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Global questions for evolution equations Landau-Lifshitz flow and Dirac equation Guan, Meijiao 2009

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Global questions for evolution equations Landau-Lifshitz flow and Dirac equation by Meijiao Guan M.Sc., Huazhong University of Science and Technology, 2003 Ph.D., The University of British Columbia, 2009 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics)  THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) July 2009  ©  Meijiao Guan 2009  Abstract This thesis concerns the stationary solutions and their stability for some evolution equations from physics. For these equations, the basic questions regarding the solutions concern existence, uniqueness, stability and singular ity formation. In this thesis, we consider two different classes of equations: the Landau-Lifshitz equations, and nonlinear Dirac equations. There are two different definitions of stationary solutions. For the Landau-Lifshitz equation, the stationary solution is time-independent, while for the Dirac equation, the stationary solution, also called solitary wave solution or ground state solution, is a solution which propagates without changing its shape. The class of Landau-Lifshitz equations (including harmonic map heat flow and Schrödinger map equations) arises in the study of ferromagnets (and anti-ferromagnets), liquid crystals, and is also very natural from a geometric standpoint. Harmonic maps are the stationary solutions to these equations. My thesis concerns the problems of singularity formation vs. global regu larity and long time asymptotics when the target space is a 2-sphere. We consider maps with some symmetry. I show that for m-equivariant maps with energy close to the harmonic map energy, the solutions to Landau Lifshitz equations are global in time and converge to a specific family of harmonic maps for big m, while for m = 1, a finite time blow up solution is constructed for harmonic map heat flow. A model equation for Schrödinger map equations is also studied in my thesis. Global existence and scattering for small solutions and local well-posedness for solutions with finite energy are proved. The existence of standing wave solutions for the nonlinear Dirac equation is studied in my thesis. I construct a branch of solutions which is a con tinuous curve by a perturbation method. It refines the existing results that infinitely many stationary solutions exist, but with uniqueness and continu ity unknown. The ground state solutions of nonlinear Schrodinger equations yield solutions to nonlinear Dirac equations. We also show that this branch of solutions is unstable. This leads to a rigorous proof of the instability of the ground states, confirming non-rigorous results in the physical literature.  11  Table of Contents Abstract  ii  Table of Contents  iii  Acknowledgements  v  Statement of Co-Authorship  vi  1  1 1 2 9 10 13  Introduction and main results 1.1 Introduction 1.1.1 Landau-Lifshitz flow 1.1.2 Model equation for Schrodinger maps 1.1.3 Nonlinear Dirac equations 1.2 Objectives 1.2.1 Global well posedness and blow up for Landau-Lifshitz flow 1.2.2 Well-posedness and scattering of a model equation for Schrodinger maps 1.2.3 Solitary wave solutions for a class of nonlinear Dirac equations 1.2.4 Instability of standing waves for the nonllnear Dirac equations .  .  .  14 15 16 16 17  Bibliography 2  Global existence and blow up for Landau-Lifshitz flow 2.1 Introduction and main results 2.2 Derived nonlinear heat equation 2.3 Proof of the main theorem 2.4 Finite time blow up  Bibliography  .  .  23 23 29 38 44 48  II’  Table of Contents 3  Well-posedness and scattering for a model equation for Schrödinger maps 50 3.1 Introduction and main results 50 3.2 Local weilposedness 56 3.3 Global small solutions and scattering states 62  Bibliography 4  67  Solitary wave solutions for a class of nonlinear Dirac equa tions 69 4.1 Introduction 69 4.2 Preliminary lemmas 73 4.3 Proof of the main theorems 76  Bibliography 5  88  Instability of solitary waves for nonlinear Dirac equations 5.1 Introduction 5.2 Spectral analysis for the linearized operator  107  Bibliography 6  90 90 95  109 Conclusions 109 6.1 Summary 111 6.2 Future research 111 6.2.1 Blow up for 1-equivariant harmonic map heat flow 6.2.2 Construction of unique ground states for nonlinear 112 Dirac equation when 0 < 8 < 1 6.2.3 Asymptotic stability for the model equation for Schrödinger maps 113 113 6.2.4 Nonlinear instability of the ground states .  Bibliography  115  iv  Acknowledgements I wish to express my sincere gratitude to my supervisors (in alphabetical order), professor Stephen Gustafson and professor Tai-Peng Tsai for their continued encouragement and invaluable suggestions during my studies at UBC. Without their inspiration and financial support, I could not finish my degree. Also I acknowledge my debt to Dr. Kyungkeun Kang and professor Dmitry Pelinovsky for their useful communications for my research. I would like to thank the department of Mathematics at UBC. Special thanks is devoted to our department secretary Lee Yupitun. Furthermore, I am deeply indebted to my friends at UBC. Throughout my thesis-writing period, they provided good advice to help get my thesis done and shared their experiences in many different ways. Lastly, I express my special thanks to my parents and my husband. Thanks for their encouragement and support during these years. This thesis is particularly devoted to the birth of my daughter.  v  Statement of Co-Authorship The second chapter of my thesis is co-written with my supervisors Stephen Gustafson and Tai-Peng Tsai. Inspired by their study on Schrödinger maps with energy near the harmonic map energy, we want to extend the anal ysis to harmonic map heat flow. Harmonic maps are the stationary solu tions to Landau-Lifshitz equations, including harmonic map heat flow and Schrödinger maps. These map equations should share some common proper ties. For this paper I obtained the key space-time estimates for the nonlinear heat-type equation in section (2.2). In section (2.3), the technical lemmas are proved by my supervisors. The manuscript was finished by me, and my supervisors made the correction. Chapter 5 is also a co-written with my supervisor Stephen Gustafson. The problem arose from the previous chapter on the nonlinear Dirac equa tion. I conjectured that the ground states are unstable. But we needed to verify it through the analysis of the linearized operator. Together with my supervisor Stephen Gustafson, we determined how to relate the eigenvalue to the linearized operator of nonlinear Schrodinger equation at its ground states (see Theorem 5.2.3). Finally I finished the manuscript.  vi  Chapter 1  Introduction and main results 1.1  Introduction  Nonlinear evolution equations (partial differential equations with a time vari able) arise throughout the sciences as descriptions of dynamics in various sys tems. The classical examples include nonlinear heat, wave and Schrödinger equations. Recently, evolution equations have emerged as an important tool in geometry. A prime example is the application of the Ricci flow to the Poincaré conjecture. For the evolution equations, once the existence of a local in time solu tion for the Cauchy problem is established, the most basic global questions concern the existence of stationary solutions and their stability. Another important question is whether or not solutions can form singularities, which may prevent the solutions from existing for all times, and may represent a breakdown of the model. This thesis addresses these basic questions for two classes of equations: Landau-Lifshitz equations which arise both in physics and geometry, and nonlinear Dirac equations coming from physics. These equations share a common feature: they are equations not for typically scalar valued functions, but rather for maps into manifolds and vector space respectively. Then the analysis of these equations become more challenging. For Landau-Lifshitz equations, the target space is a manifold, more precisely the 2-sphere. It is the nontrivial geometry of the target which creates the nonlinearity and provides many of the interesting and difficult features of this problem. In the Dirac case, the map is a four-dimensional complex vector C . Although this 4 is a linear space, the vector map significantly complicates the construction of stationary solutions and the analysis of their stability. In what follows in this chapter, a brief review, including the physical and mathematical backgrounds, as well as the most recent known results related to my thesis are provided in Section 1.1. In Section 1.2, the main objectives  1  1.1. Introduction and methodologies in my thesis are introduced.  1.1.1  Landau-Lifshitz flow  The principle assumption of the macroscopic theory of ferromagnetism is that the state of a magnetic crystal is describable by the magnetization vector m, which is a function of space and time. Magnetization m is the quantity of magnetic moment per unit volume. Suppose in a small region then the dV C , there is a number N of magnetic moments magnetization is defined as m= It is known from quantum mechanics that the magnetic moment is propor tional to the angular moment of electrons. By the momentum theorem, the rate of change of the angular momentum is equal to the torque exerted on the particle by the magnetic field H. Thus one ends up with a model which describes the precession of the magnetic moment j around the field H =  —  x H,  where -y is the absolute value of the gyromagnetic ratio. By taking the volume average of both sides, one has the following continuum gyromagnetic precession model 8m = —‘ym x H. In 1935 Landau and Lifshitz proposed the Landau-Lifshitz equation [51] as a model for the precessional motion of the magnetization, in which H is replaced by the effective field Heff. They arrived at the Landau-Lifshitz equation =  —‘ym  X  Heff.  (1.1.1)  The effective magnetic field Heff is equal to the variational derivative of the magnetic crystal energy with respect to the vector m. In equation (1.1.1), no dissipative terms appear. Nevertheless, dissipative processes take place within dynamic magnetization processes. Landau and Lifshitz proposed to introduce an additional torque term as a dissipation that pushes magneti zation in the direction of the effective field. Then, the Landau-Lifshitz (LL) equation becomes 3m =  —‘ym x Heff  A  —  iFm x (m X Heff),  (1.1.2)  2  1.1. Introduction  where ) > 0 is a phenomenological constant characteristic of the material. The effective magnetic field Heff is equal to the variational derivative of the magnetic crystal energy with respect to the vector H. In 1955, Gilbert [34] proposed to introduce a different damping term for Landau-Lifshitz equation (1.1.1). He derived an equation which is generally referred to as the Landau-Lifshitz-Gilbert (LLG) equation: 3m =  —7m x  +  X  3m  (1.1.3)  ._.  From a mathematical view point, equation (1.1.2) and (1.1.3) are very sim  ilar. The basic property of LL equation and LLG equation is that the magnitude of m is conserved since = 0. This implies that any mag netization motion, at a given location , will occur a sphere, which can be normalized to be the unit sphere. In the simplest case, the energy is just the “exchange energy” fc IVmIdx, §2 and we can take Heff = —Lm. Let u = in : f C R Th x 3 with C R I = 1, then LL equation can be considered as a linear combination of heat and Schrodinger flow for harmonic maps ,  =  —au x (u x /u)+i3u x u,  (1.1.4)  where denotes the Laplace operator in R’, > 0 is a Gilbert damping constant, /3 IR and “ x “ denotes the usual vector product in R . At first 3 sight, this equation is a strongly coupled degenerate quasi-linear parabolic system, which makes it hard to analyze mathematically. There are other equivalent forms of LL equations. Suppose there are smooth solutions, by the vector cross product formula ax (b xc)  =  (a.c)b— (a.b)c,  one can easily verify that equation (1.1.4) is equivalent to ãtu  If we let plane  U  =  cu +  u) + /3u 2 IVuI  x  LU.  (1.1.5)  denote the orthogonal projection from 1R 3 onto the tangent 2 :={R TS Ic.u=0} 3  to  g2  at n with  3  1.1. Introduction then an equivalent equation, written in a more geometric way, is tU 8  =  aPUZ.,u  + ,BJUPUL\u  where JU = ux is a rotation through 7r/2 on the tangent plane TS . 2 The borderline cases for Landau-Lifshitz equation include harmonic map heat flow when 3 = 0, a = 1: ãu and Schrödinger maps if a  =  =  0, j3  iu + =  u, 2 IVuI  (1.1.6)  1: (1.1.7)  8u=uxzu. The Dirichlet energy functional of (1.1.4) given by  E(u)  =  JIvuI2dx=  plays an very important role in the analysis of these flow equations. The energy identity, obtained formally by taking the scalar product with pu/’u = u + IVuI u and integrating over 2 x [0,t), 2 E(u(t)) +  a  ff  Iu + 2 u IVuI d xdt I=  E(u(0))  implies that for a> 0, the energy is nonincreasing, while for a = 0, the en ergy is conserved. Moreover, with respect to scaling, the energy has critical dimension n = 2. Rescaling the spatial variable, we have E(u(.))  =  where s > 0. Thus the energy is invariant under the scaling if space di mension n = 2. This suggests that n = 2 is critical for the formation of singularities for these equations. Another important feature of the domain 2 problem is the relationship between the energy and the topology, as JR expressed by a lower bound for the energy: E(u) =  2  dx 2 IVuI  4Ideg(u)I  2 R  where deg(u) is the topological degree for the map u : JR 2 space dimension 2 is particularly interesting mathematically.  2 S  —>  2•  So  4  1.1. Introduction These equations have been widely studied by mathematicians and physi cists. One of the most interesting and challenging questions is that of sin gularity formation vs. global regularity: do all solutions with smooth initial data remain smooth for all time, or do they form singularities in finite time for some data? In the following, we describe some important results for the various map equations described above. Harmonic map heat flow Among the Landau-Lifshitz flow equations (including harmonic map heat flow and Schrödinger maps), harmonic map heat flow is the best understood and studied. Equation (1.1.6) generalizes the linear heat equation to maps into Riemannian manifolds. In geometry, it arises in the construction of harmonic maps of certain homotopy types [61]. In physics it arises in the theory of ferromagnetic materials [24—26, 51] and in the theory of liquid crystals [271. A more general setting for equation (1.1.6) is to assume u a differentiable mapping from M, a compact two-dimensional Riemannian manifold (possibly with smooth boundary) to a compact manifold N, which is isometrically embedded in Rk. Eells and Sampson (see [29]) used this equation to construct a harmonic map from M to N. They proved that this equation has a smooth solution defined for all time and converging to a harmonic map under the assumption that the sectional curvature of N was non-positive. This result is not true if no curvature assumptions are made on N. Struwe in [59] proved partial regularity for global weak solutions with finite initial energy (uo H’(M, N)), with at most finitely many singular space-time points where nonconstant harmonic maps “separate”. The small energy solutions are global. Freire [31] showed that the weak solution in [59] is unique if the energy is non-increasing along the flow. Since space dimension 2 is energy critical, much effort has been devoted to the case where the domain is the unit disk in R , the target manifold is 2 2, and for a special class of solutions: u(., t) : (r, 0)  —  (cos mO sin (r, t), sin mO sin (r, t), cos (r, t))  2 eS  C  (1.1.8) for a positive integer m. (r, 0) are polar coordinates. Then the scalar func tion satisfies a nonlinear heat-type equation: trr+r_m251,  O<r<1, t>O.  (1.1.9)  One specifies initial condition (r,O) = o(r), and typical boundary condi tions are (O, t) = 0, and (1, t) = i E Il’. In fact the requirement that 5  1.1. Introduction sin (0, t)  =  0 is necessary for solutions to have finite energy E()  f  =  (r  + Sm  )dr.  Singularities can develop only at the origin. If singularity occurs at some space point r = ro 0, then the corresponding solution u(x, t) has infinitely many singular points along the circle IxI = r , which is a contradiction to 0 Struwe’s result. Using the standard notation for Lebesgue norms: we say solutions blow up at time T < oo, if  lim sup IIVu(.,t)II  =  00,  lim sup IIu(,t)IIL2  =  00,  or t-T  while u remains bounded (uI = 1). Thus, for equation (1.1.9), q blows up while remains bounded. The issue is whether all solutions eventually converge to equilibria or singularities form in finite time for some initial data. Intuitively if we choose initial data in a topological class which does not contain any equilibria, the solutions must blow up, at least in infinite time. For m = 1, global regularity is proved by Chang and Ding [17] if qo (r) I ir, 0 r 1. However the global solution in [17] may not subconverge to a harmonic map as t —> oc if i = ir, since there is no harmonic map satisfying both the boundary conditions. It must develop a singularity at t = oo. In 1992 collaborating with Ye [18], they showed (again for m = 1) that, indeed, finite time blow-up does occur for finite energy solutions if I I > ir. A result of [11] tells us that even if m = 1 and < K, finite time blow up is still possible if qo(r) rises above K for some r E (0, 1). As for m 2, the situation may be different. Grotowski and Shatah [38] observed the difference and proved that for boundary data II < 2ir, the solution remains regular for all time, so finite time blow up will not occur. Those results implying the finite-time blow-up do not predict at what rate the gradient should blow up. A direct calculation shows that one pa rameter family of finite energy, static solutions to equation (1.1.9) are given by f(r)  =  2arctan-,  \ >0.  