]) using (4.14)that etn,j∆[λjnwJn(λjnx+ xjn)] ⇀ 0 in H˙1, which concludes the proof of the case (4.41)and hence the proof of the Lemma.By (4.26), (4.38), (4.39), we getEc ≥ lim supn→∞supt∈Kmn‖∇un(t)‖2L2 = limJ→∞lim supn→∞{‖∇wJn(t)‖2L2 + supt∈KmnJ∑j=1‖∇vjn(t)‖2L2}.Taking a limit in m and employing (4.37), we see that we actually have equalityeverywhere. This implies that J1 = 1, vjn ≡ 0, ∀j ≥ 2, wn := w1n H˙1−→ 0. So un(0, x) =1λnφ(x−x1nλ1n) + wn(x), for some functions φ,wn ∈ H˙1, wn s−→ 0 in H˙1.We have shown (for that sequence of solutions un we found) that for the corre-sponding sequence of initial data un,0 : λ1nun,0(λ1nx + x1n)H˙1−→ φ. This finishes the99proof of Proposition 4.14.Now, we are in a position to prove Theorem 4.13.Proof. By the definition of such a threshold we can find a sequence of solutionsun : In × R4 → R, with In compact, so thatsupnsupt∈In‖∇un(t)‖2L2 = Ec and limn‖un‖S(In) = +∞.An application of Proposition 4.14 shows that (up to symmetries) the correspondingsequence of initial data converges to some φ, strongly in H˙1. By a further rescalingand translation we can take λ1n ≡ 1, x1n ≡ 0.Let uc : I × R4 → R be the maximal-lifespan solution with initial data φ. Sinceun,0H˙1−→ φ, employing the stability Proposition 4.7, I ⊂ lim inf In, and ‖un −uc‖L∞t H˙1x(K×R4)n→∞−−−→ 0, for all compact K ⊂ I. Thus, by (4.26):supt∈I‖∇uc(t)‖2L2 ≤ Ec (4.42)Applying the stability Proposition 4.7 once again we can also see that ‖uc‖S(I) =∞.Hence, by the definition of the critical kinetic energy level, Ec,supt∈I‖∇uc(t)‖2L2 ≥ Ec (4.43)In conclusion,supt∈I‖∇uc(t)‖2L2 = Ec. (4.44)and‖uc‖S(I) = +∞. (4.45)Let I := [0, T ∗), where T ∗ := Tmax(φ), φ the data that corresponds to the previouslyfound critical element uc.Finally, the compactness modulo symmetries (4.25) follows from another appli-100cation of Proposition 4.14. We omit the standard proof (see for example [79] or[83]).4.5 RigidityThe main result of this section is the following theorem ruling out finite-time blowupof compact (modulo symmetries) solutions. Note this is a considerably strongerstatement than we require, since it is not limited to solutions with below-thresholdkinetic energy:Theorem 4.17. If u is a solution to (4.1) on maximal existence interval I = [0, T ∗),such that K :={1λ(t)u(t,x− x(t)λ(t)) | t ∈ I}is precompact in H˙1 for some x(t) ∈ R4,λ(t) ∈ R+, then T ∗ = +∞.As a corollary, we can complete the proof of the main result Theorem 4.1 byshowing:Corollary 4.18. For any solution satisfying (4.8), Tmax(u(0)) =∞.Proof. By Theorem 4.17, the solution uc produced by Theorem 4.13 must be global:Tmax(uc(0)) =∞. But since ‖uc‖S(R+) =∞, Theorem 4.10 shows Ec = ‖∇W‖22, andthe Corollary follows.The rest of the section is devoted to the proof of the Theorem 4.17. Our proof isinspired by the work of Kenig and Koch [78] for the Navier-Stokes system, and it’sbased on classical