When blow up occurs at t = T, an appropriate rescale will reveal a stationary solution. Let R(t) > 0 be the rescale function, then locally in space lim $(t,rR(t))  =  . m 2arctanr 6  1.1. Introduction In [10] the generic blow-up behavior (and blow-up rate) was analyzed via formal asymptotics by Berg, Hulfson and King, where they observed that whether or not singularities occur appears to depend on m, as well as on the initial and boundary data. Moreover, different blow up rates were given explicitly. In [1] Angenent and Hulshof rigorously prove the infinite time blow-up rate in [10] for a special setting, where they assume that the initial data can reach only one time. When blow up occurs, the value of at the origin jumps to another value, the solution changes to another topological class, and the energy jumps by an integer multiple of 4Trm. The harmonic map heat flow splits off “bubbles”, i.e. non constant harmonic maps separate at the singular points. In [57] Qing pointed out the energy loss at a singularity can be recovered as a finite sum of energies of tangent bubbles. After blow-up, the weak solutions can no longer be expected to be unique if the energy is not assumed to be decreasing. In [11] Bertsch, Passo and Van Der Hout constructed explicit examples for non-uniqueness of extensions of solutions after blow-up. These solutions are characterized by “backward bubbling” at some arbitrarily large time and all have uniformly bounded energy. The qualitative descriptions of Struwe’s solutions near their singular points are also given in [65, 66]. In addition to the critical space dimension 2, different space dimensions for harmonic map heat flow are also widely studied. Finite-time blow up was proved by Coron and Ghidaglia in [20] for certain smooth initial data from IR or S’ to §fl, n 3. Later Chen and Ding in [14] extended this result to n 3 without the assumption of symmetries of 5fl and the initial maps. For maps between the 3-dimensional ball and 2, the nonuniqueness of solutions was proved in [12, 13] for some initial data. Grotowski [37] established blow-up and convergence results for certain axially symmetric initial data. Landau-Lifshitz flow For Landau-Lifshitz flow equation, with the Schrödinger-type term 3 0, our understanding diminishes considerably. Indeed this equation is still parabolic, but maximum principle arguments are not readily applicable. Due to the extra nonlinear term u x u, more estimates are required in the proof of regularity results compared to those of harmonic map heat flow. The first existence results for weak solutions to LL equation are due to [68], see also [2, 7]. In [2], Alouges and Soyeur established a nonuniqueness result for weak solutions when . Their approach was based on the 3 E 1R method introduced in [13] to prove the nonuniqueness of weak solutions to  7  1.1. Introduction the harmonic map heat flow. For a mapping u from a two-dimensional com pact Riemannian manifold M to g2, partial regularity result are analogous extensions of those for Struwe’s solutions in [59]. In the case of OM = 0, Guo and Hong [44] proved the regularity of weak solutions for uo E H’ (M, S ) 2 with the exception of at most finitely many singular space-time points; they are globally regular in the case of small initial energy. The arguments are based on the a priori uniform estimates for 0u and D n. For M with smooth 2 boundary, the similar results were obtained by Chen in [15, 16]. Moreover, the local behavior of the solution near its singularities was investigated in [16]. Instead of using the uniform estimates for and D u, L L and 2 W” estimates for the linear parabolic system were used to show the partial regularity of the weak solutions. The study of the weak solution near the singular points also follows from previous results in the harmonic map heat flow [11, 57]. For bounded domain in R , Harpes [48] analyzed the geometric description of the flow at isolated 2 singularities and showed the non-uniqueness of extensions of the flow after blow-up. But the formation of singularities is still open. Since maximum principle arguments (for example, sub-solution construction) do not apply, the blow-up argument for harmonic map heat flow may not provide a useful tool for the Landau-Lifshitz flow equation. —  Schrödinger maps In the case of c = 0, equation (1.1.4) becomes the Schrödinger map equation (1.1.7). The analysis of Schrödinger maps becomes more difficult, even the local-in-time theory is not yet well understood. There is a great deal of recent work on the local well-posedness problems in two space dimensions ([4], [5], [23], [49], [53], [63]), see also on [50, 55] for the “modified Schrödinger 52, local well-p osedness is established for map”. For maps u : R x R+ 0 e Vu k> n/2 + 1 is an integer, and also the global well-posedness is proved for small data when ii > 2 in [63]. Chang, Shatah and Uhlenbeck in [19] considered the global well-posedness problem for finite energy. For 0 E H’(R’) (1.1.7) has a unique global solution. While for n = 2, small U energy implies global existence and uniqueness for radial and equivariant maps. For u : R 2 , by an m-equivariant map, we mean 3 C R —,  u(r,8)  =  emOv(r)  (1.1.10)  where m e Z a non-zero integer,(r, 8) are polar coordinates on R , and R is 2 the matrix generating rotations around the u -axis. The argument in [19] is 3 based on a generalized version of the “Hasimoto transformation”. The key 8  1.1. Introduction — !JiJuRu e observation involves the vector field w(x, t) = which 52 lies in the tangent plane to at u for each x, t. For an appropriate choice of an orthonormal frame on the tangent plane, the coordinates of w satisfy a kind of cubic nonlinear Schrödinger equation. Then the standard estimates (Strichartz estimates) for nonlinear Schrodinger equations can be applied to this equation. Gustafson, Kang and Tsai in [45, 46] investigated Schrödinger flow equa tion for energy space initial data, i.e. the energy of the initial data is near the harmonic map energy 4irImI. Their results reveal that if the topological degree m of the map is at least four, blow-up does not occur, and the global in time solution converges (in a dispersive sense) to a fixed harmonic map for large initial data. This is the first general result for Schrödinger maps with non-small energy space initial data. It is still open whether or not finite-time singularities can form for Schrödinger maps from R 2 x R to 2, partly because the class (1.1.8) is no longer preserved, as in the harmonic map heat flow. This motivated us to consider a model equation for Schrodinger maps.  1.1.2  Model equation for Schrödinger maps  —* S Let u : R 2 x . For geometric evolution equations such as harmonic 2 map heat flow (1.1.6) and wave maps  PuUtt  =  Puu  a special class of solutions (1.1.8) was preserved, which allows a reduction to a single scalar equation (e.g. equation (1.1.9)). However this class is not preserved by Schrödinger maps which makes the construction of singular solutions much harder. This is our motivation to introduce a model equation. x R —* C be a radial scalar function and m > 0 be an Let 7(x, t) : integer. By analogy with (1.1.9) for the heat flow, we consider the nonlinear Schrodinger equation  () =o(x).  (1.1.11)  2 norm is The nonlinearity is designed to be gauge invariant (hence the L preserved) and to admit a conserved energy defined by  irJ  E(c,b) =  2 m (IrI2 + —-sin 2 hI)rdr. r  9  1.1. Introduction The nontrivial stationary solutions with finite energy are given by a 2parameter class of functions :=  {eQ(r/s)IQ(r)  =  ),s m 2arctan(r  > O,c  e  R}.  Thus we comes up with two questions: whether we can draw the same conclusion as in [46] (the Schrödinger maps), i.e. harmonic maps are stable under this equation for large m, or can we show finite time blow up when m = 1? Moreover, since zero is a trivial static solution, do small solutions exist for all time? This equation is related to Cross-Pitaveskii equation (Nonlinear Schrodinger equation with nonzero boundary conditions, see [47]), but it is hard in some respects.  1.1.3  Nonlinear Dirac equations  In physics, the Dirac equation is a relativistic quantum mechanical wave equation and provides a description of elementary spin-i particles, such as electrons. In 1928, British physicist Paul Dirac derived the linear Dirac equation from the relation of the energy E and the momentum p of a free relativistic particle 2 =c E p+m 2 , 4 c 2 where c is the speed of light, m is the rest mass of the electron. Quantum mechanically , E = ih is the kinetic energy and p = —ihV e R 3 is the momentum operator (h is Planck’s constant). For the wave function q5(x, t), a relativistic equation is obtained =  (V2 —  m2c2,  (1.1.12)  2)  In order to derive Dirac equation, Dirac proposed to factorize the left hand side of equation (1.1.12) as follows: =  (71  +  +  0y + 2 + 7°8t)(7’O + 7  Dz  —  + 73t),  where ‘y°, ’(j = 1,2,3) are to be determined. On multiplying out the right 7 side, one found out y , yi must satisfy 0 773  +  737 =  , 1 j 26  0 j 7  +7  =  0,  (70)2  =  I.  These conditions could be met if ‘y°, 73 are at least 4 x 4 matrices (see [64]) and then the wave functions have four components. The usual representation 10  1.1. Introduction O, f 0 i 7  is 0  where  —  n  (L2x2  \ 0  /c  I  ix)  k_I ‘  —  k  ‘  2  o  iS\U  —  ‘  are Pauli matrices:  a  (0  1”  2  o)’  (0  —  —  —  o)’  (‘1  ko  0 —1  Given the factorization and equation (1.1.12), one can obtain the linear Dirac equation which is first order in space and time:  —  Dm  =  0,  Dm  =  kOk + mc O 7 _ich 7°. 2  x IR —* C for ,b : . Without loss of generality, we always assume c = h = 1. 4 The Dirac operator is symmetric and has a negative continuous spectrum which is not bounded from below, as for which makes it harder to analyze mathematically and physically. On one hand, the energy functional is strongly indefinite. On the other hand, although there is no observable electron of negative energy, the negative spectrum plays an important role in the physics. The main interest in this thesis is to present the various results regarding the standing wave solutions for nonlinear Dirac equations. A general form of nonlinear Dirac equation is given by —,  iOt’cb  VF(b) 0 Dm’b + 7  —  =  0.  (1.1.13)  From experimental data in [58], a Lorentz-invariant interaction term F(b) is chosen in order to find a model of the free localized electron (or on another spin-i particle). So the usual assumption on F is that F e 4 ( 2 C , R) C and ) 8 F(e  =  F()  for all  0.  As pointed out in [58], stationary wave solutions of equation (1.1.13) repre sent the state of a localized particle which can propagate without changing its shape (x,t) = where  is a nonzero localized function satisfying Dmq  —  wq  —  VF() 0 7  0.  (1.1.14) 11  1.1. Introduction Different functions F have been used to model various types of self-interaction. In [58], Rafiada gave a very interesting review on the historic background of different models. The existence of standing wave solutions has been extensively studied by different methods. In [32, 33], Finkelstein et al. proposed various models for extended particles corresponding to the fourth order self-couplings like F()  :=  75  b) 5 b(y , 2+2 a()  =  70717273  (1.1.15)  where ‘z,b = (7°,?) and (.,.) is the Hermitian inner product in C . This 4 nonlinearity is a sum of a scalar and pseudoscalar terms. In these papers, they gave some numerical results of the structure of the solutions for dif ferent values of the parameters. The special case b = 0, a > 0, called the Soler model [60], is proposed by Soler to describe elementary fermions. The advantage of this model is that its solutions can be factorized in spherical coordinates:  if(r)  (1.1.16)  cos’ sin We ’ 1  Vazquez studied the Soler model in [67] and came up with a necessary condi tion for the existence of localized solutions (see also [52]). In [3] the existence of infinitely many standing waves are obtained. For the generalized Soler model when F(&)  =  G(),G E C (R,R),G(0) 2  =  0.  (1.1.17)  the existence of infinity many stationary solutions have been obtained in [21] for some function G and w € (0,m). Since the localized solutions are separable in spherical coordinates, the nonlinear Dirac equation can be re duced to a nonautonomous planar differential system. This system can be solved by a shooting method. Later on , this result was extended to a wider class of nonlinearities by Merle [54]. In the case where G has a singularity at the origin like G(s) = —ISIP, 0 <p < 1, Balabane, Cazenave and Vazquez [9] proved that for every > m, there exists a solution of (1.1.14) such that has a compact support. But there are other models of self couplings for which the ansatz (1.1.16) is no longer valid, for instance (1.1.15) and F()  =  iIc1 +  byS2  12  1.2.  Objectives  with nonzero b and c, c 2 > 0. Esteban and Séré in [28] studied this more general nonlinearity. They proved by a variational method, there exists an infinity of solutions under the assumption G’(x).xOG(x),  6>1,  xEIR  for 1 < al,a2 < None of the approaches mentioned above yield a curve of solutions: the continuity of with respect to w, even the uniqueness of was unknown. These issues are important to study the stability of the standing waves. To our knowledge, Ounaies [56] was the first one to consider the regularity of the stationary solutions. He related the solutions to (1.1.14) to those of nonlinear Schrödinger equations. The ground states of Schrödinger equations generate a branch of solutions with small parameter s = m w for nonlinear Dirac equations. He claimed that for F(s) = sla, 1 < is continuous < 2, w.r.t. w when w e (m eo,m) for some sç > 0. For a thorough review on the linear and nonlinear Dirac equation, we refer to the work by Esteban, Lewin and Séré [30]. A basic question about standing waves is their stability, which has been studied for nonlinear Klein-Gordon equations and nonlinear Schrödinger equations. Grillakis, Shatah and Strauss [35, 36] proved a general orbital stability and instability condition in a very general setting, which can be applied to traveling waves of nonlinear PDEs such as Klein-Gordon, and Schrödinger and wave equations. Their assumptions allow the second vari ation operator to have only one simple negative eigenvalue, a kernel of di mension one and the rest of the spectrum to be positive and bounded away from zero. But this method cannot be applied to the Dirac operator di rectly. Contrary to the Dirac operator Dm = —i)’° m is j+0 8 y not bounded from below. However there are some partial results about the application of this method to the Dirac equation. Bogolubsky in [6] requires the positivity of the second variation of the energy functional as a necessary condition for stability. Werle [69], Strauss and Vázquez in [62] claim that the solitary waves are unstable, if the energy functional does not have a local minimum at the solitary waves. —  —  —,  1.2  Objectives  In the following I describe the results and the methods used in my thesis for the above equations.  13  1.2. Objectives  1.2.1  Global well posedness and blow up for Landau-Lifshitz flow  §2, For the Landau-Lifshitz flow equation (1.1.4) where u : 1R 2 x the well posedness vs. blowup is studied in my thesis [39, 40]. A good starting point to analyze the flow equation is to assume some symmetry. For maps Rv(r), the energy E(u) has a minimal 8 u with equivariant symmetry u = em energy 4irImI, which is attainable by a two-parameter family of harmonic maps: ,.  ’ 5 H  =  e(mOh(r/s), /  hi(r) \\ h(r)=( 0 (r) 3 h ‘  E IR, s> 0  /  I  2 r+r  1=1  J  \  0  \  The harmonic maps are static solutions of the evolution equation. The natural question is to consider the stability of the harmonic maps under the Landau-Lifshitz flow. Our first result concerns m-equivariant maps with energy near the min imal energy 47r1m1,  E(uo)  =  47r1m1 +  ,  0  <  öo << 1.  We have shown that there is no finite time blowup for ml 4. Further more the solutions converge to a specific family of harmonic maps in the space-time norm sense. Hence we say that the harmonic maps are asymp totically stable under the Landau-Lifshitz flow. This result is a rigorous verification of [10] where the authors showed no singularity formation in finite time by formal asymptotic analysis. Our main ingredients involved in the proof are the usage of two different coordinate systems. In an ap propriate orthonormal frame on the tangent plane, the coordinates of the JuRu satisfy a nonlinear heat-Schrödinger type tangent vector field u,. equation. It leaves us with an equation with small L 2 initial data. This equation is also coupled to a 2-dimensional dynamical system describing the dynamics of the scaling parameter s(t) and rotation parametero(t) of a nearby harmonic map H(s(t),(t)). A careful choice of these parameters must be made at each time to allow estimates. The key to prove convergence of the solutions is the space-time estimates for the linear operator of the nonlinear heat-Schrödinger type equation. This can be done because of the energy inequality, the positivity of the energy and the assumption of radial functions. —  14  1.2.  Objectives  For harmonic map heat flow, i.e. /3 = 0, in the subclass of m-equivariant maps (see (1.1.8)), we proved that for m = 1, finite time singularities do occur for some initial data close to the energy of harmonic maps. This result is an adaptation of the blow up result in [18] for a disk domain D 2 in R . 2  1.2.2  Well-posedness and scattering of a model equation for Schrödinger maps  Recall the model equation (1.1.11)  sin2I4 2 m 2I  (0,r)=o(r)  where q(x, t) 1l2 x R —> C is a radial scalar function and m> 0 is an integer. We are interested in finite energy solutions which leads to sin Io(x)I = 0 both at IxI = 0 and lxI = oo. Therefore this yields the following boundary conditions ) <oo, 0 {o: [0, oc) — C, E( lim o(x) = k ir, urn o(x) = k 1 7r}, 2 IzcHO xI—oo =  ,k 1 k 2  e  (1.2.1) .  By examining the energy E(), we find that for solutions in the class with = 0, o(oo) = ir, E() has a minimal lower bound 4irImI. Harmonic maps are stationary solutions of (1.1.11). Moreover equation (1.1.11) pos sesses constant solutions kir(k E Z) with finite energy. I consider the well posedness for solutions in the form qr, t) = (r, t) + S for either S = 0 or ). In [43] I proved that for any m S = Q(r) = 2arctan(r  m  X  {:  [0,oo)  C  (rdr), 2 eL  E L(rdr)},  there exists a maximal time interval I containing 0, such that the model equation (1.1.11) has a unique solution in the class (r,t) = (r,t) + 5, satisfying j e C(I;X)flL(I;X ) T where X := {u: [0, cc)  —*  CIur  e L(rdr),  E L(rdr)}  and 2 < q < cc, + = Moreover, if S = 0 and IIco lix 5 for suffi ciently small 8, then the solution to equation (1.1.11) is defined for all time. The approach is to linearize equation (1.1.11) at the stationary solutions. Treating the linearized operator as — plus some perturbation, then + the Strichartz estimates can be applied to yield the results. .  15  1.2.  1.2.3  Objectives  Solitary wave solutions for a class of nonlinear Dirac equations  We consider a class of nonlinear Dirac equations in [41] i8 +  i7’Oj  —  m+  ii°  =  0.  (1.2.2)  Under the spherical coordinates ansatz (1.1.16), the equation for solitary waves can be reduced to a nonautonomous planar differential system for (f,g). A rescaling argument reveals that the solutions to this system is generated by the ground states of nonlinear Schrödinger equations —+v—  v=0. 