parabolic tools – local smallness regularity, backwards uniqueness,and unique continuation – though implemented in a somewhat different way. Inparticular, we will make use of the following two results, proved in [49], [50] (also see[51]):Theorem 4.19. (Backwards Uniqueness) Fix any R, δ,M, and c0 > 0. Let QR,δ :=(R4 \BR(0))× (−δ, 0), and suppose a vector-valued function v and its distributionalderivatives satisfy v,∇v,∇2v ∈ L2(Ω) for any bounded subset Ω ⊂ QR,δ, |v(x, t)| ≤101eM |x|2for all (x, t) ∈ QR,δ, |vt−∆v| ≤ c0(|∇v|+ |v|) on QR,δ, and v(x, 0) = 0 for allx ∈ R4 \BR(0). Then v ≡ 0 in QR,δ.Theorem 4.20. (Unique Continuation) Let Qr,δ := Br(0)× (−δ, 0), for some r, δ >0, and suppose a vector-valued function v and its distributional derivatives satisfyv,∇v,∇2v ∈ L2(Qr,δ) and there exist c0, Ck > 0, (k ∈ N) such that |vt − ∆v| ≤c0(|∇v| + |v|) a.e. on Qr,δ and |v(x, t)| ≤ Ck(|x| +√−t)k for all (x, t) ∈ Qr,δ. Thenv(x, 0) ≡ 0 for all x ∈ Br(0).As well, we establish the following:Lemma 4.21. (Local Smallness Regularity Criterion) There exist positive absoluteconstants 0, ck for k ∈ N with the following property: if u is a solution of equation(4.1) on Q1, where Qr := Br(0)× (−r2, 0) for r > 0, and satisfies‖u‖L∞t (H˙1x∩L4x)(Q1) < 0then u is smooth on Q 12with bounds on all derivatives,maxQ 12|Dku| ≤ ck.Proof. Assume ‖u‖L∞t (H˙1x∩L4x)(Q1) < , for small enough (to be picked). Define‖u‖2X(Q1) := ‖∇u‖2L∞t L2x∩L2tL4x(Q1) + ‖u‖2L∞t L4(Q1)+ ‖D2u‖2L2tL2x(Q1).Assuming for ease of writing that u is real-valued, differentiating (4.1) and definingu˜ := ∇u, we getu˜t = ∆u˜+ 3u2u˜. (4.46)Consider a smooth, compactly supported spatial cut-off function φ0(x) such thatsupp(φ0) ⊂ B1(0) and φ0 ≡ 1 on Bρ0(0), for some 12 < ρ0 < 1 to be chosen.Multiplying the above equation by φ20u˜ and integrating in space-time (from now on,102unless otherwise specified, t ∈ [−1, 0]):∫ t−1∫|x|≤1(u˜t −∆u˜)φ20u˜ dxdt = 3∫ t−1∫|x|≤1(u2u˜)φ20u˜ dxdt⇒ 12‖φ0u˜(t)‖2L2 +∫ t−1∫|x|≤1φ20|∇u˜|2 dxdt=12‖φ0u˜(0)‖2L2 + 3∫ t−1∫|x|≤1φ20u2u˜2dxdt+ 2∫ t−1∫|x|≤1φ0∇φ0(u˜∇u˜)dxdt.For the sake of brevity, let us define v0 := φ0u˜ = φ0∇u and thus (always on the samecylinder):‖v0‖2L∞t L2x + ‖φ0∇u˜‖2L2tL2x. ‖v0(0, x)‖2L2 + ‖u2‖L∞t L2x‖v20‖L1tL2x + ‖φ0∇u˜‖L2tL2x‖u˜‖L2tL2x= ‖v0(0, x)‖2L2 + ‖u‖2L∞t L4x‖v0‖2L2tL4x+ ‖φ0∇u˜‖L2tL2x‖u˜‖L2tL2xBy the smallness assumed on the cylinder Q1 and an application of Young’s inequal-ity, for any δ > 0 (and also using Ho¨lder and the boundedness of the domain):‖v0‖2L∞t L2x + ‖φ0∇u˜‖2L2tL2x. 2 + 2‖v0‖2L2tL4x(Q1) + δ2‖φ0∇u˜‖L2tL2x(Q1) +‖u˜‖2L∞t L4x(Q1)δ2⇒ ‖v0‖2L∞t L2x + ‖φ0∇u˜‖2L2tL2x. 2 + 2δ2+ 2‖v0‖2L2tL4xif δ is chosen small enough. Since ∇v0 = φ0∇u˜+∇φ0 u˜ :‖∇v0‖L2 . ‖φ0∇u˜‖L2 + ‖∇φ0‖L4‖u˜‖L4and so using the Sobolev inequality,‖v0‖2L∞t L2x + ‖∇v0‖2L2tL2x+ ‖v0‖2L2tL4x . 2 + 2‖v0‖2L2tL4x .Choosing small enough yields‖u‖X(Qρ0 ) . .103Define another smooth compactly supported cut-off function φ1(x) ≤ φ0(x), withsupport in Bρ0 , and φ1 ≡ 1 on Bρ1(0), some 12 < ρ1 < ρ0 < 1 to be chosen. Letvˆ := D2u, and v1 := φ1vˆ.Remark 4.22. We will be abusing notation from this point onwards. For the point-wise operations and estimates we are actually considering the mixed partial deriva-tives ∂k∂ju, j, k = 1, ..., 4 but we will be writing D2u all the same without taking careto specify the matrix element at hand. In the end, we are using standard matrixnorms.Differentiating (4.46), multiplying by φ21vˆ, and integrating over space gives12∂t∫φ21vˆ2dx+∫φ21|∇vˆ|2dx = 3∫φ21u2vˆ2dx+ 6∫φ21uu˜2vˆdx+ 2∫φ1∇φ1 · vˆ∇vˆdx(4.47)Since by the previous step, ‖∇v0‖L2L2(Qρ0 ) . , we can find −1 < t1 < −ρ20 such that‖∇v0(·, t1)‖L2(Bρ0 ) . (where the implied constant may depend on ρ0), so that‖φ1vˆ(·, t1)‖L2 = ‖φ1D2u(·, t1)‖L2 ≤ ‖∇v0(·, t1)‖L2(Bρ0 ) . .Integrating (4.47) in t from t1 to 0, and using the estimates from the previous step:‖v1‖2L∞t L2 + ‖φ1∇vˆ‖2L2tL2x. ‖φ0u‖2L∞L4‖v1‖2L2L4 + ‖φ0u‖L∞L4‖v0‖L∞L4‖v0‖L2L4‖v1‖L2L4+ ‖vˆ∇vˆφ1∇φ1‖L1tL1x + 2. 2‖v1‖2L2L4 + 3‖v1‖L2L4 + ‖φ1∇vˆ‖L2L2‖∇φ1vˆ‖L2L2 + 2where everywhere here the time interval is [t1, 0]. We have∇v0 = φ0D2u+∇φ0∇u = φ0vˆ +∇φ0u˜104and so|φ0vˆ| . |∇v0|+ |∇φ0u˜| ⇒ |∇φ1vˆ| . |∇φ1φ0||φ0vˆ| . |∇v0|+ |∇φ0u˜|.Thus‖∇φ1vˆ‖L2L2 . ‖∇v0‖L2L2 + . .By Young’s inequality once more, for some δ1 > 0 sufficiently small,‖v1‖2L∞L2 + ‖φ1∇vˆ‖2L2L2 . 2‖v1‖2L2L4 + δ21‖φ1∇vˆ‖2L2L2 + δ21‖v1‖2L2L4 +2δ21.Using Sobolev again as above, ‖v1‖2L∞L2 + ‖v1‖2L2L4 + ‖∇v1‖2L2L2 . . In particular‖D2u‖X(Qρ1 ) . .This process can be iterated a given finite number of times, to show that for givenk > 0, there are 0 = (k), C = C(k), such that if ‖u‖L∞(H˙1∩L4)(Q1) = < 0, then‖Dku‖X(Q1/2) ≤ C.We proceed now with the proof of Theorem 4.17.Proof. Let us assume that the conclusion is false, i.e., T ∗ < +∞. Note first thatλ(t)→ +∞.In fact, lim inft→T ∗−√T ∗ − t λ(t) > 0, since if √T ∗ − tnλ(tn) → 0 along a sequencetn ↗ T ∗, by the compactness assumption (and up to subsequence)vn(x) :=1λ(tn)uc(tn,x− x(tn)λ(tn))H˙1−→ ∃ v(x) ∈ H˙1.Let Tˆ > 0 be the maximal existence time for the solution of the Cauchy problem (4.1)with initial data v(x). Define wn(t, x) to be the solutions with initial data wn(x, tn) =vn(x) prescribed at time tn, and denote their maximal lifespans as [tn, Tmaxn ). By105continuous dependence on initial data, 0 < Tˆ ≤ lim inf(Tmaxn − tn). But from scaling:Tmaxn − tn = Tmax(1λ(tn)uc(tn,· − x(tn)λ(tn)))=1λ2(tn)Tmax(uc(tn, ·))= λ2(tn)(T∗ − tn)→ 0,a contradiction.By compactness in H˙1, and the continuous embedding H˙1 ↪→ L4, for every > 0,there is a R > 0 such that for all t ∈ I := [0, T ∗) :∫|x−x(t)|≥ Rλ(t)(|∇uc(t, x)|2 + |uc(t, x)|4) dx < (4.48)Fix any {tn} ⊂ [0, T ∗), tn ↗ T ∗, and let λn = λ(tn)→∞ and {xn} = {x(tn)} ⊂R4, so that (up to subsequence)vn(x) =1λnuc(x− xnλn, tn)H˙1−→ v¯, some v¯ ∈ H˙1,and also in L4 by