2 IvI  (1.2.3)  It is well known that for 8 e (0,2), equation (1.2.3) admits a unique solution called the ground state Q(x) which is smooth,positive, decreases monotoni cally as a functions of lxi and decays exponentially at infinity. We prove that when 1 < 8 < 2, there exists e 0 > 0 such that for w E (m Eo, m), there exists a solution (t, x) = and the mapping from w to is continuous. This result is different from that in [56j where Ounaies claimed it for 0 < 8 < 1. But with the restriction 0 < 8 < 1, we are unable to verify the Lipshitz continuity of the rionlinearities. Thus the contraction mapping theorem is not readily applied to construct solutions. —  1.2.4  Instability of standing waves for the nonlinear Dirac equations  The stability problem of the standing wave solutions for the nonlinear Dirac equation with scalar self-interaction is considered in [42]. It is shown that the branch of standing waves constructed in [41] is unstable. The question of stability is related to the eigenvalues of the linearized operator. To show the instability, the main goal is to show the linearized operator has an eigenvalue with positive real part. Since the linearized operator is four-by-four matrix, it is important to block-diagonalize it as in [22]. It turns out that the eigenvalue is related to that of the cubic, focusing and radial nonlinear Schrödinger equations. We use formal expansion of the eigenvalue and eigenfunction to show that there exists such an eigenvalue ). with Re A positive. 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TsAI, Global Questions for Map Evolution Equations, Centre de Recherches Mathé matiques, CRM Proceedings and Lecture Notes,Vol. 44, 2008(61-73).  [41] M. GuAN, Standing Wave Solutions for Nonlinear Dirac Equation, 2008, http://arxiv.org/abs/0812.2273, submitted. [42] M. GuAN, S. GusTAFs0N, Instability of solitary waves for nonlinear Dirac equations, 2009, to be submitted. [43] M. GuAN, Well-posedness and scattering of a model equation for Schrödinger flow, 2009, to be submitted. [44] B. Guo, M. C. HONG, The Landau-Lifshitz equation of the ferromag netic spin chain and harmonic maps, Caic. Var. 1, 311-334(1993). [45] S. GusTAFsON, K. KANG & T.-P. TsAI, schrödinger flow near har monic maps, Comm. Pure Appi. Math. 60 (2007) no. 4, 463—499. [46] S. GusTAFsON, K. KANG & T.-P. TsAI, Asymptotic stability of har monic maps under the schrödinger flow, Duke Math. J. 145, No. 3 (2008), 537-583. & T.-P. TsAI. 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ToPPING, Winding behavior of finite-time singularieis of the har monic map heat flow. Math. Z., 247 (2004), pp. 279-302. [67] L. VAzQuEz, Localized solutions of a nonlinear spinor field, J. Phys. A: Math. Gen., Vol. 10, No. 8, 1977(1361 -1368). [68] A. VISINTIN, On Landau-Lifshitz equations for ferromagnetism, Japan J. Appi. Math., 2 (1985), pp. 49C84. [69] J.  WERLE,  1981, Acta Physica Polonica B12, pp. 601-606.  22  Chapter 2  Global existence and blow up for Landau-Lifshitz flow As we introduced in the first chapter, we have obtained global result for the Landau-Lifshitz equation, including harmonic map heat flow. The following content is concentrated on harmonic map heat flow. The proof of global regularity and asymptotic stability results for Landau-Lifshitz equation is similar and is given in [11] (see the remark (*) in Section 2.2).  2.1  Introduction and main results  The harmonic map heat flow we consider is given by the equation =  where u(., t) :  C  1u +  u, 2 IVuI  u(x, 0)  =  (2.1.1)  Uo(x)  §, S 2 is the 2-sphere  R”  I lul denotes the Laplace operator in R”, IVuI 2 {u  =  ) 3 (‘ul,u2,u  =  1} CR , 3  2 ( 1 Z— . ) This equa  =  tion written in a more geometric way is Ut  where  U  =  D8u  =  puU  3 onto the tangent plane denotes the orthogonal projection from R  2 TS  :={ e R 3 Iu =0}  3 is the covariant. is the usual partial derivative, and D to 2 at u, 8 = derivative, acting on vector fields (x) e T()S 2 -.  •PU9  =  —  (0 u)u .  =  Oj + (Ou )u. .  A version of this chapter has been published. Guan, M., Gustafson, S. and Tsai, T.-P. Global existence and blow-up for harmonic map heat flow, J. Duff. Equations (2009) 246, 1—20.  23  2.1. Introduction and main results Equation (2.1.1) is the gradient flow for the energy functional E(u)  =  L1vuI2  Static solutions of Equation (2.1.1) are harmonic maps from Q to S . 2 Equation (2.1.1) is a particular case of the harmonic map heat flow between Riemannian manifolds introduced by Eells and Sampson ([8]). On the other hand, Equation (2.1.1) is a borderline case of the Landau-Lifshitz-Gilbert equations which model isotropic ferromagnetic spin systems: ut=aP/.u+buxP/.u,  a>O.  (2.1.2)  (see [14, 161). The harmonic map heat flow corresponds to the case a = 1,b= 0. In this chapter, we consider space dimension ii = 2, which makes the energy E(u) invariant under scaling, and so is in some sense a borderline case for the interesting question of singularity formation vs. global regularity: do all solutions with smooth initial data remain smooth for all time, or do they form singularities in finite time for some data? Let us recall some of the important results for n = 2. Struwe in [19] proved that weak solutions to Equation (2.1.1) exist globally for finite-energy initial data, and are smooth except for at most finitely many singular spacetime points where non-constant harmonic maps “separate”. Also, solutions are global for small initial energy. Freire showed that the weak solution is unique if the energy is non-increasing along the flow ([9]). Much effort has been devoted to the case where the domain is the unit disk in R , and for a 2 special class of solutions: u(., t) : (r, 8)  —÷  (cosm8 sin (r, t), sinm8sin (r, t), cos (r, t))  for a positive integer m, usually satisfies the equation:  m  =  1.  trr+r_m2S1, 2r r  (r,  (2.1.3)  0) are polar coordinates. Then  0<r<1,t>0.  (2.1.4)  One specifies initial conditions (r, 0) = qSo(r), and typical boundary condi tions are (0,t) = 0, and (1,t) = e R. For m = 1, global regularity is proved if Ic(O, r)I r in [4]. However even if the flow exists for all time, it may develop singularities at T = cc, so that it fails to converge asymptoti cally. In [5] the authors showed (again for m = 1) that, indeed, finite time 24  2.1. Introduction and main results blow-up does occur for finite energy solutions, if q > n. A result of [2] tells us that even if m = 1 and iI < 7t, finite time blow up is still possible if 1(0, r) rises above r for some r e (0, 1). Recently, the generic blow-up be havior (and blow-up rate) was analyzed via formal asymptotics in [1], where they observed that whether or not singularities occur appears to depend on the degree m, as well as on the initial and boundary data. One of the purposes of our study is to provide a rigorous proof of this observation. We note that finite-time singularities may also form in the harmonic map heat flow when n 3 ([3, 7]). It is worth remarking that the blow-up vs. global smoothness question is also currently studied for both the wave and Schrödinger “analogues” of the harmonic map heat flow. The possibility of finite-time blow-up for the energy-space critical (n = 2) wave maps was established recently ([15, 18]), while the problem remains open for n = 2 Schrödinger maps (the a = 0 case of (2.1.2)), though a partial answer was given in [13]: in contrast to the wave map case, high-degree equivariant (see next section for the definition) Schrödinger maps with near-harmonic energy are globally smooth. The main goal of the present paper is to address the global regularity vs. finite-time blowup question for a larger class of maps than (2.1.3), and for the problem on the plane, rather than a disk. This means that the evolution is no longer described by a single, simple nonlinear heat equation like (2.1.4), but rather by a more complex system. In particular, maximum principles are no longer available (at least directly). To be more precise, we consider m-equivariant maps u : R 2 x 2 S 2 is of the with m E Z a non-zero integer. An m-equivariant map u : —‘ form .•  u(r,  )  =  emOR v(r)  where (r,9) are polar coordinates on R , v : [0,co) 2 matrix generating rotations around the u -axis: 3  R  =  /0 1  —1 0  0\ 0  )  e°  ,  \o Do)  =  /cos c sin c  \o  —  52, and R is the  sin c cos c  —  o  0 0 1  In what follows, we will take m> 0 (the m < 0 cases are equivalent, by a simple transformation). If u is m-equivariant, we have IVuI 2 = IUrI 1u = Ivr r 2 1 2+0 2 + IRvI2 and so p00/  E(u)=lrJ  2  +—-(v+v)) 2 (IvrI r  rdr.  /  25  2.1. Introduction and main results For finite energy E(u), it is necessary to have v(0),v(oo) = ±, where k = (0,0, i)T (see [10] Section 2.2 for details). We fix v(0) = —k and denote by 2m the class of m-equivariant maps with v(oo) = k:  {u :  =  u  =  emGRv(r), (u)  <,  v(0)  =  —, v(oo) =  We measure distances between maps in m in the energy norm  lu ulI —  IIV(u ü)11L . 2  =  —  The class m contains (3.1.1) as a special case (up to a trivial reflection ‘U3 —# —‘U3, and ignoring boundary conditions). For u m-equivariant, the energy E(u) can be rewritten as follows: 2  00  E(u)  =  irf  (IvTl2 +  IJ0RvI2) rdr  =  irf  lVr_JvRhl2rth+Emin  where J 0 := vx is a ir/2 rotation on TS , and 2 p00  Emin  =  27rJ  p00  r  J’Rvrdr  =  27r1m1J  )dr 3 (v  =  27r1m1[v3(oo) + 1]  1). The number Emin, which depends only on the (using v + v + V boundary conditions, is in fact 47r times the absolute value of the degree of the map u, considered as a map from S 2 to itself by compactifying the domain R 2 (via stereographic projection); the degree is defined, for example, by integrating the pullback by u of the volume form on It provides a lower bound for the energy of an m-equivariant map, E(u) > Emin, and this lower bound is attained if and only if (2.1.5)  VT = HJVRV  If v(oo) = the minimal energy is Emin = 0 and is attained by the constant map, u —k. On the other hand, if u E so that v(oc) = k, the minimal energy is E(u) Emin = 4rlml —,  and is attained by the 2-parameter family of harmonic maps := {em0hs(r)  I  s>0, a  [0,27r)}  where  ’°(r) := e00h(r/s), 8 h 26  2.1. Introduction and main results and  h(r)  /  =  I  \  hi(r) 0 1, / \ 1 It3T) \ /  h,(r)  =  2 1 + rImI rIm  rIm r_ImI 1 1 + rHmi rim —  ,  ha(r)  =  We record for later use that h(r) satisfying (2.1.5) means  (hi)r  =  m ——h,h , 3 r  )r 3 (h  =  m 2 —h,. r  So (3m is the orbit of the single harmonic map emORh(r) under the sym metries of the energy which preserve equivariance: scaling, and rotation. Explicitly, cos(mO + c)h,(r/s) emORh8a(r) sin(m9 + a)hi(r/s) = (r/s) 3 h  / (  We begin with the energy-space local-in-time theory: Theorem 2.1.1 Let m > 1. There exist S > 0 and C > 0 such that if C Dm and (uo) = 4-,rm + S for some 5 o 5, then the following hold:  U0  (a) there exists T = T(uo) > 0 and a unique solution u(t) C C([0, T); m) to Equation (2.1.1). E(u(t)) is non-increasing forte [0,T).  (b) there exist 8(t) e C([0,T); (0,oo)) and c(t) e C([0,T);R) so that u(x,t)  —  e(m0t))?h(r/s(t))W.  , 9 <CS  Vt e (0,T).  (2.1.6)  (c) Suppose T < oo. Then T is the maximal existence time of the solution u(t) e C([0,T);Dm) if and only if liminfs(t) =0.  (2.1.7)  Remark 2.1.2 Statement (b) of Theorem 2.1.1 implies the orbital stability of harmonic maps (at least up to the possible blow-up time) under the heat flow. If the initial data u 0 is close to 0m in H’, then solutions of equation (2.1.1) will stay close to the harmonic maps in H’ (though not necessarily in H ). Statement (c) can be viewed as a characterization of 2 blow-up for energy near Emin: solutions blow-up if and only if the H’nearest harmonic map “collapses” (i.e., its length-scale goes to zero). Here s(t) and c(t) are determined by finding, at each time t, the harmonic map which is H’- closest to u(t). 27  2.1. Introduction and main results A theorem identical to Theorem 2.1.1 is established in [12] for the (more delicate) corresponding Schrödinger flow problem. The proof there uses the same geometric representation and decomposition of the solution used in the present paper, and indeed we show here that the same estimates (and more) hold for the linearized problem (Section 2.2) and the nonlinear terms (Section 2.3), and so the proof carries over with no significant alteration. For this reason, and since the full energy space local well-posedness (without symmetry or energy restrictions) is already well-understood for the heat flow (in particular, Struwe [19]), we will not provide the details. We remark that the blow-up characterization (2.1.7) corresponds to the “separation” of a harmonic map at a singularity in [19]. The next theorem, our main result, shows that when the degree is at least 4, singularities do not form, and we can describe precisely the asymptotic behavior: Theorem 2.1.3 Let m > 4. There exists ö 1 e (0, ó) and C> 0 such that if < 6, then the existence time T = T(uo) in Theorem 2.1.1 can be taken o 8  to be T = cc. One also has  IIV(u(x,t)  —  emSRh(5(t)  t)))IIL2Lc,flLoOL2  < Co . 0  (2.1.8)  Moreover there exist c, and positive s such that (s(t), c(t))  —  (s, ) as t  —*  cc.  (2.1.9)  1. Not only is the solution global, but (2.1.8) and (2.1.9) show that u(., t) converges to a fixed harmonic map as t — cc (at least in a time-averaged sense) — in particular, this gives asymptotic stability of the harmonic maps for m 4.  Remark 2.1.4  2. For the cases m = 2, 3, we conjecture that solutions are still global, but this is presently beyond the reach of our methods. The technical reason is that we need r (rdr), which requires m > 3. 2 hi(r) e L 2 The final theorem shows that when m = 1, finite-time blow-up does occur within our class of solutions. Theorem 2.1.5 If m = 1, for any 6 > 0, there exists u0 E >J with 0 < < 62 such that the corresponding solution of the harmonic map E(uo) — heat flow blows up in finite time, in the sense that IIVu(., t)ILc,o —‘ cc.  28  2.2. Derived nonlinear heat equation Remark 2.1.6 Our result that blowup occurs for degree one, but not for higher degree, is consistent with the formal asymptotic analysis of [1].  The paper is organized as follows. In Section 2 we derive, from the harmonic map heat flow equation, a related nonlinear heat equation, by a choice of frame on the tangent space. We also establish space-time es timates (including “endpoint” -type estimates) and weighted estimates for the linear operator which comes from the perturbation about the harmonic maps. Even though the potential appearing in the linear operator behaves like l/1x1 2 both at the origin and as lxi —* oo, we can treat it by an energy inequality to avoid the difficulty. In Section 3, we obtain explicit equations for the parameters (s(t), c(t)) by a choice of suitable orthogonality condi tion which only works for m 3. On the basis of the space-time estimates obtained in Section 2, we give the proof of Theorem 2.1.3. In Section 4, we construct an example to show that finite time singularities really occur for energy close to the harmonic map energy when m = 1, proving Theo rem 2.1.5. Throughout the paper, the letter C is used to denote a generic constant, the value of which may change from line to line.  2.2  Derived nonlinear heat equation  In this section we derive a nonlinear heat equation associated to the har monic map heat flow. We use the technique introduced in [6], obtaining an equation for the coordinates of the tangent vector field v,- — mJvRv with respect to a certain orthonormal frame. Under the m-equivariance assumption that the solution to equation (2.1.1) has the form u(x, t) = emG?v(r, t), v satisfies the evolution equation: Vt  =  (D +  — ---)(vT  —  JVRV) tm  where, recall, D is the covariant derivative, acting on vector fields tangent to 2 at v. Let e E TS 2 be a unit tangent vector field parallel transported along the curve v(., t) De=O. Then {e,J’-’e} is an orthonormal frame on TS . Let q(r,t) = qj(r,t) + 2 iq2(r, t) be the complex coordinates of the vector field ‘vj,. — !!2JvRv 2 TS in this basis m JVRV Jve. 2 = qie + q v. — r  29  2.2. Derived nonlinear heat equation qj. e + q JU e for convenience. Define  We sometimes write qe  JvRv=vie+v J 2 ve  2 + 1 v=v . izi  Now it is a straightforward matter to show that the complex function q(r, t) solves the following nonlinear heat equation with a non-local nonlin earity (1 —mV3) 2 m(v3)rq (2.2.1) qN(q) q = —  —  —  where N(q) =  f°°  Q  Q(r’)dr’,  :=  i Tm  (rT  + mP  [r  +  We will use equation (2.2.1) to obtain estimates on q. Given an m-equivariant map u(x,t) E with E(uo) = eme?v(r,t) would like to write the solution u with: v(r,t)  =  1 —my 3  —  e t)R(h(r/s(t)) +(r/s(t),t))  4irm  q])  we  <  (2.2.2)  where (r/s(t), t) is a perturbation. Using the explicit orthonormal basis of , 2 TS  3=  (  1  3 jh  and  =  \oJ  ( \  0 h 1  it is convenient to decompose the perturbation term gential and normal to S 2 at h(p): (p,t)  = z(p,t)3-{-  J into components tan  7 ( 2 z ( p,t)Jl3+ p,t)h  (2.2.3)  for p = r/s(t). This decomposition defines the complex-valued function z +iz Using v3(p,t) = 3 1 z . 2 (p,t), h ( p)+ we find (v )r = 3 which we substitute into (2.2.1) to obtain qt=—Hq—2m —  (p) 3 h  —  (r) 3 h  :=  q  2m 3 h 2 3 +m —2 2 m3q  (2.2.4) —  m(3)rq —  qN(q),  where H is the operator H  =  —  + V(r),  V(r)  =  1+m 2  —  2mh ( 3 r)  30  2.2. Derived nonlinear heat equation So q(r, t) satisfies a nonlinear heat equation with linear operator H. Now the difficulty comes from the singular potential V(r) which behaves like const./r 2 as r —* 0 and as r — oo (with different constants). To some extent, H is like the heat operator with an inverse-square potential which is studied in [21]. But their arguments only work for dimension n 3 because of the lack of Hardy inequality in dimension 2. In this chapter we need to obtain space-time estimates for e, the one-parameter semigroup generated by —H. We know that for the free heat operator the following inequalities hold ([10]):  lie $‘llLL < CIlIlL2, I  Jo  (2.2.5)  \ 3 e (t—s)t jfS)uS  where (r,p) is an “admissible pair”  fY  LL  S  ‘—‘  t  i.e., 1/r + 1/p = 1/2, (i,) is the conjugate exponent pair of another admissible pair (i, ), excluding the case r = = 2. But in a following Lemma, we prove that not only do estimates like (2.2.5) hold for the operator H, but the “endpoint” version (r = i = 2) also holds for radial functions and f. A preliminary lemma ensures that H is self adjoint.  Lemma 2.2.1 ([17] pp.161 R\{0} satisfying  )  —  Let V(r) be a continuous radial potential on  3 . V(r)+ Then  —L  °°(R\{0}). 0 + V(r) is essentially seif-adjoint on C  On the basis of Lemma 2.2.1, we have the following: Lemma 2.2.2 The operator H extends to a positive self-adjoint operator on a domain D(H) with C8°(R (R L ) , hence —H is the \{0}) C D(H) C 2 2  infinitesimal generator of a contraction semigroup {e_t}t>o on 2 (R L ) . Furthermore, D(H) C L°°(R ). 