Sobolev embedding.We also make and prove the following claim as in [78]Claim 4.23. For any R > 0,limn→∞∫|x|≤R|uc(x, tn)|2dx = 0.Proof. ∫|x|≤R|uc(x, tn)|2dx =∫|x|≤R|λnvn(λnx+ xn)|2dx=1λ2n∫|y−xn|≤λnR|vn(y)|2dy = 1λ2n‖vn‖2L2(BλnR(xn)).106Denoting Br := Br(0), for any > 0,1λ2n‖vn‖2L2(BλnR(xn)) =1λ2n‖vn‖2L2(BλnR(xn))∩BλnR) +1λ2n‖vn‖2L2(BλnR(xn))∩BcλnR).Using Ho¨lder’s inequality, and the compactness, we get1λ2n‖vn‖2L2(BλnR(xn)) .1λ2n‖vn‖2L4(R4)|BλnR|12 +1λ2n‖vn‖2L4(BλnR(xn)∩BcλnR)|BλnR(xn)|12. 2R2‖v¯‖2L4(R4) +|BλnR(xn)|12λ2n(‖vn − v¯‖2L4(R4) + ‖v¯‖2L4(BλnR(xn)∩BcλnR)). 2R2‖v¯‖2L4(R4) +R2‖vn − v¯‖2L4(R4) +R2‖v¯‖2L4(BcλnR). 2R2 +R2‖vn − v¯‖2L4(R4) +R2‖v¯‖2L4(BcλnR).The first term is arbitrarily small with , the second one goes to zero (as n → ∞)because of the compactness, and for a fixed , the last one goes to zero since λn →∞.We also prove that the center of compactness x(t) is bounded:Proposition 4.24. sup0≤t<T ∗|x(t)| <∞.Proof. We will first make the assumption thatE := inft∈[0,T ∗)E(uc(t)) > 0, (4.49)and later show that this is indeed the case for compact blowing-up solutions, withoutany size restriction. Note that under the assumptions of our Theorem 4.1, i.e., inthe below threshold case, we certainly have that E > 0. This can be easily deducedby the variational estimates in Lemma 4.9 and the small data theory.The energy dissipation relationE(u(t2)) +∫ t2t1‖ut‖2L2 ds = E(u(t1)) ≤ E(u(0)) (4.50)107for t2 > t1 > 0 will be of use. We will assume for contradiction that there is asequence of times tn ↗ T ∗ : |x(tn)| → ∞.Choose a radial smooth cut-off function ψ such thatψ(x) ={0 if |x| ≤ 11 if |x| ≥ 2and define ψR(x) := ψ(|x|R). Choosing any t0 ∈ (0, T ∗), we can find R0 ≥ 1 such that∫R4(12|∇uc(t0)|2 − 14|uc(t0)|4)ψR0(x)dx ≤14E. (4.51)Since |x(tn)| → ∞ and λ(tn) → ∞, for any > 0, B Rλ(tn)(x(tn)) ⊂ Bc2R0 for n largeenough, and so by (4.48):limt↗T ∗∫R4(12|∇uc(t)|2 − 14|uc(t)|4)ψR0(x)dx = E,hence we can find a t1 ∈ (t0, T ∗) such that∫R4(12|∇uc(t1)|2 − 14|uc(t1)|4)ψR0(x)dx ≥12E. (4.52)Combining (4.51) and (4.52):∫ t1t0ddt∫R4(12|∇uc(t)|2 − 14|uc(t)|4)ψR0(x)dxdt ≥14E. (4.53)On the other hand:ddt∫R4(12|∇uc(t)|2 − 14|uc(t)|4)ψR0(x)dx =∫R4(∇uc · ∇(uc)t − u3c(uc)t)ψR0(x)dx=∫R4(∇uc · ∇(uc)t − ((uc)t −∆uc)(uc)t)ψR0(x)dx= −∫R4(uc)2tψR0 dx−∫R4(uc)t∇uc · ∇ψR0 dx .∫R4|(uc)t||∇uc| dx,108since |∇ψR0(x)| .1R0≤ 1. So by Ho¨lder,∫ t1t0ddt∫R4(12|∇uc(t)|2 − 14|uc(t)|4)ψR0(x)dxdt. ‖∇uc‖L∞t L2√t1 − t0 ‖(uc)t‖L2L2[t0,t1]×R4. ‖(uc)t‖L2L2([t0,T ∗)×R4)(4.54)where we have uniformly bounded the kinetic energy of uc by once more employingthe compactness. Combining (4.53) and (4.54) yields:0 <14E . ‖(uc)t‖L2L2([t0,T ∗)×R4) → 0 as t0 ↗ T ∗ (4.55)by the energy dissipation relation (4.50), a contradiction.Now we show (4.49). Choose a smooth radial cut-off function φ such thatφ(x) ={1 if |x| ≤ 10 if |x| ≥ 2and define φR(x) := φ(|x|R), andIR(t) :=∫|uc(x, t)|2φR(x)dx, t ∈ [0, T ∗).We then haveI ′R(t) =∫φR(|uc|4 − |∇uc|2)dx− 1R∫uc∇uc∇φ( xR)dxand by Sobolev, Hardy and the compactness, we can immediately deduce that|I ′R(t)| ≤ C,109C a universal constant. Integrating from t0 to T∗ > t > t0 ≥ 0:|IR(t)− IR(t0)| ≤ C(t− t0).By Claim 4.23, we get that IR(t)→ 0 as t→ T ∗, for all R > 0. Hence|IR(t0)| ≤ C(T ∗ − t0).Since this bound is uniform in R, by taking R → ∞, we conclude uc(t0) ∈ L2, andso indeed uc(t) ∈ L2, t ∈ [0, T ∗). Defining then I(t) := 12∫|uc(t, x)|2dx, by directcalculation we getI ′(t) = −∫ (|∇uc|2 − |uc|4) dx = −K(uc(t))(K is defined below). Now for any sequence {tn}n ↗ T ∗, let (up to subsequence)1λ(tk)uc(x− xkλ(tk), tk)H˙1−→ v¯ ∈ H˙1.SetK(u) :=∫ (|∇u|2 − |u|4) dx = 2E(u)− 12∫|u|4dx.Proceeding by contradiction, we suppose E ≤ 0. If so,K(v¯) = limk→∞K(uc(tk)) = 2E − 12∫|v¯|4 ≤ −12∫|v¯|4 < 0,since v¯ ≡ 0 would contradict the assumption T ∗ <∞. SoI ′(tk) = −K(uc(tk))→ −K(v¯) > 0Thus I ′(t) > 0 for all t sufficiently close to T ∗; otherwise, we could find a subsequencealong which I ′ ≤ 0, and the preceding argument would provide a contradiction. So110I(t) is increasing for t near T ∗. But we have also shown thatlimt→T ∗I(t) = limt→T ∗limR→∞IR(t) = limR→∞limt→T ∗IR(t) = 0,a contradiction. Thus we have shown that E > 0, completing the proof that |x(t)|remains bounded.Since |x(t)| remains bounded while λ(t) t→T ∗−−−→ ∞, by the compactness we canfind an R0 > 0 large enough such that for all x, |x| ≥ R0 :‖uc‖L∞t H˙1x∩L∞t L4x(ΩT∗ ) < 0,where ΩT ∗ := (0, T∗)×B√T ∗(x0).By an appropriate scaling and shifting argument, the Regularity Lemma 4.21shows that uc is smooth on Ω := (R4 \ BR0(0)) × [34T ∗, T ∗], with uniform boundson derivatives. Since u is continuous up to T ∗ outside BR0 , Claim 4.23 implies thatuc(x, T∗) ≡ 0, in the exterior of this ball. Since uc is bounded and smooth in Ω, anapplication of the Backwards Uniqueness Theorem 4.19 implies that uc ≡ 0 in Ω.Define Ω˜ := R4× (34T ∗, 78T ∗]. Applying the Unique Continuation Theorem 4.20 on acylinder of sufficiently large spatial radius, centered at a point of Ω, implies uc ≡ 0in Ω˜. By the uniqueness guaranteed by the local wellposedness theory we get thatuc ≡ 0, which contradicts (4.45).Note that our results extend to any dimension bigger than d = 2 with obvious mod-ifications to the proofs. The profile decomposition statement we used is valid forall d ≥ 3, and we refer to [124] for a proof of the stability result (4.7) in higherdimensions.4.6 Blow-upIn this section we give various criteria on the initial data so that the correspondingsolutions blow-up in finite-time.111The following result is well-known [95, 4, 17] but we give the proof for the convenienceof the reader.Proposition 4.25. Solutions ofut = ∆u+ |u|p−1u, 1 < p ≤ 2∗ = 2dd− 2u(x, 0) = u0(x) ∈ H1(Rd)(4.56)withE(u0) :=∫Rd(12|∇u0|2dx− 1p+ 1|u0|p+1)dx < 0must blow-up in finite-time, in the sense that