2 Proof. We first consider m  2. Since V(r) (1 2 /r by Lemma m) , 2.2.1, H is essentially self-adjoint on C8°(R \{•0}), and so its closure (which 2 we still denote by H) can be uniquely extended to a self-adjoint operator on a dense domain D(H) C 2 (1R L ) . When m = 1, the operator H is simplified as 4 4 Hr+ 2 r 2 1+r —  31  2.2. Derived nonlinear heat equation By the above argument H = extends to a seif-adjoint operator on + (R L ) 2 , and under the bounded perturbation it remains seif-adjoint by the Kato perturbation theory ([17]). So, H generates a semigroup e_tH on 2 (R L ) . The non-negativity of V immediately implies H 0, and so e_tH is a contraction semigroup on —  (R L ) 2 . The final part of the lemma, the L°° estimate, is more delicate. Let e C8°(R \{0}). We have 2  f  {lV + V1y1 } 2  =  (,H)  and since for a fixed disk DR centered at the origin  V(lxl) 2 lxI  V(lxl)  1 on DR, and  bounded on D,  (2.2.6)  we conclude +  lxi Now multiply Hço  =  —  + Vço by  ). 2 IlHyll  (2.2.7)  and integrate by  parts to obtain  C(ltii2  L2  /jxl  +  f2f  lxi  llHil  —  0 p27r  2 L  =  p00  J J 0  )drd8 2 +V  —  0  0  ( + /r  V 2 + (r  1  —  / —) ) 2 drd8 r 2  and so by (2.2.6) again, and (2.2.7),  Vli + lIlxl_3/2li2 lllxI” 2 Fix p e (1,4/3), and set q  llliLP(DR)  :=  llHllLP(DR)  2 C(Jlll,  +  ). 2 ilHll  (2.2.8)  (2,4). Using (2.2.8), we have  + llVIlLP(DR)  C(H + I 1x1 312 VIILQ(DR) Ii ixi col1L2(DR)) 312 <C(IlHllL2 + 11s011L2) which combined with  llsoilL2(D)  llHcoIlL2 + llVllL(D) ISOIIL2  C(liHiiL2 + IlOilL2)  and a Sobolev inequality, yields the required estimate  IkolILco <CIlçoilw2,P(D)flH2(Dc)  G(liHllL2 + li(,OllL2).  (2.2.9)  32  2.2. Derived nonlinear heat equation Remark 2.2.3 The Kato perturbation theory for seif-adjoint operators is  not applicable here, since V(r) is too singular at the origin. Next we establish some properties of the semigroup well-known semigroup eta.  e_tH  satisfied by the  Lemma 2.2.4 Let {e_tH}t>o be the semigroup generated by the operator H  (R L ) . Let 1 <a in 2  <  b  <  6 IIe_tHcDWL  ), 2 oo. For ço e Laçll  c  t_(h/a_h/b)  IIYIILa  for all t  >  0  (here e’ can be defined by density). = b = 2, the statement just follows from the fact that {e_tH}to is a contraction semigroup on 2 (R L ) . Now let \{0}) C D(H). 2 C°(R e_tp Then u(t) D(H). Thus Hu(t) = He_tHço = e_tHHy, and so IHu(t)IIL2 IIHsoI. Using (2.2.9), we find  Proof. If a  IIu(t) IL°°  2 + IIS2IIL2), C(IIHoIIL  C(JIH’u(t) 11L2 + IU(t)1IL2)  so there exists M> 0 such that sup, Iu(x,t)I <M. Now we want to apply the maximum principle in R . We first assume p 2 0, so that e_tFço> 0 et is positivity preserving (see a.e. for each t > 0, since the semigroup = e_to solves Since 246 in [17)). u(t) pp. Ut  =  —  V(x)u,  ° 0 C \ {0}), °(R we have Ut < Lu, which and V(x) > 0, given any 0 < o E 2 subsolution the means e ço is a 1 to heat equation. Since sup Iu(x, t) I M, it follows by the maximum principle that 0 < e_tIcp  eço  for all t> 0, and hence  ) 2 e_tHLb(  <  c  t_(h/a_1/b)  I!ILa(R2).  For general ço 0, we can rewrite p = where p+ = ma.x{, 0} 0, = max{—,0} 0 and reach the same conclusion since Ie_tI etII. The proof of the inequality for y L follows by the density of °{o}) 0 C \ 2 °(R in La. —  Recall that the potential is singular at the origin. Fortunately, precisely because of its inverse-square form and positivity, it yields “endpoint”-type 33  2.2. Derived nonlinear heat equation space-time estimates. Consider the mixed space-time Lebesgue norms: for an interval I C R+,  If IILL(R2xI)  :=  f (L 2  If(x,t)IPdx)dt.  Theorem 2.2.5 Let the exponent pairs (r,p) and (i) be “admissible” (i.e. 1/r + 1/p = 1/2), but exclude the case r = = 2. Then we have  lift IILL + t IIe  e_(t_8)Hf(s)dsIILrL < C(IiIIL2 + IifiIL’Ls’)  for all functions ,,f(.,t) on R . If, in addition, m > 2, and ,f(.,t) are 2 radial functions, then the estimate holds also in the “endpoint” case r = = 2, and we have weighted estimates Iie_tIiLL + Ii(e )TIiLL2 t  ft  e  22 t_s)Hf(s)ds11  +  li(f  e t f(5)d5)rIIL2L2 <CIifIILfL51.  Remark 2.2.6 it is easy to check that the above “endpoint” estimate fails for the free heat operator eta, even for radial functions (see [20] for the Schrodinger case). Proof. Let’s first prove the non-endpoint estimates. For the homogeneous estimate, we establish a more general result, following [10]:  Cçoja  iie_tFcoIiLL with + = , r> a> 1. For fixed p By Lemma 2.2.4 So F is of weak type (1,  ---),  e  (1, ], define F(t)  =  iIe_tSoiILp.  On the other hand,  Ie_tIiLrL  CIROIILP,  so F is of strong type (p, oo). By the Marcinkiewicz interpolation theorem, F is of strong type (a,r) with = — , r a>1, and  iIetFoIILLp  GRLa. 34  2.2. Derived nonlinear heat equation Now we turn to the nonhomogeneous estimate. For 1 have  f1t  f(  e_(t_8)Hf(s)dsiiL  ‘  p  oo, we  s)h/’h/P)iif(s)IiLwds,  so by the Hardy-Littlewood inequality,  t if  e_(t_8)Hf(s)dsipLrL  <CIIfIiLfLl  with + + + 1, provided 0 < 1/ + 1/r < 1. In particular if are admissible (excepting r = (r,p), = 2), we obtain the desired space-time estimate. Now we will prove the “endpoint” case (r,p) = = (2, oo) for radial functions, if m 2. Our method relies on the energy inequality. Let , f(s) be radial functions and u(x, t) = e_tHço. By the imbedding inequality  IiUiILoo <C(IiuTiIL2 + Iiu/rIIL2)  (2.2.10)  for radial functions in two-dimensions (see [10]), we have IiUiILL  + IIlILL  2 C(11ur11L2L  (2.2.11)  Since in two dimensions the Hardy inequality does not hold, we change variable u(r, t) 9 v(x, t) := e so. that we have IIUTIIL2L2 + IiIILL since  2 IVvi  =  2 + li2. irl  Now Vt  =  v(x,t)  —  (2.2.12)  solves the equation:  2 m  LV  CIIVvIIL2L2  —  3 2mh V  2 r  (2.2.13)  with initial data eiecp. Multiplying equation (2.2.13) by V, taking the real part and using m 3 > 0 (since m 2mh 2 2), then integrating over space and time, we arrive at —  IiViioLl + IiVvlIL  Iiv(0)II  = 2riI’JI2.  35  2.2. Derived nonlinear heat equation Note that using (2.2.11)-(2.2.12) gives us Ie_t IL1 CIIp!1L2 (which O 4 in any case is covered by the above argument). Similarly we get:  =  II ILLl+IIUrIILL  <CIIVVIIL2L2  Now let (i, 3) = (2,oo). If w(r,t) := f e_(t_8)Hf(s)ds, then w(r,O) 1 By the above arguments we get, for v = e w, 8  IIWIJL  0.  =  2 Ifvldxds 6IIvIILo + C(E)IIfII cff Li 2  CIIVvIIL —  IIWIILLo +  ()IIfIILLl  Taking e = 1/2, we obtain IWIIL2L Cf IIL2L1, the “endpoint” estimate we were seeking. We also get the weighted estimate, since  IIILL + IPWrIILL  IIVVL  ff  Ifvldxds  CIIVILrLPIIfIIL/L! As a direct result of this proof, we have: Corollary 2.2.7 Theorem 2i.5 also holds for e_tH where H is any opera tor of the form  1<a(r)<C. Remark (*). A theorem similar to Theorem 2.2.5 is obtained for the lin earized operator of Landau-Lifshitz equation. Let a = = 1. A Schrödinger heat type equation similar to equation (2.2.1) is obtained, where the linear part is (2.2.14) qt = (1 + i)( V(r))q. —  We can space-time estimates for the linear evolution operator out using maximum principle.  e_(1i)tH  with  Theorem 2.2.8 Let the exponent pairs (r,p) and (i,) be “admissible” (i.e. 1/r + 1/p = 1/2), including the case r = = 2. Then we have  lie_ti  LL  +  I  f  e_(t_s)  f(S)d5IILrLP  C( +  36  2.2. Derived nonlinear heat equation for m hold  2, and radial functions  ,  f(., t). Moreover the weighted estimates  IIe_t(1IILL + II(e_t(1+)rIlLL 1  i  f  e_(t_8)(1+f(s)dsIIL2L2 +  II(f  e_(t_8)  f(S)d5)rIL2L2  <CIIfIILIL1.  Proof. The goal is to estimate u(x, t) a solution of the linear inhomogeneous initial value problem w(x,O)=q(Ix)  ut+(1+i)Hu=f,  First we prove the basic L 2 estimate. Multiplying this equation by ü, taking the real part, and integrating in space and time, then we obtain the following estimate 1UIIL00L2 + IIVuIIL?L C(IIIIi + IIfIILL).  Then we can obtain the weighted space-time estimates. Since the hardy inequality is not true in dimension 2, by using the change of functions w(x,t)  =  then  u(x,t), 8 e U  IVWIHI+IUrI r and w(x, t) solves the equation Wt = where function u,  + (1 + i)(— + /)w  —h m m 2 3 2  . 8 fe  2. By embedding theorem, if radial  0 for m  IIUIIL0O  =  IIUrIIL + HIIL2  =  IIIIL  Therefore IIUIIL2 L°° + IIUrIL2L2 +  IIIILfL  CIIVWIIL2L2.  Similarly multiplying the equation for w by D, taking the real part, and integrating in space and time (using the positivity of V),  IIVwIILL  IIIIL2 + IwfILL. 37  2.3. Proof of the main theorem Then  IIUIL2L + IIUTIIL2L2 +  CI +eIIuIILL +C(e)IfIIL2Li.  IHJIL2L2  If e is small, we arrive at the endpoint estimate  C(IIIIL2  IIfIILLo + IIUrIILL + IIIILL  + IIfIILL ). 1  By Holder inequality and interpolation yields all the desired estimates.  2.3  Proof of the main theorem  In this section we prove Theorem 2.1.3 which gives the global well-posedness for the equivariant harmonic map heat-flow with near-harmonic energy when the degree m is at least four. This is done through the study of a coupled system of ODEs for the parameters s(t) and a(t), and a nonlinear heat-type PDE for the deviation of the solution from the harmonic maps. In particular, we will show that the length-scales s(t) of certain “nearby” harmonic maps, stay bounded away from zero, and, in fact, converge as t —* oo. For an m-equivariant solution u(x, t) E 2m of (2.1.1), with initial energy  E(uo)=4nm+5,  so<<l  the decomposition (2.2.2)-(2.2.3) of u into a harmonic map with time-varying parameters, and a controlled correction, is established in [13]: Lemma 2.3.1 /13J If m> 3 and 6 is sufficiently small, then for any map E m with (u) < 4nm + ö, there exists s > 0, a R, and a complex  ‘U  function z u(r, 8)  =  1 + iz, such that z  =  )R[(1 + 7(p))h(p) + zi(p)3 + z 8 e(m (p)J], 2  p  :=  r/s,  with z satisfying (z,hl)L2(pdp)  =  0  and  IzIIc Here X  :=  {z: [0,oo)  —*  <C((uo)  4’rrm).  (pdp), 2 C I z 1 E L co  2  —  zx.—  1  I  Jo  ,‘  2  (pdp)} with the norm 2 L  2 Z i)pup ,  p  38  2.3. Proof of the main theorem  is the natural space for z, corresponding to the energy space H’ for u(x). As above, for u e , we define v(r)  =  =:  e_mORu(x)  =  (1 + ‘y(p))h(p) +  (p)J 2 z1(p) + z  h(p) + (p).  The pointwise constraint lvi = 1 gives 1 shall prove) remains pointwise small,  1z1 + 2  =  , 2 171 Ciz1  —1o, 2 7= V1—izi  (1 +  7)2  and since (as we  7piCIZpZl.  Making the decomposition given by Lemma 2.3.1 for u(x, t), at each time t, yields the complex function z(p, t), and time-varying parameters s(t), a(t), which together give a full description of the solution map u(x, t). Since we will use the estimates of the previous section to estimate q(r, t) rather than z(p,t), we need to know that z can be controlled by q. This fol lows from the lemmas below. For convenience, we introduce spaces X, 2 p < cx), with the norm:  iIzIl so that X  =  100 p  :=  /  (izI +  p  . 2 X  Lemma 2.3.2 /13J Let 2 <p < oo. If (z,  h,)L2  lIzIlx  =  0 and m> 3, then  CIILOzIILP  IlIzI/p+ izi/p IiL2 2  CiILoz/pIiL2  where L 0 is the operator Lo:=(a+h ( 3 p))=h,(p)8_1 h,(p) p  Since, modulo nonlinear terms, (Loz)(p)  (2.3.1)  sq(sp), we also have:  Lemma 2.3.3 /13J Under the assumptions of Lemma 2.3.2, if  lizlix  << 1,  then Z Xp(pdp)  ,—  1—2/p  —  IIIIL2(pdp) + Il-11L2(pdp)  <  q  LP(rdr)  Cs  WrWL2(rdr)  39  2.3. Proof of the main theorem  So the original map u(x, t) can be fully described by the function z(p, t) and the parameters s(t) and (t), while z(p,t) can be controlled by q(r,t). So we are going to derive the equation for z(p, t) in order to estimate s(t) and c(t), and then use Lemma 2.3.3 and estimates on the equation for q to complete the proof of Theorem 2.1.3. Rewriting equation (2.1.1) in terms of the vector v(r, t) yields Vt  =  MrV +  2+ (IvrI  IRvI2)v,  Mr :  ã+  +  R2.  (2.3.2)  Inserting the decomposition v(r,t)  into (2.3.2),  =  et[h(p) + (p,t)J,  p  =  r/s(t)  we arrive at (Mp+Iap(h+)I2+IR(h+)I2)(h+). (2.3.3)  Now using the further decomposition of the perturbation into tangent and normal components, = z + z J+7 2 h, we find, after routine (though somewhat involved) computations, that the tangential components of (2.3.3) yield our desired equation for z: Zt 2 S  =  —Nz + (ims  —  (2.3.4)  6)hi + 1 2 s F+F 2  where N denotes the differential operator N := —8  —  +  -(1  —  2h)  =  (2.3.5)  0 LL  (here L5 is the adjoint of L 0 in L (pdp)). We will not write the nonlinear 2 terms F 1 and F 2 explicitly, but only give the necessary estimates (we are omitting many of these details since very similar calculations are presented in [13]): Co>OsuchthatIIzIIx<Co  ==  IF ( 1 p,t) + sâpz C(Is I + sI)Pz, 2 + z)(Jp2 1 (p,t)I C(h 2 1F + 2 /p 1z1 ) . From (2.3.4), we see that the linearized equation (setting now) for z(p, t) is = —Nz.  (2.3.6)  s(t)  1 for  (2.3.7) 40  2.3. Proof of the main theorem We would like to impose some orthogonality condition on z which ensures: (a) solutions of (2.3.7) decay; (b) certain norms of z(p, t) are controlled by norms of q(r, t). Since N is seif-adjoint in L (pdp), and from (2.3.5)- (2.3.1) 2 we see that ker(N) = span{hi}, it is natural to impose  (z, hl)L2 =  f  z(p)hj(p)pdp  0.  (2.3.8)  Lemma 2.3.3 then gives the desired control of z by q. Now we may explain the source of our restriction m 4. Firstly, in the energy space we have z X, and in general z L . But we have 2  (z,hl)L21  1 zxph  IIIIL2IPhlIL2  and so to make sense of the condition (2.3.8), we require z/p E L , which 2 leads to the restriction m 3. The further restriction m > 3 is needed for (seemingly) technical reasons in the second estimate of Lemma 2.3.2 (and hence also in Lemma 2.3.3). The next step is to estimate the parameter velocities: Lemma 2.3.4 If z satisfies (2.3.4),  IzIIx 2  J<G 2 saI+ls Proof.  <<1, and +  (z,hl)L2  0, then  2  P  2 L  Differentiating (2.3.8) with respect to t, and using (2.3.4), yields:  a—ims)IIhiII (s 2  =  ,F +F (h 1 ) 2  and by the estimates (2.3.6), we obtain  IsI +  I 2 5  ,F 1 j(h ) j) C((hi,Fi)I + 2  G(s + Is aI)Khi,pz+ IzW +Cfhi(hi + IzI)(IzpI 2 2+2 /p 1z1 ) CIjzx(Js +  s2I)  +  2+ C(IIz/pII  II2) 2 Iz/p (2.3.9)  where we used • 11z11L00  <CIIzIlx  (i.e. (2.2.10))  • I(hi,pz)I = p(8+ )(phi),z/p)I ing m 3)  CIIphlIIL2IIz/pIIL2  <CIIzIIx  (us  41  2.3. Proof of the main theorem  • 2 lip hliiLoo  <C (true for m  2).  Absorbing the first term on the r.h.s. Lemma 2.3.4.  of (2.3.9) completes the proof of  The next step is to estimate the function q(r, t). Without loss of gen erality, we will rescale the solution so that SO := s(O) = 1. Let q(r, t) be the corresponding complex-valued function derived from the map u(x, t) in Section 2.2. We use Lemma 2.3.3, together with Lemma 2.2.5, to prove the following estimate for q: Lemma 2.3.5 For cr > 0, set I := [0,u), Q := spacetime norm  iill  +  x  I and define the  + iITiiLL(Q)  If IlqoiIL2 = 6// is sufficiently small, we have  iiqii  C (iiqoliL2 +  lJs’iiLr([o,])ils ‘ilLr([o,u]) iiqIl + lilJ + ilil).  Proof. Note that  iiqll2(rdr)  = iIvr  —  —J RvilL2(d) = (E(u)  <(E(uo)  -  —  4irm)  4rm) =  can be taken small. We start by rewriting equation (2.2.4) in integral form:  q(t) = e_tHqo  +  f  e_(t_8)H(F(q(s)) +  —  (p) 3 h  q(s))ds  (2.3.10)  for 0 < t < u where F(q) : =  —  33 + m h 2 2m 2  —  m3q 2  —  )r 3 m(  q  —  qN(q) =: I +11+111. (2.3.11)  Due to Lemma 2.2.5 (including the endpoint case) we have  liqily  C(ilqoilL2  + 1 (r) 3 h  -  (p) 3 h IiLL  + ilF(q) 1IL4I3L4/3)•  (2.3.12)  42  2.3. Proof of the main theorem Note  I  ha(r)  —  (p) 3 h  CIIqI2I  IILL <  CIIqI2  (r) 3 h  (r/s) 3 h  —  r2  IILi’oIIs 1 Is  (2.3.13) —  lIILi’°  where the last estimate comes from expressing h (r)—ha(r/s) as the integral 3 of its derivative with respect to s. Next we use Lemma 2.3.3 to estimate IIF(q)IJ4/34/3(Q) term by term. Note that the estimate of F(q) = I + II + III has some overlap with the nonlinear estimates appearing in [9], and so we only give brief computations. Recall v3(r,t) = h (p,t) +3(p,t) and = z 3 1+ h 2 3 for p = r/s(t), hence h 7 2  34 C(s  IVIIL4’3  IqIILs +  PL  PL8  <C (IiqIi8 + IqIl (1+  qIIL4)  (‘+  IIHLf)).  So we obtain  II’11L4/3L4/3  C  (iiqiii  + IIILLf)  Next we estimate  :=  —  =  C  (IIqII + IqII).  II(m(3)T/r)qM4/34/3.  2 z 3 h 1 mh  [12  3+ +h  (2.3.14)  Compute  m7h]  Again using Lemma 3.3, we arrive at  III’HIL4/3L4/3  C(q  C IIqIILL 4 (1+  +  IIqI)  (2.3.15)  Finally, using the Hardy inequality  (L (L  ‘1 fd) XP_ldx)  i/p (f(Yf(u))PP_’dY)  for f 0, inthecasep=p=2, wefind  IIN(q)IILf and since  i-’  CQI[L2  4 + C(Iq  <1, we obtain  II”UL4I3L/3  IIqULLfHN(q)WLL C  +  ) 2 IIq/rII  (2.3.16)  (IIqII + IIqII). 43  24. Finite time blow up Combining (2.3.12)- (2.3.16), completes the proof of Lemma 2.3.5. Now we are ready to finish the proof of Theorem 2.1.3. Completion of the proof of Theorem 2.1.3. have =  From Lemma 2.3.1, we  e(me+a[(1 +7 (r/so))h(r/so) + (zo)i(r/so)+ (zo) 0 (r/so)J’3] 2  with (zo,hl)L2 = 0 and lizolix C5o. Let (r,t) u(sor,st). Then ü is also a solution to the heat flow equation (2.1.1) with initial data (r,0)  =  e(mO  1 o)R[(  (r))h(r) + (zo)i(r)3+ (zo) 0 +7 (r)J], 2  and time-dependent decomposition =  e(me(t))R[(1 + 7(r/s(t)))h(r/s(t)) + zi(r/s(t),t) + z (r/s(t),t)Jj 2  with s(t) E C([0,T);Rj, (t) E C([0,T);R), s(0) = 1, and c(0) = cEO. If 5 o is sufficiently small, the estimates of Lemma 2.3.4 and Lemma 2.3.5 together yield 118 sIlL + IIlIL Cllql G8, and in particular that s(t) Co > 0. Hence the solution extends to T = and as t —* 00, s(t) — s > 0, and a(t) —> a. Furthermore, for any pair (r,p)with+=,2roo,  IIV(u  —  em R 9 h8 (t  t))IILrLP  CS’IIZlILXp  Finally, undoing the rescaling u(r, t) Theorem 2.1.3.  2.4  =  CIIqIIr  CIIqy  Coo.  ü(r/so, t/s) completes the proof of  Finite time blow up  In this section we give the proof of Theorem 2.1.5 by constructing a 12 with near-harmonic energy. equivariant finite-time blow-up solution in 1R The proof is a variant of that of [5], adapted to the plane R . 2 subclass of 1-equivariarit solutions One special is given by  u(., t) (r, 8)  —  (cos 8 sin (r, t), sin 8 sin q(r, t), cos (r, t)),  (2.4.1)  44  2.4. Finite time blow up where (r, 8) are the polar coordinates on the plane, and /(r, t) is the angle with the ‘u -axis. If u solves the harmonic map heat flow, then, as is easily 3 checked, (r, t) satisfies 1 =  sin2ql 2 2r  O<r<oo, t>O,  (2.4.2)  O(r).  The energy of u can be written in terms of q: E(u(t))  =  e()  + sm2 )rdr.  :=  In order to have a degree-i solution with finite energy, we impose the bound ary conditions lim (r, t) = ir. (O, t) = 0, (2.4.3) r—oo  We take C’ initial data: tlo  e C’([O,oo)),  limo(r)  r—*O  =  0,  lim o(r)  r—oo  =  r.  (2.4.4)  Local existence of a classical solution is straightforward: there is T> 0 such that (2.4.2)—(2.4.4) admits a unique solution (r,t)  ([0,oo))) flC°°((0,oo) x (0,T)) 1 e C([0,Tj;C  (one way to see this is to solve the full harmonic map heat-flow with the initial data corresponding to co, locally in time in classical spaces, use the fact that the form (2.4.1) is preserved ([4]), and then recover (r, t)). Next we establish an extension to R 2 of the comparison principle of [4] for the unit disk (where they observed that although equation (2.4.2) is singular at r = 0, the maximum principle may still be applied).  ((0,oo) x (0,T)) be 2 q1,12 E BC([0,oo) x [0,T]) fl C solutions of the problem (2.4.2)— (2.4.4) with initial data If ç5j then bi(r,t) (r,t), (r,t) [0,oo) x [0,T]. 