there is no global solution u ∈ C([0,∞);H1(Rd)).Notice that we can always find such initial data, e.g., if u0 = λf, f ∈ H1(Rd) ∩Lp+1(Rd) we can force the negative energy assumption picking λ large enough.Proof. We first derive some identities satisfied as long as a solution remains regular.Multiplying the equation (4.56) first by u and then by ut and integrating by partswe obtainddt(12∫Rd|u|2dx)=∫Rd|u|p+1dx−∫Rd|∇u|2dx (4.57)and∫Rd|ut|2dx = ddt(1p+ 1∫Rd|u|p+1dx− 12∫Rd|∇u|2dx)= − ddtE(u(t)), (4.58)the energy dissipation relation. For convenience we define J(t) := −E(t) and henceby (4.58) we have that J ′(t) :=∫Rd|ut|2dx ≥ 0 and by the assumption on the energyJ(0) > 0. It will be also useful to write J(t) asJ(t) = J(0) +∫ t0∫Rd|ut|2dxdt (4.59)DefineI(t) =∫ t0∫Rd|u|2dxdt+ A (4.60)112with A > 0, to be chosen later. With this definitionI ′(t) =∫Rd|u|2dx (4.61)andI ′′(t) = 2(∫Rd|u|p+1dx−∫Rd|∇u|2dx)(4.62)Since p > 1, δ := 12(p− 1) > 0; a comparison with the energy functional yieldsI ′′(t) ≥ 4(1 + δ)J(t) = 4(1 + δ)(J(0) +∫ t0∫Rd|ut|2dxdt). (4.63)We can also rewriteI ′(t) =∫Rd|u|2dx =∫Rd|u0|2dx+ 2Re∫ t0∫Rdu¯utdxdt.For any > 0 the Young and Ho¨lder inequalities give(I ′(t))2 ≤ 4(1 + )(∫ t0∫Rd|u|2dxdt)(∫ t0∫Rd|ut|2dxdt)+ (1 +1)(∫Rd|u0|2dx)2(4.64)Combining (4.63),(4.59),(4.64), for any α > 0 we obtain:I ′′(t)I(t)− (1 + α)(I ′(t))2 ≥ 4(1 + δ)[J(0) +∫ t0∫Rd|ut|2dxdt] [∫ t0∫Rd|u|2dxdt+ A]− 4(1 + )(1 + α)[∫ t0∫Rd|u|2dxdt] [∫ t0∫Rd|ut|2dxdt]− (1 + 1)(1 + α)[∫Rd|u0|2dx]2.(4.65)Choose α, small enough for 1 + δ ≥ (1 + α)(1 + ). Since J(0) > 0 picking Alarge enough we can ensure I ′′(t)I(t) − (1 + α)(I ′(t))2 > 0. But this is equivalenttoddt(I ′(t)Iα+1(t))> 0 which in turn implies I′(t)Iα+1(t)> I′(0)Iα+1(0)=: a˜ for all t > 0.113Integrating I ′(t) > a˜Iα+1 gives1α(1Iα(0)− 1Iα(t))> a˜t ⇒ Iα(t) > Iα(0)1− Iα(0)αa˜t →∞as t → 1Iα(0)αa˜= 1Aααa˜=: tˆ. This in turn implies that lim supt→tˆ−‖u‖L2 = ∞, show-ing that the solution cannot be globally in CtH1. Note also that (4.57) implieslim supt→tˆ−‖u‖Lp+1 =∞.We present a refinement in the critical case which includes some positive energydata, and in particular establishes Theorem (4.2). So consider now equation (4.9),for whichE(u) =∫Rd(12|∇u|2 − 12∗|u|2∗)dx.Proposition 4.26. Let u0 ∈ H1(Rd) and δ0 > 0 such thatE(u0) < E(W ) and ‖∇u0‖L2 ≥ ‖∇W‖L2 . (4.66)Then the corresponding solution u to (4.9) blows up in finite time. That is, Tmax(u0)(coming from the H˙1 local theory as in Theorem (4.6)) is finite.Proof. We will give a sketch of the proof, which is largely a modification of the proofof the previous proposition.By the Sobolev inequality (4.5),E(u) =12∫Rd|∇u|2dx− 12∗∫Rd|u|2∗dx ≥ 12‖∇u‖2L2 −12∗‖W‖2∗L2∗‖∇W‖2∗L2‖∇u‖2∗L2 (4.67)We define f(y) :=12y − 12∗C2∗y2∗2 , C2∗ = ‖W‖L2∗‖∇W‖L2= ‖∇W‖−2dL2 , so that by energydissipation and (4.66),f(‖∇u‖2L2) ≤ E(u) ≤ E(u0) < E(W ). (4.68)114It is straightforward to verify that