2 b  Lemma 2.4.1 Let  Proof.  Let o  :=  —  i.  Then =  rr  satisfies +  + p(r, t),  (2.4.5)  45  2.4. Finite time blow up where p(r,t)  sin 22 sin 2r 2 @2—&) —  := —  =  1 ——cos(h r  2 sin(  —  +2)  Fix T 1 e (0,T). Since 1 (0,t) = 2 (0,t) = 0, there exists S > 0 such that , c E [—7r/8,lr/8] in (0,6) x (0,Tij (continuity of and 2), and hence p < 0 on this set. Therefore, there is K 0 such that p K on (0,oo) x (0,7’i). Setting v(r,t) := e_(4)tp(r,t), (5.1.5) yields Vt  And setting find w  =  w  we(r,  t)  +  +  =  := v(r,  (p(r,  + (p(r,t)  +  Vrr  t) + f(T 2  t)  —  K  —  —  K  —  1)v.  — t)_1eT /(4(T2_t)) 2  1)(we  —  2 f(T  for some T 2  >  — t)_1eT /(4(T2_t))). 2  , we 1 T (2.4.6)  Now suppose inf[ ) [0, 0 6 <0. By the boundary conditions and bound ’i] w 7 eness of , this implies wE(r,t) = inf[O)[OT ] w < 0 for some r > 0, 1 t (0,Tij. Hence w(r,t) 0, w(r,t) = 0, and wr(r,t) 0. This contra wE —* dicts (2.4.6). So we have 0, and sending e 0, we recover 0 in 1 <T was arbitrary, we are done. ]. Since T 1 [0, oo) x [0, T The proof of Theorem 2.1.5 is a combination of methods from [5] (subsolution construction and comparison principle) and [2] (use of comparison principle on a sub domain). The following Lemma is proved in [5]: Lemma 2.4.2 [5] Let a, A , 0  t  R and  (0, 1). Let A(t) be the solution  ii  of  If TA  :=  sup{t  f(r,t) :=  >  arccos  A’  =  ’, 1 aA  0, A(t)  >  0} (the “blow-up time of  2 A(t) (A(t)2  —  t> 0,  /,2  2 r  )+arccos 2  2  A(0)  —  =  . 0 A  f”),  then the function  r2(1+v)’\ 2(1+v))  (r,t)  e (0,1)x(0,TA)  the following properties: (i) f E C°°([0,11 x (ii) limr_o f(r, t) = 0 for t [0, TA), 1imro f(r, TA) = (iii) there exists i > 0 such that for every t > i we can find such that 1 sin2f (r,t) (0,1) x (0,TA). ft TT + f 2 2r for all a satisfies  ii)  —  46  2.4. Finite time blow up Proof of Theorem 2.1.5. Given any small 6> 0, let the initial data to equation (2.4.2) be of the form o(r) =  arccos  /s_r2  + r)  +  qlo(r)  3  where (ir) is a non-negative C°° function supported in [, j. We can ensure It < (1) by choosing so = s(6) small enough (depending on 6) so that arccos  ()  6, and choosing (1) > 10 (thus  can be chosen independent of 6). It is then easy to check that e(qSo) 4ir+ 062. Moreover, 0 can be chosen such that e(O)(7/8<r<) < 32/10, which means C(r) and s that the energy is concentrated in a neighborhood of the origin. Now let (r, t) be the unique classical solution with initial data oQr), and let T be its maximal existence time. Since (1) > t, by continuity there exists T* (0,T) and y > it such that (1,t) > y for 0 < t < T. It  —  Let h(r) arccos  =  arccos  ()  It  /  —  2_r2(l+v)\  2+2(1+v)).  <7—  Choose  t  sufficiently large to ensure h(1)  =  Let f(r,t) be the function from Lemma 2.4.2, and choose cr small enough so that property (iii) if Lemma 2.4.2 holds. And finally, choose )o small enough so that TA <T*, and so that the energy of f(r, 0) on [0, 1] is <4it + 062. So f(r, t) is a subsolution for the problem it.  1 sin2g gt=grr+—gr— 2 2r g(0,t) = 0 g(1,t)  0<r<1 (2.4.7)  =‘.  Now select a smooth, bounded function o(r) satisfying (O) 1’o(r)o(r),  =  0, and  r[0,oo),  o(r) > f(r,0),  r  e  [0,1],  e(o) <4it+C6 2  (the last property can be achieved since it holds for both o(r) and f(r, 0), the latter on [0, 1]). Let T 0 be the maximal existence time of the classical solution (r,t) with initial data co(r). By the comparison principle Lemma 4.1, (r, t) > /(r, t) for (r, t) E (0, oo) x (0, min{To, T}), and in particular, (1,t) > (1,t) > for 0 < t < T*. So (r,t) is a supersolution for prob lem (2.4.7), while f(r, t) is a subsolution. So by the maximum principle for this problem on the disk ([4]), for 0 < r < 1, we have (r, t) f(r, t). Hence (r, t) blows up (in the C’ sense) at or before time TA. 47  Bibliography [1] J.B. BERG, J. HuLsH0F, & J.R.KING, Formal asymptotics of Bubbling in the harmonic map heat flow, SIAM J. APP1. MATH. Vol. 63, No. 5, pp. 1682—1717.  12] M.  BERTsCH, R. D. PAsso & R. V. D. HouT, Nonuniqueness for the Heat Flow of Harmonic Maps on the Disk, Arch. Rational Mech. Anal. 161(2002) 93-112.  [3] Y. CHEN, W.-Y. DING, Blow up and global existence for the heat flows  of harmonic maps, Invent. Math  99. 567—578 (1990).  [4] K.-C. CHANG, W.-Y. DING, A result on the global existence for heat flows of harmonic maps from D 2 into S , Nemantics, J.-M. Corori et al. 2 de., Kluwer Academic Publishers, 1990, 37 48. —  [5] K.-C. CHANG, W.Y. DING, & R. YE, Finite-time blow-up of the heat flow of harmonic maps from surfaces, J. Differential Geom. 36 (1992), rio. 2, 507—515. [6] N.-H. CHANG, J. SHATAH, & K. UHLENBEcK, Schrödinger maps, Comm. Pure AppI. Math. 53 (2000), no. 5, 590—602. [7] J.-M. CoRoN, J.-M. GHIDAGLIA Explosion en temps fini pour le flot des applications harmonique, C.R. Acad. Sci Paris Ser. I 308 (1989) 339-344.  [8] J. BELLS, J.H. SAMPSON, Harmonic mappings of Riemannian mani folds, Am. J. Math 86 (1964), 109-160.  [9] A. 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ZuAzUA, The Hardy Inequality and the Asymptotic Behavior of the Heat Equation with an Inverse-Square Potential, Journal of Functional Analysis, 173, 103—153(2000).  49  Chapter 3  Well-posedness and scattering for a model equation for Schrödinger maps  3.1  Introduction and main results  Some geometric evolution equations from R° x R to harmonic map heat flow (HMHF)  g2  (the unit 2-sphere):  Utr=PLU ([1, 2, 5—7, 10]), Schrödinger maps (SM)  JPIU ([4, 8, 9]) and wave maps (WM) PU  =  PLU  ([12, 13]) in critical dimension d = 2 have been studied (here P denotes the orthogonal projection from R 3 onto the tangent plane of g2 and J = U x). For HMHF and WM equations, much effort has been devoted to a subclass of m-equivariant maps U(x, t) : R 2 x R —* S 2 U(., t) : (r, 8)  —*  (cosm8 sinu(r, t), sin mO sinu(r, t), cos u(r, t))  (3.1.1)  A version of this chapter will be submitted for publication, Guan, M. Well-posedness and scattering for a model equation for Schrodinger maps.  50  3.1. Introduction and main results where u(r, t) stands for the longitudinal angle. Then the map equations are reduced to scalar PDEs for u as ut(orutt)  =  u— -sin2u.  (3.1.2)  However this subclass is not preserved by Schrodinger maps which makes the construction of singular solutions for Schrödinger flow much harder. This is our motivation to study for this model equation: 2 sin(21u1) m  2 jUt+U  21u1  u=O,  u(x,O)=uo(x).  (3.1.3)  x IR — C is radial and m is a nonzero integer. Without loss where u(x, t) : of generality, we assume in > 0 in this context. Equation (3.1.3) is a non linear Schrodinger equation with smooth, but spatially varying nonlinearity. It is invariant under “gauge” rotation: u —* eiau, c E R. This equation is a natural Schrödinger analogue of equation (3.1.2). We observe that an energy is formally preserved by equation (3.1.3): E(u(t)) =0, where the energy is defined by  poo  E(u)  =  7t  J  2+ (uJ  2 !_  2 u)rdr. sin  Finite energy solutions require sin Iuo(x)I  = 0 both at lxi Therefore this yields the following boundary conditions  uo  :  {uo: {0,oo) —* C,E(uo) <cc, urn uo(x)=kiir, lirn uo(x)=k r, 2  =  0 and  xl  =  cc.  =  IxI—O  IxI—oo  2 , 1 k e Z}. k  (3.14)  Equation (3.1.3) is energy critical in space dimension 2 in the sense that the energy is invariant under the scaling since =  Throughout this chapter, we are interested in the global existence and long time behavior of the solutions.  51  3.1. Introduction and main results Rewriting the energy E(n), we find out for solutions in the class u(O) 0, uo(oo) = ir, E(u) has a minimal lower bound 4irm E(u) =  f  =ir f Jo  00  2 m  2+ (Iui u,.  —  —-  =  2 Iuhrdr sin  msinu 2 1°°sinluj üurdr I rdr + 2irm Re .‘oI r u ui  —  where 2m Ref o  sin  Iui  üurdr  =  mf  =  —27rm  lul (ittI2)dT iul dr  o  J  dr  —  cos(iuhdr  =  4irm.  Therefore  E(u) > 4irm. The minimal energy is attainable when —  Ur  rnsiniul r ui  =  0.  (3.1.5)  The solution to equation (3.1.5) is obtained by the 2-parameter family of harmonic maps: :=  {eQ(r/s)iQ(r)  =  2arctan(r ) m ,o E IR,s > 0}.  These harmonic maps are the stationary solutions of (3.1.3). Thus there are two natural questions: whether we can draw the same conclusion as in [9], the Schrödinger flow case, i.e. harmonic maps are stable under this equation for large m; and whether we can construct singular solutions when m = 1? Moreover equation (3.1.3) possesses constant solutions kir(k e Z) with finite energy. It is also natural to consider the stability of these static solutions. Compared to equation (3.1.2), the analysis of equation (3.1.3) is more complicated. It turns out to be related to the Gross-Pitaveskii equation(NLS with nonzero boundary conditions, see [11]). Let u(r, t) = S(r)+(r,t), where S(r) can be taken as either kir or Q(r), equation (3.1.3) becomes =  2 m 2 sin 2S(r) m 2 m —-cos2S(r)Re,7+i---Im+ N(i), 2S(r) iio(r), ?Jo(0) = ?70(oC) = 0.  —77+  i(r,0)  =  (3.1.6)  52  3.1. Introduction and main results where N(ij) represents quadratic and higher order terms of ii N(ii)  =  sin2lii±SI  —  (ii + S)  sin2S —  cos 2S Re ii  sin2S  —  Tm ii.  The convenient way to study equation (3.1.6) is to write the linear operator as a matrix operator acting on (Re ii Tm )T: =  Lîj  + nonlinear terms  (3.1.7)  where  (-  L=  (3.1.8)  ‘h-)  2 m sin2S 2 m L=-+---cos2S, L=-+-r r 2S The operators L+, L_ are seif-adjoint with continuous spectrum [0, oo). For the non-seif-adjoint operator L, the imaginary axis is the continuous spec trum. Dispersive estimates for this kind of operator were obtained under various decay assumptions on the potential and the assumption that zero is neither an eigenvalue nor a resonance of L. But unfortunately these as sumptions do not hold for L if S = Q(r). Since on one hand Q(r/s) is the stationary solution to equation (3.1.3), then  -  (irs  -  !sin2Q(r/s)) is=i  =  0.  This yields LsinQ=0. Thus L has sin Q as the unique ground state if m 2(sin Q e L ) and as a 2 resonance if m = 1(I sin QIIL2 = oo). On the other hand, for the stationary solution eQ(r),  gives a resonance  (er  -  sin 2Q(r)e) =0  Q for L_, L_Q  2 m  =  -Q+ —sin2Q 2 2r  =  0  since Q 0 L 2 for m 1. So it is unknown at this point how to get the dispersive estimates for the evolution operator et. 53  3.1. Introduction and main results Because of the difficulties described above, in this chapter we will only consider finding local in time solutions with finite energy and global solutions with small energy. When E(uo) is small, we must have that uo(O) = uo(oo) = 0 and then zero is the static solution. Rewriting equation (3.1.3), we have m sin I2u 2 —1)u=0. r 12u1  2 m  (3.1.9)  This equation resembles cubic NLS with inverse square potential. boundedness (smallness) of  The  m  )rdr 2 I (ILr + —-juI r Jo implies boundedness (smallness) of E(u). Hence the natural space for u is u e H’ and e L . This is different from usual cubic NLS equations where 2 or u E H’. u E Let us introduce some Banach spaces. Define X := {u: [0, cc) —* Cpu,- E (rdr), 2 L (rdr)}, with the norm 2 L  IIII :=  f°°  {1ur12 + m2} rdr.  :=  For 2 < p < cc, also define X’ LP(rdr)}, with the norm  IIuIIcp :=  [  {u  {ir  [0, cc)  —*  CIUr  E L?(rdr),  E  + mJ-} rdr.  So X = X . For radial complex-valued function u(r), v(r), define the follow 2 ing inner product, (u, v)L2 =  and (u,V)x  For an interval I  =  f  f  ãvrdr  (ür(r)vr(r) +  ü(r)v(r)) rdr.  c R, define space-time norms:  IUI(I,x)  f (f(lurIP+mP)rdr)dt.  In this chapter, we will use the following Strichartz estimate and Sobolev type embedding theorem. 54  3.1. Introduction and main results Lemma 3.1.1 (Strichartz estimate /3]) In space dimension  a pair of exponents (q,r) is admissible if + = have (i) < CilllL2 lie  r  ,  <  we say that oo. Then we  (ii)  f where  ‘,  <Cllgil/!  ei(t_T)g(.,r)dr  0  t  LL  i are the conjugate exponents of admissible pairs  Lemma 3.1.2 (Sobolev-type Embedding /8]) Suppose  and  fr,  X  f/r E L ), lim f(r) R 2  E H(R ) is radial 2  0. Then  C (ffrl2 +  If il(R2)  f  (, ix).  m2rdr).  One can find this Lemma in [8]. Here we provide a very elementary proof. For any radial function f e 2 °°(R 0 C ) , 00  (r) 2 f  =  _2f  00  f(s)f’(s)ds  =  _2f  -f’(s)sds.  Thus by Holder inequality  If 112)  C  (f(ifrl2  + m24)rdr).  The lemma follows by approximating f E X by functions in C fl X. Now we are in a position to state our theorems. Theorem 3.1.3 (Local weilposedness) Let m  1. Let either S = 0 or S = Q(r) = 2arctan(rm). For any jj E X, there exists maximal time interval I = (Tmin, Tmax) containing 0, such that the integral equation of (3.1.6) has a unique solution in the class satisfying r(r, t)  =  u(r, t)  —  S(r)  e  C(I; X)  n L(I; X ) T  where2<q<oo and+=. Moreover, the function u(r,t) = S(r) + ri(r,t) solves equation (3.1.3) in a distribution sense and u(t) e C(I, ) and E(u(t)) = E(uo) for all time tel. 55  3.2. Local weilposedness We know from Lemma 3.2.1 in the next section, E(u) is finite if E X. The energy is not necessarily near the harmonic map energy. For maps with energy close to the harmonic map energy E(u) = 4irm + 62,0 < 6 << 1, the local existence can be extended to global solutions if we have a space-time estimate for the linear operator £. We hope to develop the global existence theory in forthcoming research. Theorem 3.1.4 (Global weliposedness and scattering) For m there exists Eo > 0 such that when uo e , k 1 = k 2 = 0 and  Iluo lx  1,  Eo,  then the solution u to equation (3.1.3) given in Theorem 3.1.3 is defined for all time u E C(R;X)flL”(IR;X ). T Furthermore, there exist unique functions u+, u E X such that  Iie_t_u(t) uIix —  0,  as  t  This result is consistent with Schrodinger flow in [4] (small energy implies global weilposeness). Since u(t) is radial, the operator acting on + = u 6 eim (r,t). u(t) is like acting on function of the form v(x,t) It suffices to prove that v is global and scatters in X. Notation. In this chapter we use the notation A B whenever there exists some constant C > 0 so that A GB. Similarly, we use A B if A 5 B A. A << B means that for some constant c> 0, which may be choosen arbitrarily small, A cB. The letter C is used to denote a generic constant unless specified, the value of which may change from line to line. The proofs of Theorem 3.1.3 and Theorem 3.1.4 are given in section 3.2 and section 3.3 respectively. —  -  —  3.2  Local wellposedness  In this section, we aim to get local in time solutions for ij equation. Let ). Substitution u = S + i into equation m S = 0 or S = Q = 2arctan(r (3.1.3) yields 2 m 2 sin 2S m 2 m Imi+ N 2 (ij), it=—+-cos2SReij+’t2S (r,0) = m = uo S(r) .  (3.2.1)  —  56  3.2. Local weliposedness where  N()  =  sin2S  ) 775 sin2frl±SI(  are nonlinear terms. Let  +  =  ,  —cos2SRe—i I 2 m  and  sin 21S + f(171,u2)=  2IS+uj  ui (S+u), f  then using Taylor expansion for the function  IN(ui)I  u2),  we have  (Ifxx(1,2)7??I + Ify(1,2)’I + Ifxy(1,2)u717?2I)  2 + Iu1 I 7 C(Su ). 5 where i is between 0 and u, 2 is between 0 and u. It is worth remarking that on the right hand, the role of S(r) is very important since we need S(r)/r to be bounded (seen the reason from the estimate (3.2.13)). Because of this, for the constant solution S(r) = r, even the local existence of the solutions to equation (3.1.6) is hard to obtain. In order to prove local exis tence, we need to have dispersive estimates for the linear operator near the origin. As r —* 0, £ is basically like —/. + - since hmcos2S(r)  1,  =  r—O  lim  r—*O  sin 2S(r) 2S(r)  =  1.  We treat the operator —L + ! as the linear operator and the differences !(cos 25— 1) — 1) as the perturbations. Then the natural space for u is X. In fact, u X implies E(u) is finite in the following lemma. Lemma 3.2.1 Suppose  o  and u(r,t)  Proof.  =  If S =  2 m 2 )rdr < co,i (0,t) 7 0 (hr! 2 +--H f °°  =  =  u(oo,t)  5(r) + u(r,t), then u E D. =  Q,  rewriting the energy as roo  E(u)  ‘1I,2  4irm + I  m2  .2 (IuI 2 + 2QrReuir + —-(sm IQ + iI r  —  .2 sin Q))rdr.  Using integration by parts  f  2QrRerkrdr  =  _f  co  2  -sin2QReurdr.  (3.2.2)  57  =  3.2. Local weilposedness For the difference of the squares, writing the first few terms in the Taylor series for sin 2 (Q + ) , we get 2 n  IQ +  where R()I  —  . 3 G  E(u) < 4irm+f  2Q sin  sin 2Q Re ?7 + cos 2Q(Re )2 sin2Q (Imp) 2 +R(?7). + 2Q =  (3.23)  Using (3.2.2), (3.2.3), the energy can be estimated as  (177r12+- cos 2Q1 Re?712+- sm2Q  ImI2+CII3)rdr.  To show E(u) is finite, it suffices to prove that the integral on the right hand is finite. Since II1IIx < oo, we claim that f(Iii2 +  cos2Q)Re7)l2 +  51IImj2)rdr <  The claim is true since cos2Q T  (  and sin2Q 2 m  2 r From Lemma 3.1.2,  2Q  2 —,-  f —  HI L,co <CIlx,  IIiiII  asrO asr—÷oo  asr—*O as r —* oo we have  CIIIILIIII  C.  Therefore E(u) <4rm +  C(IIII3 + IIJI3)  <°°  and u satisfies the boundary conditions uo(O) = O,uo(cx) = ‘r. If S = 0, then u = ?7 and u has zero boundary conditions. Using sin Cj, we have E(n)  =  E()  =  f  2+ (IrI  )rdr < 2 1 sinI  The proof is complete. 58  3.2. Local weilposedness Now we are ready to prove Theorem 3.1.3. The operator + is basically conjugated by eImO when acting on radial functions. By changing of variable from to with (x,t) = &meq(r,t), we get V’ = 11. Thus i e X implies IrIP+!cIIP,2 <p < oo and ) and 2 E I’(R —  —  L(R). For convenience, let us define two notations :  := {v  —  CIVv  e L, v/ri  } 2 L  (3.2.4)  and (P  =  {v :  CIVv E L”,  v/ri  E L,2 <p < oo}.  (3.2.5)  Proof of local well-posedness. Let S = Q(r) in equation (3.1.6). The proof of S = 0 is similar. Since = eimSri, then solves the equation = —  -(cos2Q  +  —N(rl)eim8, + 1  o  —  =  1)(Re)eme +  —  1)(Im)eime  e”°r1o.  (3.2.6) We use this equation to construct a contraction mapping. By Duhamel’s formula, it is enough to find solutions to the integral equation (t)  =  eto + i  f  ei(t_F((s))ds  (3.2.7)  where F(i)  =  -(cos 2Q —1) Re eimO +  .m (S1n2Q  —1)  jim8  +  N()em6.  Define the solution map by M()(x,t)  =  etto +  if  ei(t_5F((s))ds  We want to find fixed point of the map M in the set D  {(x, t)  =  (r, t)eimoi  L°(I; ) fl L(I; 4);  iIiIL (3.2.8)  59  3.2. Local weilposedness for 6 > 0 to be specified later. Let I = (—T,T) that for some 6> , 1 L satisfies E H  IR with 0  E  e I.  Suppose  <6.  (3.2.9)  26 + CIIVFIIL4/3(IL4/3)  (3.2.10)  Then <  IiM()IILr(J.)flL(I,4)  where we have used Strichartz estimate Lemma 3.1.1 and IIIIL CII VIILP. Then we estimate IIVFIIL4/3L4/3 term by term. Let f (r) = (cos2Q— 1 (r) 2 1), f  — m sin2Q — — —-(--—  t  S  1). Both functions are smooth. Since  IV(fi(r)Reeimo)I CIfiIIVI+IfirUD, By using of Holder inequality in space and time, we have  HV(fi Ree imO’ I  <CT1/2(IIVIIL4L4 + =  II-IILL4) t T  (3.2.11)  CT/IIIIL4)4.  Similarly we get  IIV(f I 2 me mU )“L’L’  <  CT” I 2 lIIL44.  (3.2.12)  It is left to estimate IIVN(i)II L4/3 L 4/3. Recall that IN(7l)I ) 5 + 2 <C(QII II 5 then  IVI  +  I()rI  +  so that  3 IIVN()IIL/aL/  4 C(T”  II44  +  III44)  (3.2.13)  where we have used and  r  r  The combinations of (3.2.11), (3.2.12), (3.2.13) and (3.2.10) give  lIM()lILnL <26+ 2 C(T” l IL4 4  S  +  44 I 4 T’ III  +  3  IIIIL44) t S  <2ö+C(Ti6+T62+63) <36 60  3.2. Local weilposedness  if 6 and Tare small enough such that C(Ti+Ti6+62) 1/2. Then choosing the time interval I sufficiently small, we can ensure equation (3.2.9). Hence M maps to itself. Then we need to prove the contraction under the metric ) 2 , 1 d(  =  IIi  —  2IILooflL44.  We will use the two inequalities  II2  I 2 I2I  —  I+ 1 C(I  I2I)I1  —  2I  and  IIiI2i  —  2I 2 I2I  2 + I22)I1 j 1 C([  —  2I.  Let T be small enough. Then by Strichartz estimates  ),Jv1( 1 d(.A4( ) 2 )  ) 1 CV(F(  ))IJL 2 F( / L 3 / 4 a  —  —  V2IIL4L4  (IIV + 4 +T’ I 1  -  +  II + IIVlII 1 (IIV 2 )LLfIIVl 2  <  12 + T1/46 + C(T’  —  IIL4L4 2 V  IL4L 2 V ) 4  if 6, T are chosen sufficiently small. By the fixed point theorem, we get a solution on (—T, T). The regularity property of the solution follows from Strichartz estimates. It remains to establish the blowup alternative. We show the blow up alternative by contradiction. Suppose Tmax < co and III’L4((oT )4) < Let 0 < t < t + T < Tmax. It follows that eitV(r)  =  V(t +  Strichartz estimate and \ e it/,f çiT  r)  111,4  —  i  ei(tt’)VF((r + t’))dt’.  IlVIILf yield r  ((O,Tmaxt),X L ) 4  \ e itLr7,-f vc-r)  ((O,Tmaxt),L L ) 4  <2PlVIlL4((tTmax)L4) + +  T  +  IIIl4((t,Tmax),L4)) IIIIL((t,Tmax),L)  lIViIL4((tTmax)L4)  61  3.3.  Global small solutions and scattering states  Therefore for t sufficiently close to Tmax(r  —>  + T’/ IIVIIL4((t,Tmax),L4) + 2  0) such that  IVII4((t,Tmax),L4))  and  o  HIL4((t,Tmax),L4) 7 Ii we obtain ye itL.,-i çi )  X7  By the existence theorem, tion. This shows  ((O,Tmax_t),L L ) 4  can be extended past Tmax which is a contrac “‘RL4((O,Tmax),4) = 00.  The local well-posedness theory for is a direct result of(r, t) = e_tmO(x, t). The energy E(u(t))(t e I) is conserved since E(u(t)) = 4m +  f  (ITI2  + 2QrRer +  (sin2  I(Q + )I  —  2 Q))rdr. sin  This integral is a conserved quantity which follows formally from multiplying equation (3.2.1) by integrating over R , and taking the real part. This can 2 be rigorously justified following along the line of [3]. The proof is complete.  3.3  Global small solutions and scattering states  In this section we prove Theorem 3.1.4. We first show that for small energy solutions, the local solutions and be extended to global solutions in time, then construct the scattering states and the wave operators. Recall equation (3.1.9) 2 m  2 sin m  12u1  I2uF There exists  E  such that if  1G(u)l  mlix =  EU,  si2 I 2u I  —1)u=O,  then u  -  <  u(x,O)=uo(x).  , E(u) 0 CE  <  0 and CE  ii <Cu.  We show that the corresponding maximal solution given by Theorem 3.1.3 is global in time, i.e. Tmin = Tmax = DO.  62  3.3. Global small solutions and scattering states Let v(x, t)  =  eimGu(r, t), then v solves the equation ivt + v +  G(u)v  =  0,  vo(x)  =  8 uoeim  (3.3.1)  By Duhamel’s formula,  v(t)  =  etvo + i  L  e(t_8)G(u)vds,  t  I.  For 0 <t < Tmax, we define  g(t)  =  IVIIL((ot)) +  IIVIL4((o,t),x4)  It follows from Strichartz estimates 2  CIIVvOIIL2 + 3 L/ IIVG(u)vIIL/  g(t)  <CIvoIIx Cilvolix If  EO  +  I  IIVVIIL4L L IIIL ILL4 + 4  + Cg(t) . 3  is sufficiently small such that E < 1, 3 (2C)  then  2CIIvo lix for  g(t) Letting t  all  0  <  t < Tmax.  Tmax, we obtain in particular that  IIVliL4((O,Tmax),.4)  <00,  so that Tmax = co by the blow up alternative. This implies g(t) is bounded as t —*00 and v e L((0,oo),Xr). Next we construct the scattering states. Let w(t) = e_itIv(t), we have  w(t) = vo + Therefore for 0 < t <  ei8G(u(s))v(s)ds.  T,  =  w(t)  f  —  w(r)  if  eG(u(s))v(s)ds.  63  3.3. Global small solutions and scattering states If follows from  Hw(t)  -  is unitary  w(r)II  f IeIv(s)I2v(s)IIids I  II4((t,r);L4) IIHL4((t,r);L4) +  I II4((t,r);L4)  By the global existence for v in LX , 4  IIw(t)  —  w(r)II  as  0  —  r,t  —*  00.  t  —*  oo.  Therefore there exists v+ E X such that  Ie_ttv(t)  vII  —  —*  as  0  One can show as well that there exists v  IIetv(t)  —  vIIi  E X such that  0  —  as  t  —  —oo.  We now construct the wave operators for v, which completes the proof of Theorem 3.1.4. Lemma 3.3.1 (1) For every v+ E X, there exists a unique that the solution v e C(R, X) of equation (3.3.1) satisfies  IIetv(t)  —  vIl.  —*  0  t  as  E X such  00  (2) For every v e X, there exists a unique L’ e X such that the solution V E C(R, X) of equation (3.3.1) satisfies  IIe_itv(t) vIL —  —*  0  as  t  —>  —00.  Proof. We only prove (1), the proof of (2) is similar. Consider T> 0, by  Duhamel’s formula on [T, oo), v satisfies the equation v(t)  =  etv+  —  if  et8G(u(s))v(s)ds  (3.3.2)  for t > T. Now the idea is to solve equation (3.3.2) by the fixed-point theorem. Taking admissible pair (q, r) = (4,4) and applying Strichartz estimate, we get <r + e itL+ V — V X 4 L Let  ‘T  =  [T, oc) and =  e  V  64  Global small solutions and scattering states  3.3. So CT  —*  0 as T D  —*  cc. Define the set  {v E 4 (IT,X flL°° (IT,X) : IIVIIL4(IT4) L )  =  2C}  and T(v) by T(v)(t)  =  if  2  00  ei(t_TG(u(s))v(s)ds.  Then IIT(v)IIL(IT) + IIT()IIL4(IT,5C4) C(2C.  C(IIv/rII4(ITL4) + IVv4(JT,L4)) We see if T is sufficiently large, such that C(2CT) 3  CT, then  IIT(v)IL(JT) + IT(v)IL4(IT,4)  CT.  Thus the solution map M defined by  M(v)(t)  =  ettv+ + T(v)(t)  t > T  for  maps D to itself if T is large enough. We can easily verify that  ) 2 d(Mvi,Mv So M has a fixed point v = v(T) e X, and then  v(t + T)  =  ), 2 d(vi,v  V1,V2  E D.  e D satisfying equation (3.3.2) on [T, cc). Let  ft ei(t_G(u(s  eit(T) +  + T))v(s + T)ds.  Therefore v is the solution of the equation (3.3.1) with v(T) global existence solution of equation (3.3.1), we know v(0) defined and e_2tv(t) Since v  —  =  if  =  .  By the  e X is well  e18G(u(s))v(s)ds.  D, then  je_itv(t) therefore, v(0)  =  —  vII 3 00 ((t, IIVII 4 )  —*  0  as  t  —,  cc.  satisfies the conclusion of the lemma. 65  3.3. Global small solutions and scattering states Finally, we show uniqueness, let i,b , i,b 1 2 E X and v ,v 1 2 be the corre sponding solutions of (3.3.1) satisfying  IIe_itvj(t) vII —  —  0  as t co for j = 1, 2. It follows by the above arguments that v is a solution of equation (3.3.2), and satisfies vj (R, X 4 L ). By the routine 4 argument we obtain vi(t) = v (t) for t sufficiently large. The uniqueness 2 of the Cauchy problem at finite time gives . The proof of Theorem 2 = b 3.1.4 is complete. —  66  Bibliography [1] J.B. BERG, J. HuLsH0F, & J. R. KING. Formal asymptotics of Bub bling in the harmonic map heat flow. SIAM J. APP1. MATH. 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On the formation of singularities in the critical 0(3) a-model, arxiv:math.AP/0605023  68  Chapter 4  Solitary wave solutions for a class of nonlinear Dirac equations  4.1  Introduction  A class of nonlinear Dirac equations for elementary spin-i particles (such as electrons) is of the form  —  m + F()  =  (4.1.1)  0.  Here F : R —‘ R models the nonlinear interaction, : 1R 4 — C 4 is a fourcomponent wave function, and m is a positive number. 8, = 8/ax, and ‘y are the 4 x 4 Dirac matrices:  =  where  k  ()  k)  k 7 =  k  (_k  =  1,2,3  are Pauli matrices: 1  (0 i  —  1”  2  o)’  —  (0  i’\  o)’  —  (1  0 —1  We define =  70 =  ()  =  ()  A version of this chapter has been submitted for publication. Guan, M. Solitary wave solut ions for a class of nonlinear Dirac equations.  69  4.1. Introduction where (.,.) is the Hermitian inner product in C’. Throughout this chapter we are interested in the case F(s)  =  IsI°,  0 < 8  <  oo.  (4.1.2)  The local and global existence problems for nonlinearity as above have been considered in [5, 8]. For us, we seek standing waves (or solitary wave solu tions, or ground states solutions of (4.1.1)) of the form b(xo,x) where x 0  =  t, x  =  =  e_t(x)  (x,, X2, x ). It follows that 3  —  m+  :R 3  —+  + F()  Cz solves the equation =  0.  (4.1.3)  Different functions F have been used to model various types of self couplings. Stationary states of the nonlinear Dirac field with the scalar fourth order self coupling (corresponding to F(s) = s ) were first considered by Soler [11, 12] proposing them as a model of extended fermions. Subsequently, existence of stationary states under certain hypotheses on F was studied by Balabane [1], Cazenave and Vazquez [3] and Merle[6], where by shooting method they established the existence of infinitely many localized solutions for every 0 < w < m. Esteban and Séré in [4], by a variational method, proved the existence of an infinity of solutions in a more general case for nonlinearity F()  =  S2), 7 + bI  75  =  70717273  for 0 < ai, c 2 < . Vazquez [15] prove the existence of localized solutions ob tained as a Klein-Gordon limit for the nonlinear Dirac equation (F(s) = s). A summary of different models with numerical and theoretical developments is described by Ranada [10]. None of the approaches mentioned above yield a curve of solutions: the continuity of with respect to w, and the uniqueness of was unknown. Our purpose is to give some positive answers to these open problems. These issues are important to study the stability of the standing waves, a question we will address in future work.  70  4.1. Introduction Following [11], we study solutions which are separable in spherical coor dinates,  (x)  =  where r = lxi, (‘.1’, ) are the angular parameters and f, g are radial func tions. Equation (4.1.3) is then reduced to a nonautonomous planar differ ential system in the r variable fI+f=(g2_f26_(m_w))g  (414) =  2 (1g  _1218_  (m+w))f.  Ounaies in [9] studied the existence of solutions for equation (4.1.3) using a perturbation method. Let e = m By a rescaling argument, (4.1.4) can be transformed into a perturbed system — .  If  =  u’  +  v’  + 2mu  —  v+v 28 1v1 —  E(1 + iv 2  —  —  2 (iv  —  I  2 Eu  l°)u 2 eu  =  —  )v 28 iv1  =  0,  (4.1.5)  0  0, (4.1.5) can be related to the nonlinear Schrodinger equation _+v_ivl20v=0, 2m  (4.1.6)  u=—--—.  2m  It is well known that for e E (0,2), the first equation in (4.1.6) admits a unique positive, radially and symmetric solution called the ground state Q (x) which is smooth, decreases monotonically as a function of xl and decays exponential at infinity(see [13] and references therein). Let U 0 = want then we to continue to yield a branch of bound states Uo (Q, —Q’), with parameter 6 for (4.1.5) by contraction mapping theorem. Ounaies carried out this analysis for 0 < < 1 and he claimed that the nonlinearities in (4.1.5) are continuously differentiable. But with the restriction 0 < 8 < 1 we are unable to verify it. The term 1v 2 eu i° has 2 Along this cone, the first derivative of a cancelation cone when v = is unbounded for 0 < 8 < 1. But Ounaies’ argument may go 2 2 Eu Iv through for 9 1, which gives us the motivation of the current research. However we can not work in the natural Sobolev space H’ (1R , 1R 3 ). Since 2 ) 3 H’(1R (1R 6 L ) 3 , we lose regularity. To overcome these difficulties, we —  —  ‘—*  71  4.1. Introduction want to consider equation (4.1.5) in the Sobolev space W’P(IRS, II ),p > 2 2 and 8> 1. To state the main result, we introduce the following notations. For any 1 < p < oo,L = L(1R ) denotes the Lebesgue space for radial functions 3 on R . 3 = W,’’(R) denotes the Sobolev space for radial functions on . Let X? = 3 R x Y,? = L? x Li?. Unless specified, the constant C is generic and may vary from line to line. In this chapter, we assume that m = , since after a rescaling b(x) = (2m)W(2mx), equation (4.1.1) becomes +F()=O. We prove the following results: Theorem 4.1.1 Let e = m For 1 < 8 < 2 there exists €j = 60(8) > 0 and a unique solution of (4.1.4) (f,g)(e) E 2 )1R,R 0 C((0,e ( 4 ) ,W,’ ) satisfy ing —  .  f(r)  =  6(-Q’(/r) +  g(r)  =  S(Q(vr) + ei(r))  e2(r))  with IIejiIWi,2  Ce  for some  C  =  C(8) > 0,j  =  1,2.  Remark: The necessary condition w m must be satisfied in order to guarantee the existence of localized states for the nonlinear Dirac equation (see [15], [7]). The solutions constructed in Theorem 4.1.1 have more regularity. In fact, they are classical solutions and have exponential decay at infinity. Theorem 4.1.2 There exists C  =  C(e) > 0,o  Iej(r)I + Iörej(r)I  T Ce  =  u(e) > 0 such that  j  =  1,2.  Moreover, the solutions (f,g) in Theorem 4.1.1 are classical solutions  f,ge  fl  w,’P.  2<p<+oo  72  4.2. Preliminary lemmas Remark. From the physical view point, the nonlinear Dirac equation with  F(s) = s (Soler model) is the most interesting. Theorem 4.1.2 are both true for the Soler model. can find out that (v 2 eu ) v 2 2 = —eu 2 which is adaption of the proofs of the above theorems will —  —  In fact, Theorem 4.1.1, In fact, from (4.1.5) one Lipschitz continuous. An yield:  Theorem 4.1.3 For the Soler model F(s) = s, there is a localized solution of equation (4.1.3) satisfying Theorem 4.1.1 and Theorem 4.1.2.  Next we proceed as follows. In section 2, we introduce several prelim inary lemmas. In section 3, we give the proof of Theorem 4.1.1, Theorem 4.1.2.  4.2  Preliminary lemmas  We list several lemmas which will be used in Section 3. Lemma 4.2.1 Let g : R  g(a +  )  —  g(a)  —p  R be defined by g(t)  (26 + 1)IaI29u  —  where 2 ,0 depends on 0 and C 1 C 1  =  =  28 t t,  ’+ 29 (CiIa  0  >  0, then  C2uI201)2  0 if 0<0  Proof. We may assume that a> 0 in our proof. It is trivial if a  assume that a  0. If a  =  0. So we  21a1, then a + °i <31o1 and  <  g(a + a) <Jg(a +  —  g(a)  a)I +  —  (26 + 1)a29a  g(a)I + (20 +  1)1a28a1  <  If a  2aI, then a+a21a1+aIaj>0,  so that g(a+a)  =  ’. 20 (a+a)  Taylor’s theorem gives g(a + a)  —  g(a)  —  (26 + 1)a a 26  =  73  4.2. Preliminary lemmas where then  is between a+ and a. Since g”()  g”()I If 28  —  =  28(28+1)29_1, if 28—1 <0,  ’. 2 C 8  1 > 0, we have < Cmax{(a + u) ’} ’,a 28  C(a°’ +  o1281).  Hence we prove the lemma. Lemma 4.2.2 For any a, b E IR, 8> 0, we have  Ia  —  8 bI  IaI <CiIaI°’IbI + C 1b1° 2  —  where Ci,C 2 depends on 8 and C  =  0 if 0<8  1.  Proof. The proof is basically similar to that of the lemma as above. It is trivial if b = 0. So we may assume that b 0 and a> 0. If a < 2IbI, then -  8 b1  aI  -  21b1, then a  On the other hand, if a mean value theorem  Ia bI° -  where t is between a  C(Ia°  -  b and a. If 8  —  ItI°’  —  IbI°) <Cb°. a  bI  —  IaI  +  =  —  IbI  IbI.  So by using the  OItI ’ 8 IbI  1 > 0, then  < C(a°’ +  IbI°’),  hence  Ia bI°  -  -  If 8  —  1 < 0, then  ’ 8 ItI  IbI + 1b1 8 IaI <CiIaI . 8  , so that we conclude 1 CIbI°  Ia  -  -  IaI  <CjbI°.  The proof is complete. Lemma 4.2.3 For any a,b,c e R, ifl  a + b + c1 9  —  a+  9 b1  —  8<2, then  a + cI° + IaI°  C(IcI°’  +  IbI’)IbI,  where C depends on 8. 74  4.2. Preliminary lemmas Remark. This inequality is symmetric about b, c, so the right hand side can be equivalently replaced by C(lcI’ + lbI°’)lcl. Without loss of generality, we assume that bi Id in the following. Proof. For simplicity, let  L It is trivial for 8  =  =  Ia + b + Cl 9  Ia + bI°  —  lal  1, since if  ILl So next we consider 8 > 1. If 4.2.2, we have  a + c1 8 + al . 9  —  5IbI then L  =  0. If  la  51b1,  C(IbI + Id),  <  51b1,  al  1 + LI <C(Ia + c  by triangle inequality and Lemma  1 Ial°  +  Ibl°’)lbl  <C(lcI°’ + IbI°’)Ibl. If  lal  5lbI,  by using Taylor’s theorem  ILl  =  c 2 CI(a+tib+t l bcI. )I°  where t 2 , 1 t E (0,1) and  (a + tib +  c)l 2 t  >  lal  —  2lbl  —  id  Id.  Soifl<8<2,wehave  ILl  <  1b1. 1 CIeI°  The proof is complete. Lemma 4.2.4 Let 2 <p < cc, f 3 R —* R be radial and bounded. Suppose L. 1ff T + f E L, then E L and fr +  f  2  CIiãrf + fIILP  IIlIL Proof. We begin with p  =  cc. Using integration by parts  f(r) 2 r  =  (f T af + P  f)p2dp.  0  Hence  fI 2 Ir  <  iIf + fIIL  f  ds 2 s  =  (4.2.1)  IIfr + fIIL 75  4.3. Proof of the main theorems  which gives  IIi Next let us consider p ri < lxi <T2} and  =  I  <C3rf +  2. Let 0  =  fIlL. Denote D  1 < r2 < cc. < r  =  {x  ,0 < 3 R  + f)Lr2dr.  2ir  By Holder inequality, I  (D)lIfr + fiIL2(D). 2 CIILIIL  <  On the other hand, we have 1 Since  (r > f 2 r )  llil2(D) +  (r2) 2 r2f  0 we have  Illi2(D) <C (lIlIL2(D)II(fr + Let  T2  —p  cc, r 1  —>  f)IiL2(D)  +  rlf2(T1)).  0, we obtain  f  lIliL2 The intermediate case 2 <p  4.3  (ri)). 2 rif  -  < cc  2 C8,. + flIL2. is a direct result of interpolation  Proof of the main theorems  Similar to [9], we use a rescaling argument to transform (4.1.4) into a per turbed system. Let e = m (remember m = ). The first step is to introduce the new variables —  f(T) =  E  20  u(/T),  g(T)  620  where (f,g) are the solutions of (4.1.4). Then (u,v) solve u’ + ‘  +  1v1 + v (1v v 28 2 eu i° 2 e(1 + 1v 2 8 2 eu )u = 0.  —  —  —  —  —  )v 20 1v1  =  0,  (4.3.1)  —  76  4.3. Proof of the main theorems  Our goal is to solve (4.3.1) near  E  =  0. If e  0, (4.3.1) becomes  =  2  u’ + —u Iv v+v 20 r VI + U = 0. —  =  0  =  —v”  (4.3.2)  This yields the elliptic equation —v +  v  =  v, u 28 1v1  (4.3.3)  It is well known that for 0 < 6 < 2, there exists a unique positive radial solution Q(z) = Q(IxI) of the first equation in (4.3.3) which is smooth and exponentially decaying. This solution called a nonlinear ground state. Therefore U = (—Q’, Q) is the unique solution to (4.3.3) under the condition that v is real and positive. We want to ensure that the ground state solutions Uo can be continued to yield a branch of solutions of (4.3.1). Let v(r) = Q(r) + ei(r), u(r) = -Q’(r) + e2(r). Substitution into (4.3.1) gives rise to e(r)  +  e(r) +  1 +e  e2(r) e2(r)  —  1 e 29 (26+ 1)Q  =  ) 2 Ki(e,ei,e  (4.3.4)  =  where ) 2 Ki(e,ei,e  (Q+ei)—(26+1)Q 2 IQ+e 1 e 20 8 2 + (v  (e, ci, e2) 2 K  —  28 6U  e(1 + 1v 2  —  29 v )v  )u. 8 J 2 eu  —  Define L the first order linear differential operator L : X 1 e  =  28 (1—(26+1)Q  e2)  1 + 28 _Q  Or  —  Y,? by  8r+ (ei 1 )e2  Then we aim to solve the equation Le=K(E,e)  (4.3.5)  )T(c,e). Let 1= (O,),u>0. We say 2 )T,K(e,e) = (Ki,K 2 where e = (ei,e e(e) is a weak X-solution to equation (4.3.5) if e satisfies e  for a.e. e  e  =  L’K(e, e)  (4.3.6)  I. L is indeed invertible as we learn from the following lemma. 77  4.3. Proof of the main theorems Lemma 4.3.1 Let 0 < 8 < 2, the linear differential operator  8r+ 1  L_(1(28+1 is an isomorphism from Xf onto Y?? for 2  p  00.  Proof. First we prove that L is one to one. Suppose that there exist radial functions el, e2 e such that  L  ( e1 J  0.  =  \\ e2  Then 1 —rel + e  —  (28 + 1)Q 1 e 26  0,  =  e  —e.  =  (4.3.7)  It is well known (see, eg. [14]) that e 1 = 0 is the unique solution in H . 1 Next we prove that L is onto. Indeed L is a sum of an isomorphism and a relatively compact perturbation: L=  (  28 8r-i-)+(_(20+1)Q(r)  )  =L+i.  M is relatively compact because of the exponentially decay of the ground state at infinity. So we only need to prove that L is an isomorphism from X, to Y,°, i.e. for any (1,2) e L? x L, there exist (el,e2) e W, x W,’’ such that ‘e2J  \2  It is equivalent to solve 1 + (ôr + )e2 e  =  2 +e  =  ôrel  and show that ei, e2  e  By eliminating  (— + 1)ei Define  G(x)  =  1  =  —  e2 we know  (ãr +  (4.3.8)  that ei satisfies (4.3.9)  (47r)_hIxI_1e_khi. (4.3.9) has the solution ei  =  =  G(x)  * -  (+  1 +8,G(x)*b G(x)*q1 . 2 78  4.3. Proof of the main theorems  Here we have used the property of convolution and the fact (ãr + )*f(r) . By Young’s inequality and G,0rG L’(R 3 —8rf(r) in JR ), we have 3  IIelIILp  IIGILLi 11111LP  =  + IIDrGIIL1 II2IIL  which implies ei E L.  Similarly  e2  satisfies (—r+1+)e2= — 2 8r&.  Let H(x)  =  G(x), then e2  =H(x)* — 2 H(x)*(8rcii) =H(x)* + 2 (arH+H)*i eL  since H, (ar + )H e L’(R ). 3 To improve the regularities of ei, e2, we go back to (4.3.8). Since =  we have  1 e  —  e E L,  Regarding the regularity of  E  —e 1 = 2 (aT+)e  e2, we know that  EL.  By Lemma 4.2.4  II8re2ILp  <C(IIIILP + II(Ur + )e2IILP) <Cii(ar + )e2IILP  1 C  —  eIILP.  Hence we have e 2 Now we are ready to construct solutions of (4.3.6) by using the contrac tion mapping theorem. Proof of Theorem 4.1.1. To prove Theorem 4.1.1, we prove there exists > 0 such that for every 0 < e < e, there is a unique solution to equation (4.3.6) e = L’K(E,e) 79  4.3. Proof of the main theorems  in a small ball in X. First we must ensure that K(e, e) is well defined in Y,? if e e Xi?. Recall that Ki(e, e , 1  e2)  =  28 + ei) 1 e Q + (Q  2 + (v (,ei,e 2 K )  =  —  2I9  E(1 + v 2  —  —  eei 2 (28 + 1)Q  —  —  Q + 2 l  )v 2 v 8  eu ) 8 1 2 u.  Let us consider K , the estimate for 2 1 K is similar. Since  (IK 1 e,e)I  +  where C is a real constant depending on e, 8, it suffices to show that ’ + IuI 20 (Iv ’) E Li’. By Sobolev’s embedding 20  ) 3 W”P(1R  —÷  ) 3 L(R  for any q if p> 3. We choose p = 4 in the following. The same argument is available for K . From Lemma 4.3.1, we know that L’K Xi?. 2 Fix 3, to be chosen later. Consider the set =  and suppose e  {e E X;  =  6},  Q. We know that  11L’K(E, e) Let Ki(e,e)  IIeIIx  C(IKl(E, e)11L4 + 2 1K ( e, e)IL4).  K(e,e) + K(e,e) where  K(e,e)  =  and K(E,  e)  =  ((Q + ei)2  -  (QI + e2)29  -  l(Q + ei)126) (Q + ei).  Thus  IIK1WL  IIKIIL  +  IITIIL  For 2 IKjIjL = , b= (Q+ ) inLemma4.2.2, 2 4,leta= ru el) v =E(—Q’+e then  ) 1 201 IQ + ei1 Q + e212 + Cge(Q + IIeH,’) G  IKflIL4 <  C(QI  —  +6)  —  IQ + eiIB 28 Q’ + e2I  6/4 80  4.3. Proof of the main theorems  6  1 and e is small enough such that  Ger(IIQII + 6) For IIKI’11L4, let a  =  Q(r),o  IIKNIL  1 in Lemma 4.2.1, then e  =  <  Ce(IlQ ’ 2 eIlL4 + II61tIL)  <  1 ( 8 C e IIi,4  <C(6 +  if 6 < on  6,  +  IIei II)  620+1)  2 c 0 <2C 5 < 6/4  A similar argument can be applied to K 2 (with similar condition 6) to obtain that -.  WK2(e,e)IIL Hence we obtain  L’K(e,e) E ft Next we want to show that for any e, f E Q, and ö,e as above,  11L’(K(6,e)  -  IIe  K(E,f))IIy  -  fItx,  i.e. L’K is a contraction mapping. We have  IK(e,e)  —  K(e, f)I  IK(e)  —  K(f)I + Kj(e)  —  K(f)I + 2 1K ( e)  We compute the r.h.s. term by term. After rewriting K(e)  —  —  I(2(f)I.  K(f),  IQ+e1I28(Q+e1)_tQ+f1I29(Q+fi)_(26+1)IQ+fi28(ei_fi)  K(e)-K(f)I  +(2O+1)IQ+flI28(el_fI)_Q20(el_fl)=D+D. For D, let a  =  Q + fi, g <  =  1 e  —  fi and by use of Lemma 4.2.1, then  C(IQ + fI ’ 20  +  Iei  —  ’)Iei 29 fi1  —  . 2 fiI  By Sobolev embedding and Holder inequality, we have  IDIIL  < Co(Hei  —  f1II,,i,4  Co(ö+6 I 2 ei  —  +  IIei  —  f’IIw  IIei  —  flIIw’4 81  4.3. Proof of the main theorems if ö  Using Lemma 4.2.2, we find  <  DI  + I2Qfi +  ’f?1 8 )I2Qfi + f?Uei  —  fiI.  Hence  1 Ce  IDIIL Then let us study Kjs(e) K(e)  -  K(f)  —  -  IIei fiII•  fiIlw,p  -  Kj(f)  (1Q + ei)2 e(-Q’ + e2)210 I(Q + ei)216) (e f) + (J(Q + e e(-Q’ + e2)219 )(Q + fi) 8 1 2 I(Q + ei) (I(Q + f) 2 E(-Q’ + f2) 9 I(Q + fi) 1 2 )(Q + f’). 8 1 2 =  -  -  -  -  -  -  -  -  Notice that the first line in the r.h.s. is easy to estimate since  (1Q + ei)2 e(-Q’ + e2)210 -  e 9 <G  +  I 292 ei  —  -  Q’ +  I(Q + ei)218) +  —  (ei -  fi)WLp  )(e1 9 Q’ + e2 —  <llei  -  for e sufficiently small. For the second and the third line, let us define E(e,  f)  =  ((Q + e e(-Q’ + e2)20 (I(Q + f’) 2 e(-Q’ + f2)218 -  -  -  -  -  )(Q + 28 (Q + ei)  fi)  I(Q + f’) 1°)(Q + f’). 2  We discuss the contractive property for two different situations 0 > 1 and 9 = 1 separately. For 0 > 1, we use Lemma 4.2.3. Set a = (Q + f,) ,b 2 —(Q+fi) c = —e(Q+f2) (Q+ei) , 2 2 (notice that b, c can be taken sufficiently small), and rewrite E(e, f) to get  Ia+b+cI  JE(e, f)I +  I(Q + e,)2 C(b +  +  (Q + ei)2  —  -  9 Ia+b1  —  Ia+cI + IaIi  E(-Q’ + e2)219  I(Q + ei)2 r(-Q’ + f2)2I8 -  -  IcI)IbI -  E(-Q’ +  e2)28 -  (Q + ei)2  -  r(-Q’ + f2)2I8 (4.3.10)  where for the last line, we applied Lemma 4.2.2. We obtain  IE(e,f)IIL  C(e° +  ö)IIe  -  fIIw  Ie  -  fIIw 82  4.3. Proof of the main theorems  for e, 6 sufficiently smail. Hence we have for 1 <8 < 2,  11K ( 1 e)  —  1  Iie  4 Kl(f)ilL  fliw’,4.  —  Next we prove that E(e, f) is contractive for 0 = 1 directly. Lemma 4.2.3 can not be used since ib’ = IcI°’ = 1. In (4.3.10), if IaI max{51bI, 5IcI}, then  ia+b+ci  —  Ia+bI— ia+ci+ lal  =  0.  Thus  iiE(e,fHIL  C E 8 e  —  1  Iie — fII ,”4. 1  fiiw”  Hence we only need to consider E(e, f) if lal is small, i.e. if at < 5 max{51b1, Slci}, a+b+  cl-Ia + bi  Simply assume that ci WE(e, f)IiLP  Therefore if 8  -  Ia + ci +  al  C(ibi + Id).  Ibi, we have C 1 6 8 12 lie  fIiw’,4  -  < iIe  - fWwl4.  1, IIKl(e,e)-Kl(E,f)ilL  iIe-fIiw.  Similarly, we can prove that  11K ( 2 e)  -  K ( 2 f)Iy4  Iie fii’. -  Note we can satisfy all the condition above by choosing 6 = C e and taking e 8 sufficiently small. Then the contraction mapping theorem implies L’ K has a unique fixed point e(e) E 1 which is a weak solution of equation (4.3.5). The continuity w.r.t. follows from the continuity w.r.t e of the map L’K and its contractibility. This completes the proof of Theorem 4.1.1. LI  Let us see why a solution of equation (4.3.5) which is in X’ 4 has more regularity. This is done by using a standard bootstrap argument and the following standard lemma: Lemma 4.3.2 (see /2]) Let F : C there exists a 0 such that  IF(v)  —  F(u)I <C(IvI +  —*  C  satisfy  F(0)  Iui)Iv uI for —  =  all  0, arid assume that u,v E C.  83  4.3. Proof of the main theorems Let  1 r  -=+-,  It follows that if u  p  q L, Vu E L, then VF(u)  IVF(u)IILr  T and L  <  Proof of Theorem 4.1.2. First we can prove that  fl  el,e2 E  4 p <oo  Recall that el, e2 satisfy  (  1 e —  e2) and L is an isomorphism from X  IK1I  L  —  (  1 K  Y’. We know that  ’ 20 C,(IvI  +  ’) 28 IuI  and  1 2 1K Since el,e2 e 4 W,’ ( R) Lemma 4.3.1, we have  e(IuI  ’ + IuI 2 vI ) 26  +  L°°(R ) 3 , then K 1 x K 2  ) E 2 (ei,e  fl  x L for p  4. By  14’’ x  4 p <+oo  Next from Lemma 4.2.1, Lemma 4.2.2 and Lemma 4.2.3  ) 2 K(ei,e  , 1 K(f  f2)  <C(Q +  28 + 28 + Iei Q’1  ° 2 e2  +  I+ 1 f  ) 28 f2  (Iel—flI+1e2—f2D. So by Lemma 4.3.2, 2 x 1 VK e VK  fl  LxL.  4 p <oo  This gives that  (el,e2) E  fl  x W,’’  4 p <oo  84  4.3. Proof of the main theorems and je2p2P  <G€.  Going back to equation (4.3.5), we know that el,e2 E W’’ C C . So (f,g) 2 are classical solutions. Moreover we show that ei, e2 have exponential decay at infinity. We know el, €2 are classical solutions and eu, 1e21 < C€ by Sobolev’s embedding theorem. Taking derivatives in (4.3.2) and after tedious computations we find  f  ” 1 e e2  —  —  1 e  =  e2  =  (r)eç + 6 2 (r)Q 3 Si(r)eu + 6 cri(r)e + u 2 (r)e + 3 2 (r)Q  for r large for r large  (4 3 11)  where uj,6 E W ’ and 1o’I, 6 2 Ce(i = 1,2,3) for r large. We conclude that there exist constants r , v(e), C(e) positive such that 0 T for r el(r)I + e2(r)I < Ce”  >  . 0 r  (4.3.12)  We prove it by an application of the maximum principle. Without loss of generality, suppose e (ro) = 2€ (TO is sufficiently large). Let h(r) where g=e  +  =  0 is arbitrary and 0 then h, g satisfies >  —  Since h’  = v(_e_(r_r0)  =  el(ro)  ii  /3e1(7’_T0))  <  —  (1 + 3)  <  ii  1 is to be determined later.  zih and  0,g(oo)  g(r)O if  <  (1 + i6 )g + 52g + (1  g” with g(ro)  +  <  eT_T  for  <  —  Q  If  h, then  1)h 3 + öu + 5  (4.3.13)  0. Thus we claim that rro,  is small enough such that 1  —  +  +  II  0.  If the claim is not true, then g(r) obtains maximum at r = r 1 and g(r ) > 0. 1 Thus g”(r) < 0, g’(ru) = 0. But this contradicts with equation (4.3.13) since  85  4.3. Proof of the main theorems the right hand side of (4.3.13) is positive evaluated at r claim is true if z-’ < /1 — C and then  h(r)  ei(r)  =  . Therefore the 1 r  if r is large enough.  Then similarly we can show that  ei(r)  —h(r)  if r is large enough.  Thus ei(r)  Letting /3  —+  0  ,  <h(r)  = e_T_  +  /3eu)(T_T0).  we have < Ce” . T  for r large enough. The exponential decay estimate for e2 can be obtained in a similar way. Once we have (5.2.5), it is obvious that Iãrej(r)I < Ce and e E H . This completes the proof of Theorem 4.1.2. L 2 Remark. For 0 < 8 < 1, our method does not work since Lemma 4.2.3 is not valid, Let us consider a special example. Suppose e2 = f2 = 0, then E(ei, fi)  =  —E(Q’) 10_ IQ+ei 120_ 2 (Q+ei) (2 I(Q+fi — + 8 1 IQ+f e(Q’) ) i 29 )(Q+fi)  We want to know whether or not the following inequality is true  IE(ei(r),fi(r))I  Iei(r)  -  fi(r)t,  r  E (0,oo)  if E small enough. Letting r 0 large enough and s mined later, we assume that  Q(ro) + ei(ro) Q(ro) + fi(ro)  =  =  ‘/IQ’(ro)I(1 +  =  i/Q’(ro)I.  e, o  (4.3.14) >  0 to be deter  =  g(s)h ’ 2 ,  s),  Then under this ansatz, IE(ei(ro),  Iei(ro) where h  =  —  fi(ro))I =  fi(ro)  =  2 + 2s)° [(s  —  e 2 ((1 + s)  —  ’ 2 1)]h  sh  ,,/jQ’(ro)l. Then  E(ei(ro),fi(ro))I  =  h28Iei(ro)  -  fi(ro)I. 86  4.3. Proof of the main theorems We claim that if cx>  then  s  h > 8 >  ,  2  as  —*  0.  In fact, we have g(s)  9 Cs  since 2 + 2s)° > Cs (s 8 and (1 + s) ° 2  —  <C(s + ) 20 <<Cs s . 8  So > 28 ’h 8 Cs  =  28 CIQF( _ 0 + 8 E l) ro)j >>  1  as  e  ‘.  0  since 8 + cx(8 — 1) <0. The claim is proved and consequently, (4.3.14) does not hold for every r (O,oo).  87  Bibliography [1] M. BALABANE, T. CAzENAvE, A. D0uADY, F. MERLE. Existence of excited states for a nonlinear Dirac field. Commun. Math. Phys. 119, 153176 (1988)  [2] T. CAzENAvE. Semilinear Schrödinger equations, Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathe matical Sciences, New York; American Mathematical Society, Providence, RI, 2003 [3] T. CAzENAvE, L. VAzQuEz. Existence of localized solutions for a clas sical nonlinear Dirac filed. Commun. Math. Phys. 105, 35-47 (1986) [4] M. J. EsTEBAN, E. Stationary solutions of the nonlinear Dirac Equations: A Variational Approach. Commnu. Math. Phys. 171, 323-350 (1995) [5] M. ESc0BEDO, L. VEGA. A semilinear Dirac equation in H(R ) for 3 s> 1. SIAM J. Math. Anal. 2 (1997), 338-362, [6] F. MERLE. Existence of stationary states for nonlinear Dirac equations. J. Duff. Eq. 74(1), 50-68 (1988) [7] P. MATHIEu, T. F. MoRRIs. Existence condition for spinor solitions. Phys. Rev. D V.30, No. 8, 1835-1836, 1984 K. NAKANISHI, T. OzAwA. Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation. Rev. Mat. Iberoamericana 19 (2003), 179 -194  [8] 5. MAcHINHARA,  [9] H. OuNAJEs. Perturbation method for a class of noninear Dirac equa tions. Differential and Integral Equations. Vol 13 (4-6), 707-720 (2000) [10] A. F. RANADA. Classical nonlinear Dirac field models of extended parti cles. In: Quantum theory, group, fields and particles (editor A. 0. Barut). Amsterdam, Reidel: 1982 88  Chapter 4. Bibliography [11] M. S0LER. classical, stable nonlinear spinor field with positive rest en ergy. Phys. Rev. Dl, 2766-2769 ( 1970)  [12] M. S0LER. classical electrodynamics for a nonlinear spinorfield: per turbative and exact approaches. Phys. Rev. 3424-3429 (1973) [13] C. SuLEM, P.-L. SuLEM. The Nonlinear Schröodinger Equations: SelfFocusing and Wave Collapse, Springer-Verlag, Berlin, 1999. [14] S.-M. CHANG, S. GusTAFs0N, K. NAKANIsHI AND T.-P. TsAI. Spec  tra of linearized operators of NLS solitary waves. SIAM Journal on Math ematical Analysis 39 (2007), no 4. 1070—1111. [15] L. VAzQuEz. Localized solutions of a nonlinear spinor field. J. Phys. A: Math. Gen., Vol. 10, No. 8, 1977(1361 -1368).  89  Chapter 5  Instability of solitary waves for nonlinear Dirac equations 5.1  Introduction  In this chapter we show the instability of solitary waves, or standing waves for the nonlinear Dirac equation 3  (5.1.1) j=’  where (x,t) : R 3 x /12  \o  —  C , 4 m>0 and  0’\ k —12) ‘Y  k  =  1,2,3,  are Pauli matrices: 1  /0  iN  2  o)’  /0 =i  ZN  o)’  3  /1  =o  0 —1  We define (Q)  = =  =  ()  where (.,) is the Hermitian inner product in C’. By solitary wave, or standing wave we mean a solution of the form (x,t)  =  e_t(x)  (5.1.2)  with a real parameter and (x) decays as lxi —* co. The existence of such solutions are extensively studied in [2, 4, 5, 9—12] for a large variety of non linearities. While by shooting method [2, 4, 10] and variational method [5], A version of this chapter will be submitted for publication. Guan, M. and Gustafson, S. Instability of solitary waves for nonlinear Dirac equations.  90  5.1. Introduction the authors proved there exists infinity many localized solutions for every w e (0, m). Guan in [9] refined the result in [11] to show that for nonlinear ities with the form 1 8 < 2, or (‘)&, there is a continuous curve of solitary waves for w e (m o, rn) with some ej > 0. These are of the form g(r) —  ‘  if(r)cos if(r) sin ‘Pe  (5.1.3)  where f, g are real functions, and r, , are standard spherical coordinates in R . 3 We left an open problem about the stability of the solitary waves con structed in [9]. Stability of a solitary waves means that any initial datum sufficiently near this state gives rise to a solution which exists and remains near that state (at least up to symmetries) for later times. Otherwise it is unstable. This issue has been extensively studied for nonlinear KleinGordon equations and nonlinear Schrödinger equations. Grillakis, Shatah and Strauss [7, 8] proved a general orbital stability and instability condition in a very general setting, which can be applied to traveling waves of nonlin ear PDEs such as Klein-Gordon, and Schrodinger and wave equations. Let E() be the energy functional, and L() be the charge related to the sym metry that gives the solitary wave dynamics (in the case of solitary waves of the form (5.1.2), L() = f IkIdx). Their assumptions allow the second variation operator JI = E”(b) L”(q) to have only one simple nega tive eigenvalue, a kernel of dimension one and the rest of the spectrum to be positive and bounded away from zero. Then sharp instability condition is d”(w) <0 where d(w) = E() wL(). (5.1.4) —  —  Shatah and Strauss also came up with this as the instability criterion in [13]. But this method cannot be applied to Dirac operator directly. Contrary to the Dirac operator Dm —i yJt3j + m 0 y 0 is not bounded from below. The spectrum of Dm is (Dm) = (—oo,m] fl {m,+oo). However there are some partial results about the application of this method to Dirac equations. Bogolubsky in [1] required the positivity of the second variation of the energy functional as a necessary condition for stability, just like (5.1.4). Werle [161, Strauss and Vázquez in [14] claimed that the ground states are unstable if they exist, since the energy functional does not have a local minimum at the ground states. The condition (5.1.4) suggests that the ground states in [9] are unstable. Recall for e = m—> 0, —,  91  5.1. Introduction the ground states for the problem with nonlinearity (x)  (  -  i(x)  (x)  I°&have the ansatz  (E(x) e(x)  -  )  I  -  515  where  ( ) Q(IxI)  +°  (  =  =  and Q(x)  =  re’  )  + 0(e),  satisfies the nonlinear elliptic equation 1<6<2.  29 rQ+Q= Q, Q  Q is smooth and positive, decreases monotonically, and decays exponentially at infinity (see [6] and the references therein). We have II&11L2 Since d’(w)  Ce + 0(e),  =  —L()  II2II2  =  0(E).  —(II1I)2 + ll2II,2), then if 0>  =  ,  we have  as  2 d”(w)=—I(& < O, Il  But this is not a rigorous proof. The question of stability is related to the eigenvalues of the linearized operator. In general the stability analysis of nonlinear Dirac equations is harder than that of nonlinear Schrodinger equations. For equation (5.1.1), the solitary waves have the form g(r,t) t)  = e_t  t) cos if(r, t) sin 1!et  if(r,  Here f(r, t), g(r, t) are C-valued functions. Then equation (5.1.1) is reduced to a coupled system  f -wf  -  iOtf  0rg  -  2 (lgI  -  If I )f + 2  (516) mf.  In [9], we used rescaling and perturbation arguments. Let e = m w, p = Introducing two new functions (u, v) such that f(p, t) = Eu(/r, t), g(p, t) v(/gr,t), then (u,v) solve the coupled system 2 eh/ —  I iaV  =  E(0U  +  —  2 —eIuI IvI )u, 2 2 2 (IvI EIuI ) v + v).  517  —  92  =  5.1. Introduction Where without loss of generality, we take m = (uo(p), vo(p)) satisfy  f  .  The ground states solutions  8uo+—(v—eu)vo+vo=O, 8vo + 2mu 0 (1 + v eug)uo = 0.  j  —  518  —  When e is small enough, then  , 2 vo=Q+e  , 1 uo=—Q’+e  11e11i, IIe2IIHl and Q(x) = Q(xI) is the ground state of the cubic, focusing and radial non linear Schrödinger equation. Linearizing the system (5.1.7) around (u , vo), 0 we obtain the following linearized operator  A  —  1  —  0 e(1 + vg  0 —e(9r +  +  —  3eu)  —1+e(1+v—Eu) 0  —  ôr  E(8r+) 0  2uOvO)  —E(1  —  0 2 e uovo 0 3vg + eu)  0 0  To prove the instability, it needs to show that A has an eigenvalue A = A(s) with Re A > 0. Then we can say that solitary wave is linearly unstable. Furthermore solitary wave can be proved to be nonlinearly unstable if we can verify that A satisfies Theorem 6.1 in [8]  IeMI  <Cet  for some  0  <  <  2ReA.  The idea to find A(e) with positive real part is first to formally expand eigenvalues and eigenfunctions as a power series of e. In order to proceed with this expansion, it is useful to block-diagonalize AE into A with A=(02 —A+  A) °2x2  where A, A are two-by-two matrices. The formal expansion implies that the eigenvalue A is related to the eigenvalues of the cubic, focusing, radial NLS with the linearized operator around its ground states £=  (_  —)  (5.1.9)  where  L=-L , 2 , L+=—z+1-3Q +1-Q 93  5.1. Introduction which is known to be unstable [15]. Notice that A is not seif-adjoint, so the formal expansion of the eigenvalue and eigenfunction can not easily be used to conclude that there exist such an eigenvalue. Rather, we will verify the result by the “Lyapounov-Schmidt reduction” method from bifurcation theory. Before introducing the main result of this chapter, let us recall the main theorem in [9]: the nonlinear Dirac equation with nonlinearities 1 8 < 2 or (b’)’, admits the solitary waves of the form (5.1.3) with (f,g) satisfying Lemma 5.1.1 [9] Let e = m  solution (f,g)(E)  w. There exists Eo = eo( ) 9 4 C((O,e ( R3,R2 o),W )) satisfying —  f(r)  =  e(-Q’(vr) +  g(r)  =  e(Q(/r) + e1(/r))  >  0 and a unique  e2(/r))  with ilejilW2 <Ce  for some  lxi  and e 2 , 1 e have exponential decay as  C —  =  C(O)  >  0,j  =  1,2,  00.  