f(y) is concave for y ≥ 0 and attains its maximumvalue f(‖∇W‖2L2) = E(W ) = 1d‖∇W‖2L2 at y = ‖∇W‖2L2 . Furthermore, it is strictlyincreasing on [0, ‖∇W‖2L2 ] and strictly decreasing on [‖∇W‖2L2 ,+∞). Denote theinverse function of f on [‖∇W‖2L2 ,+∞) ase = f−1 : (−∞, E(W )]→ [‖∇W‖2L2 ,+∞),strictly decreasing. By (4.68) and (4.66) then,‖∇u(t)‖2L2 ≥ e(E(u(t)).By the definitions of K = K(u) and the energy E = E(u)−K(u) = −∫Rd|∇u|2dx+∫Rd|u|2∗dx = 2d− 2∫Rd|∇u|2dx− 2∗E(u)≥ 2d− 2 (e(E)− dE) =: g(E).Note that g(E(W )) = 0 and for E < E(W ), g(E) > 0 and g′(E) = 2d−2e′(E)− 2∗ <−2∗. Defining I(t) as in (4.60):I ′′(t) = −2K(u) ≥ 2g(E(u)) > 0.By the Fundamental Theorem of Calculus and the energy dissipation relation,2g(E(u)) = 2g(E(u0)) + 2∫ t0|g′(E(u(s)))|∫Rd|ut|2dxds.One can now repeat the proof of Proposition 4.25 replacing (4.63) byI ′′(t) ≥ 4(1 + δ)J(t) = 4(1 + δ)(2g(E(u0)) +∫ t02|g′(E(u(s)))|∫Rd|ut|2dxds)(4.69)Since g(E(u0)) > 0, we can proceed exactly as in the proof of the previous Propo-sition to conclude that if Tmax = ∞, then we must have lim supt→tˆ−‖u(t)‖L2 = ∞ for115some tˆ < ∞, which by (4.61) implies lim supt→tˆ−‖u(t)‖L2∗ = ∞, and so by Sobolev,lim supt→tˆ−‖∇u(t)‖L2 =∞, contradicting Tmax <∞.4.7 Directions for future researchIn recent work, Collot, Merle and Raphael [27], provided a complete classificationof solutions near the stationary solution W for d ≥ 7. In particular, they provedthat for 0 < η 1, if u0 ∈ H˙1(Rd) with ‖u0 − W‖H1 < η, the correspondingsolution u ∈ C((0, T ), H˙1 ∩ H˙2) follows one of the three regimes: (i) “Soliton”: thesolution is global and asymptotically attracted by a “solitary wave”: there exist(λ∞, z∞) ∈ R+∗ × Rd such thatlimt→∞∥∥∥∥∥u(t, ·)− 1λ d−22∞ W (· − z∞λ∞)∥∥∥∥∥H1= 0,(ii) Dissipation: the solution is global and dissipates to zero in both H˙1 and L∞, or(iii) Type I blow-up: the solution blows-up in finite time 0 < T < +∞ in the ODEtype I self-similar blow-up regime near the singularity. There exist solutions associ-ated to each scenario. Moreover, the scenaria of dissipation and Type I blow up arestable in the energy topology. Note that Type II blow-up is ruled out in d ≥ 7 near W.In continuation of this work, there are several questions one can ask in orderto help complete the picture. An interesting direction is provided by the followingproblemProblem: Classify all solutions in a neighbourhood of W for dimensions 3 − 6.Different behaviour is expected since we already know [118] that for d = 4 Type-IIblow-up can occur.For radial solutions, Merle and Matano have recently announced a soliton resolu-tion result for global solutions and a soliton decomposition for blowing-up solutions,along with a two-soliton construction of a global solution in d ≥ 7.116Problem: Can we establish such decompositions for general solutions, for any data?What about constructions of multi-solitons in lower dimensions?The non-radial case of the soliton resolution is open but it is probably withinthe reach of recent technology. We expect that careful energy estimates will beable to provide the missing ingredient in the absence of the one-dimensional toolsused in the proof of the radial case (in particular, the intersection number). 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Long-time dynamics for the energy-critical harmonic map heat flow and nonlinear heat equation Roxanas, Dimitrios 2017
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Title | Long-time dynamics for the energy-critical harmonic map heat flow and nonlinear heat equation |
Creator |
Roxanas, Dimitrios |
Publisher | University of British Columbia |
Date Issued | 2017 |
Description | The main focus of this thesis is on critical parabolic problems, in particular, the harmonic map heat from the plane to S2, and nonlinear focusing heat equations with an algebraic nonlinearity. The focus of this work has been on long-time dynamics, stability and singularity formation, and the investigation of the role of special, soliton-like, solutions to the asymptotic behaviour of solutions. Harmonic Map Heat Flow: Flow: we consider m-corotational solutions to the harmonic map heat flow from R2 to S2. We first work in a class of maps with trivial topology and energy of the initial data below two times the energy of the stationary harmonic map solutions. We give a new proof of global existence and decay. The proof is based on the "concentration-compactness plus rigidity" approach of Kenig and Merle and relies on the dissipation of the energy and a profile decomposition. We also treat m-corotational maps (m greater than 3) with non-trivial topology and energy of the initial data less than three times the energy of the stationary harmonic map solutions. Through a new stability argument we rule out finite-time blow-up and show that the global solution asymptotically converges to a harmonic map. Nonlinear Heat Equation: we also study solutions of the focusing energy-critical nonlinear heat equation. We show that solutions emanating from initial data with energy and kinetic energy below those of the stationary solutions are global and decay to zero. To prove that global solutions dissipate to zero we rely on a refined small data theory, L2-dissipation and an approximation argument. We then follow the "concentration-compactness plus rigidity" roadmap of Kenig and Merle (and in particular the approach taken by Kenig and Koch for Navier-Stokes) to exclude finite-time blow-up. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2017-05-12 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0347430 |
URI | http://hdl.handle.net/2429/61612 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2017-09 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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