An important Lemma addressing the instability of the ground states Q of nonlinear Schrödinger equation is as follows. Let Q = Q(lxi), x E W satisfy the nonlinear elliptic equation LrQ+Q=Q’Q,  where Q is smooth and positive, decreases monotonically, and decays expo nentially at infinity, then Lemma 5.1.2 [6] Suppose ). is an eigenvalue of £ (5.1.9) with correspond  ing eigenfunction U = _)2 and let If 1 + eigenvalue i are  /11  <0, 112  < p  =  <  1 +  =  ,  =  then  j  0, n+2  is real and the lowest  >  0.  Since )± = +/‘. Therefore 1 < 0, the first pair eigenvalues of £, (A we say the ground state is linearly unstable. In this chapter, the space dimension n = 3. Now we are ready to state the main theorem in this chapter. 94  5.2. Spectral analysis for the linearized operator Theorem 5.1.3 For the nonlinear Dirac equation (5.1.1), the ground states  in Lemma 1.1 are linearly unstable forE sufficiently small, i.e. the linearized operator A has an eigenvalue A with positive real part. In this chapter, we only consider Dirac equations with nonlinearity but in fact this theorem is still true for 1 < U < 2. The readers can carry out the same argument to prove it. Ui  In this chapter, we use the following notations. U  =  U2  denotes  U U4  a column vector in C . Sometimes we also write the column vector U as 4 (u1,’u2,u3,u4) We denote by (U,V) the Hermitian product of two vectors . T U, V in C . The usual Hermitian product in L 4 2 (1R ,C 3 ) is denoted by 4 (f,g)L2 =  5.2  f  (f(z),g(x))d x 3 .  Spectral analysis for the linearized operator  In this section, we present a block-diagonal representation of the linearized operator A. Then a formal expansion of eigenvalues and the bifurcation theory tell us that A e\/E + o(E) is one of the eigenvalue of A. To study the stability of the ground state solution, one linearizes equa tion (5.1.7) at (uo, VO). Let u(p, t) = uo(p)+e(p, t), v(p, t) = vo(p)+h(p, t), e, h are C-valued functions, then the perturbation (e, h) satisfy the system =8h + e  —  e(1 +  E((ivo + hi 2 i-a-- =th,e +  —  eiuo + e1 ) 2  + Eh  E(IVo + hi 2  —  ivo + hi 2 —  —  —  eiuo + 2 e1 ) e— (vg  e(ivo + hi 2  —  )h— 2 EIUo + ei  2 eiuo + e1 )vo + e((vo) 2  —  E(Uo) ) 2 vO.  A convenient way to study equation (5.2.1) is to write functions the real and imaginary parts. Let U  =  e,  h in their  T (Ree,Ime,Re , h,Imh)  then equation (5.2.1) becomes a system with a four-by-four matrix operator dU =  AU + nonlinear  terms  (5.2.2)  -—  95  5.2. Spectral analysis for the linearized operator where  AE=  0 1E(1+V—3EU) 0 —e(a++2euovo)  —1+E(1+v—eu) 0 0  0 Op2euovo 0 —e(1—3v+u)  —8,, 0 e(1—v+eu) 0  A is a first order, non-seif-adjoint operator. From the phase invariance of the dynamics, we know zero is an eigenvalue and 0  AE  =0. VO  Because of its connection to the stability problem, our interest is in the spectrum of this non-seif-adjoint operator. The analysis of the spectrum u(A), especially of the eigenvalues is hard because of its four-by-four format. For such kind of operator AE, the block-diagonalization is often used in fast numerical computations of eigenvalues with the Chebyshev interpolation algorithm [3]. We adopt this method to reduce A to a new operator which is more convenient to study. Theorem 5.2.1 There exists an orthogonal similarity transformation M,  such that M’  =  MT  =  M, where 1000 0 0 1 0 M— 0100 0001  that block-diagonalizes the operator A, ),fl A  IV.IL  ?,f  I’  —  I  I’  ‘‘—As \ +  —  0 /  where A  —  (—1+e(1-i-v—eu) e(8,,+)  —9,, e(1—v+eu)  and — (—1 + e(1 + v — 3eu) +i\ e(8,,++2euovo)  —0,, + 2Euovo E(1—3v+eu)  Moreover, AE and AE has the same spectrum, u(A)  =  cr(AE). 96  5.2. Spectral analysis for the linearized operator Proof. M is nonsingular since IM = 1. We can apply the similarity trans formation to A to yield A. The eigenvalue problem  AeU=AU,  , 4 UEC  is equivalent to MAM’U  =  AeU  =  ,XU.  Since M is nonsingular, then  AV=AV Then we conclude that u(AE)  V=M’U.  g(A).  =  o Because of Theorem 5.2.1, we change from studying to studying u(A) to g(AE). For the continuous spectrum u(A), by Weyl’s lemma we can drop the localized terms VO, o since they have exponential decay at infinity. AE is a sum of the operator A and a relatively compact perturbation, —  —  A! ‘-‘  (‘J2x2 A! I  n  \Zi  U2x2  with  ( i(1—e) ‘\—iE(0p +  A’— —  iãp —E  The following theorem identifies the continuous spectrum of A covering almost the entire imaginary axis. Theorem 5.2.2 The spectrum of A’ =  i[1  —  ,  co) U i(—oo, —e], 0 <e << 1.  Thus, ) 6 u(A  =  u(AE)  Proof. The eigenvalue problem diagonalized eigenvalue problems  1 U 2 (A’) with U 2 ,U 1  (  Zt  )  =  , 1 ?U  =  {ir, r e IR, r >  AEU  =  U 2 (A’)  1 u U is equivalent to the block =  , 2 u  . Therefore it suffices to find the spectrum of A’. Suppose 2 C  e D(A’) are the eigenfunctions with eigenvalue A such that A’(  u ,  (5.2.3)  )  —  then  I  i(1 e)u + i8v = Au, —ir(Du + u) iev = Av. —  (5.2.4)  —  97  5.2. Spectral analysis for the linearized operator Multiplying the second equation by A i(1 e) and using the first equation, we have (A+ie)(Ai(l6)) 1v= —  Since  —  has only continuous spectrum (—oo, 0], let (A+iE)(A—i(1 —e)) =  6  then we have A  —  i(1  —  2e) ± iV’l + 4ev 2  We claim that the spectrum of A’ is on the imaginary axis and =  i{1  —  e, oo) U i(—oo, —e].  This is the first part of the theorem. Define =  On one hand, if A  i[1  —  e, co) U i(—oo, —e].  E, it implies that (A+ie)(A-i(1-e))  For any  f  =  (  ‘  =  then v  2 )eL  +  (-oo,0).  x d (p), 2 L p let  +  -‘  -  (A  -  i(1  -  H’ by the standard regularity argument. Since A u=  and u, v  A  il  )(aPvf1)  i(1  —  e), then  , 2 EL  satisfy  (A’-AI)(  u  Hence (A’ AI)’ is invertible, which is a contradiction. Thus u(A’) ç a On the other hand, if A e D, we want to show that A e o(A’). We are —  seeking  (  ‘  )  2 x L 2 satisfy the following E L 98  5.2. Spectral analysis for the linearized operator  a)II(  Ui  b)II(A c)(  )IL2xL2  3 V  Ui 3 V  /  —A)(  Ui 3 V  =  1,  )IIL2XL2—*OasJ--*oo,  )—*0weaklyas--oo.  Since A e , then 3 E (—oo, 0] = o-(). By Weyl’s lemma, there exists 2 with IIjIL2 = 1 such that eH and  3 IR— ) jUL2-—O /  Let v then  /i +  =  and u  ( 3 V  as  j—O =  j—oo.  E H’(A  A  i(1  —  )eD(A’)  and  II(  )tI2XL2  =  (IIiIIL2  + A  -  i(1  -  E)j2  II8PJIIL2).  Since  IIOpjIIL2  =  (,—qj)  =  (, (- + /3)) + (j, -/ v) 3  — Hence  II(  (1  )II2XL2  +  =  IA  1.  Moreover, we have  II(A—A)( Therefore  Ui 3 V  )II—>0 as j—*oo.  c u(A’) and the claim is proved. By relation (5.2.3), we know E i[E,  co) fl i(—oo,  —El.  The continuous spectrum of 6 A covers almost the pure imaginary axis. In order to prove the instability, we will show A 6 has at least one eigen value with positive real part.  99  5.2. Spectral analysis for the linearized operator Theorem 5.2.3 For e sufficiently small, the operator A admits an eigen value of the form ) =eii+O(e)  and eigenfunction UE  where v  7 /E  (_i,m , 2 ) T+o m (l). i,_  =  is taken from Lemma 5.1.2, and  (  —  v)  (  ‘io 7o2  =  J  ioi, 7o2  satisfy  0.  Proof. Recall  A 0  0 A— 1 —A To find the eigenvalue A such that  AEU€  e  U  =  , 4 C  (5.2.5)  we first proceed with a formal expansion of eigenvalue and eigenfunction. can be written as a power series of e,  A  =  0 + EA A 1 +e A+ 2  (5.2.6)  ...  where  = 0 A  0 0 —1 000 1 0 000  —9 0 0 0  0 0 2QQ’ 2 —1+3Q  1-+-Q 2 O+ 0 0  and  1 A =  0 0 2 —1—Q —O—  0 1—Q 2 0 0  But A is not a standard perturbation of A 0 = AEIo : the kernel of A 0 is infinite dimensional and the essential spectrum is dramatically different from that of AE. Hence A 6 is a singular perturbation of A . It turns out there is 0 a more subtle relation with the linearized operator around the ground state of cubic, focusing and radial NLS  ( 0 —L  L_ 0 100  5.2. Spectral analysis for the linearized operator where L=, 2 . L=-z z+1 +1-3 -Q Q It is known that £ is unstable (see, e.g. [6, 15]). From lemma 5.1.2, there exist v  ( ) iio  with Rev>0 and io  =  =  (L  =  —  E C 2 such that  0.  (5.2.7)  Formally expanding ), tIE (5.2.8)  u  U0+EU1 2 +E U 2+”  (5.2.9) Substituting into equation (5.2.5), and upon collecting the powers of e, the following sequences of problems are obtained =  0 (A 0 (A 0 (A  —  —  0 Ao)U  1 + (A ?o)U 1  2 ) 0 ,\ U + (A 1  =  —  0,  (5.2.10)  A ) 1 Uo  )U + (A 1 ..\ 2  =  0,  )U = 0. 2 A 0 We will determine .\o, A 1 from those sequences of equations. 04 from equation (5.2.10), we have U ,, 01 (U 02 , U 03 )T, U —  —  (5.2.11) (5.2.12)  —  Let U 0  =  4 0 u 0 A +u 03 +u = o, 1 ç =  0,  =o +u — 2 0 u 0 1 3 A ç , =  0.  Then A 0 can be taken as either 0 or ±i. But we will ignore A 0 concentrate on A 0 = 0. Then U 01 = —U ,U 2 03 = —U . We take 4  0 U  =  ±i and  =  for some functions , i H’(p dp) to be determined later. From equation 2 14 we obtain U (5.2.11), let U 1 = (U ,, 12 ,U 11 13 )T, U (1 + 2 13 + Q ) 7 + Aiãp = U —r + (1 Q )17 = 2  , 14 au  —  (1 +  Q ) 2 8p  + 2QQ’ + Ai 11 pil = U 0 ) = —Ai? 2 + (1 3Q . 7  —  —  101  5.2. Spectral analysis for the linearized operator From the second and fourth components, we have  /  0 —L  L/ it i Ojj  =i  i =rAt  It allows us to choose from (5.2.7)  A = 1 v,  ()=m.  For the eigenvalue problem (5.2.5), the leading order term of eigenvalue is with Rev > 0, with leading order eigenfunction (—i , 701 1 We will not push the perturbation theory further since the eigenvalue is determined by the leading order. Next a rigorous proof shows indeed there exists a small eigenvalue ) EY+O(E) El)  and the corresponding eigenfunction  U  =  (—,mi, 2’ 1702)  + o(1).  The proof depends on the “Lyapunov-Schrnidt reduction” method from bifurcation theory. Let A = A 0 + EA, A = A, and AU = A U. Then it is 6 equivalent to solve U= 0 A A)U. (5.2.13) —  As we mentioned before, A 0 is a very degenerate operator. It has an infi nite dimensional kernel and cokernel and produces only the first and third nonzero components. The idea to solve equation (5.2.13) is to project equa tion (5.2.13) into the range space and complementary space of N, where 0 0  N_Il —  ‘\0  0 1  0 0  The projections split equation (5.2.13) into two equations:  U 0 NA  =  EN)’,  A)u,  (5.2.14)  and  0  =  U 0 NA  =  eN().  —  A)U  (5.2.15)  where N_/0 \0 —  1 0  0 0  0 1 102  5.2. Spectral analysis for the linearized operator We will first solve equation (5.2.14). Let U can be reduced to  ) (  u1+4 4 + 3 11 U  T , 3 (ul,u2,u , u4)  =  —u — 3 t4  =  )  then (5.2.14)  N(A)U  with —1 0  (0 i\1  J is invertible and J 1 (fl  =  —J, then we get  ) ( ) =  +  eJN(A  )U.  -  (5.2.16)  -  Now we use this equation to write uj, U3 in terms of NU=(U:=a,  u2, tL4, e  and A. Since  NU=(U2:= \tL4J  and U  =  NTNU + NTNU,  then equation (5.2.16) can be written as  (I  —  eJN(A  )N ) T o  —  (—8 + eJN(A  =  —  Now it is ready to solve c in terms of 3 and , The operator JN(A— )NT is bounded because of the exponential decays of u, v . So for small enough 0 .  It6JN(A A)N II <1, T eJN(A .)NT) exists, —  and the inverse of (I  —  —  00  (I  Let BE  =  -  eJN(A  Zk=o  k 6  -  1 ) T )N  (JN(A  —  =  )NT)k =  Ek  (JN(A  -  (8 + 6 JN(A  )NT)k.  —  A)NT),  we have  BL3,  with B 0 = —8,. This is the solution for equation (5.2.14). Now plugging = (NTo (NT + NTB)I3 into equation (5.2.15), we have U + NTI3)  N(A  —  0 ( NT  + NTBE)/3  =  0.  (5.2.17) 103  5.2. Spectral analysis for the linearized operator Let us identify the leading order,  A  1 + A  =  6, A  BE  =  p 8 _  +  Then equation (5.2.17) becomes =  —  EN ((S  —  E T A)N  —  T + NTB)) A(AT  :=  (5.2.18) For this equation, after computation, the l.h.s. can be simplified as l.h.s.(5.2.18)  =  (t.  —  Recall that (L  —  0 v)  =  0,  hence we can take Th+E/ 3 1 1 .  AV+EA,  (),3) solve the equation (1 To solve  (, ,8),  —  v)/i  + )(o +  =  (5.2.19)  E/3).  we use a solvability condition. Since ker(L_v)*=  (m2  )  =.  Equation (5.2.19) is solvable if  (, (]  + )(77o + e))  =  0,  (5.2.20)  (C L ) . Denote by (.,.) is a standard inner product in 2 projection onto , then /31 where  Q  =  /3i  I (i  —  -  Q.  =  (L  -  v)’(] + )(77o +  Q  -orthogonal 2 the L  E/3i)  Therefore  e(  -  v)’( +  ))  =  (  -  v)’Q(E +  104  5.2. Spectral analysis• for the linearized operator It yields a function in terms of r, A, and/3 1 A) is bounded, for sufficiently small e,  =  +  -  3 ( 1 e,  )II  .), since (—v)’Q(N+  <1.  Next we use the solvability condition (5.2.20) to obtain A by implicit function theorem. Let  G(e,A)  =  We first show that there exists A 2 such that G(0, A ) 2 computations, we find  G(O,A) We claim that (,o)  L_ is nonnegative and we have  (i,J ) 0 .  0 since  >  =  A(?,17o) +  =  0. After a series of  ) 2 2Re(oi,o  Q  =  2(Re  is the ground state [6], L_Q 7 , (L_io ) 2 o i  >  =  0, since  io2  CQ,  0.  Therefore, if we choose  2 A then G(0, A ) 2  =  (1oY’(?7,]o?7o).  0. Finally if (0, A ) 0, we conclude by implicit function 2 theorem, for e small, there exists A = ( 5 e) with A(0) = A . In fact, 2 =  ÔG  ) 2 —-(0,A  *  =  0  (71o,7o)  Therefore the eigenvalue problem (K  —  AE)U  =  0  has an eigenvalue of the form A =Eli+E , A 2  2 A  =  and eigenfunction  U  =  =  (jT  + NTB)(o +  (—,mi,  E/ i 3 )  102 02 )T  + o(1). 105  5.2. Spectral analysis for the linearized operator The formal asymptotics expansion is verified. The linearly instability is a direct result from Theorem 5.2.3 and Theorem 5.2.1.  106  Bibliography [1] I. L.  B000LuEsKy,  1979, Phys. Lett. A 73, pp. 87-90.  [2] M. Balabane, T. Cazenave, A. Douady, F. Merle, Existence of excited states for a nonlinear Dirac field, Commun. Math. Phys. 119, 153-176 (1988).  [3] M. CI-iucuNovA, D. PELIN0vsKY, Block-Diagonalization of the Sym metric First-Order Coupled System, SIAM J. APPLIED DYNAMICAL SYSTEMS, Vol. 5, No. 1, pp. 66-83. [4] T. CAzENAvE, L. VAzQuEz, Existence of localized solutions for a clas sical nonlinear Dirac filed, Commun. Math. Phys. 105, 35-47 (1986). [5] M. J. ESTEBAN, E. Stationary solutions of the nonlinear Dirac Equations: A Variational Approach, Commnu. Math. Phys. 171, 323-350 (1995). 16] C. SuLEM AND P.-L. SuLEM, The Nonlinear Schrodinger Equations: Self-Focusing and Wave Collapse, Springer-Verlag, Berlin, 1999. J. SHATAH, W. STRAUSS, Stability Theory of Solitary Waves in the Presence of Symmetry, J*, Journal of Functional Analysis 74, 160-197(1987).  [7] M. GRILLAKI5,  J. SHATAi, W. STRAUSS, Stability Theory of Solitary Waves in the Presence of Symmetry, 11*, Journal of Functional Analysis 94, 308-348(1990).  [8] M. GRILLAKIS,  [9] M. GuAN, Soliatry Wave Solutions for Nonlinear Dirac Equation, sub mitted, http://arxiv.org/abs/0812.2273. [10] F. MERLE, Existence of stationary states for nonlinear Dirac equations, J. Duff. Eq. 74(1), 50-68 (1988). [11] H. OUNAIBS, Perturbation method for a class of noninear Dirac equa tions. Differential and Integral Equations. Vol 13 (4-6), 707-720 (2000). 107  Chapter 5. Bibliography [12] M. S0LER, Classical, stable nonlinear spinor field with positive rest energy, Phys. Rev. Dl, 2766-2769 ( 1970).  [13] J. SHATAH, W. STRAuss, Instability of Nonlinear Bound States. Corn mun. Math. Phys. 100, 173-190 (1985). [14] W. STRAuss, L. VázQUEz, Stability under dilations of nonlinear spinor fields, Phys. Rev. D 34(2) pp. 641-643, 1986. [15] M. I. WEINsTEIN, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), no. 3, 472—491. 35Q20 (78A45 82A45). [16] J.  WERLE,  1981, Acta Physica Polonica B12, pp.601-606.  108  Chapter 6  Conclusions 6.1  Summary  This thesis focuses on the stability analysis of localized solutions Landau Lifshitz flow and nonlinear Dirac equations. There are two major contri butions to the field of study. For Landau-Lifshitz flow including harmonic map heat flow, most of the current research is focused on the compact 2dimensional manifold, or the bounded domain, and for a special class of maps. My thesis studies a bigger class of maps in the whole plane. Thus the flow equations can not be described by a single equation for a scalar function, but a more complicated coupled system. In my thesis, a rigorous proof of global existence for equivariant near-minimal energy higher degree Landau-Lifshitz flow is given, and finite time blow-up for degree m = 1 in R 2 is proved as well. For the nonlinear Dirac equations, my research is among the few to consider the uniqueness and continuity of the stationary solutions. A linearized Dirac operator around the ground states is explicitly given to analyze its spectrum. It turns out an unstable eigenvalue is related to that of the linearized operator of the Nonlinear Schrödinger equations around the ground states. In Chapter 2, we studied Landau-Lifshitz flow with equivariant symme try. The main achievement of this chapter is to use two different “coor dinate systems” in the energy space of maps to study the flow equation. Compared with the single equation for a scalar function (2.1.4), these two different coordinate systems yield more complicated equations. One is using the “generalized hasimoto transform” (see [5, 8—10]), which will remove the harmonic map component, and produce an equation (2.2.4). In this equa tion the double end points space-time estimates for H with a critical-decay potential 1/r 2 are obtained, due to the energy inequality and the positivity of potential. Usually the double end points estimates do not hold even for on radial functions. For Schrodinger maps [9], the weighted space-time estimates are used to replace the double endpoints estimates. The other decomposes the map into a nearby harmonic map plus a perturbation. This system (2.3.3) can be used to tracking the time-varying parameters s(t), c(t) —  109  6.1. Summary  of the nearest harmonic maps. But the restriction m 4 must be placed to make an appropriate choice of s(t), a(t). Finally we relate the two coor dinate systems to show that time-varying length-scale s(t) has a limit away from 0. For m = 2, 3 we conjecture that the solution is still global. For m = 1, we restricted ourselves to study (2.1.4). As we learn from [3, 4], in the disc domain, the borderline for singularity formation is that the bound ary condition q(1, t) = r. For the domain R , the requirement (oo, t) = 2 is a necessary condition for finite energy solutions. Hence a special choice of the initial data must be made to construct finite time blow-up solution. In Chapter 2, maximum principle and comparison theorem are developed as adaptations of disk in R 2 to the whole plane. In Chapter 3, a model equation for Schrodinger maps is studied. Since the subclass (3.1.1) is not preserved by Schrödinger maps, the construc tion of singular solutions becomes harder. The main goal to study equation (3.1.3) is to understand more about the Schrodinger maps when m = 1. But compared to equation (3.1.2), this hyperbolic equation is more complicated. We are looking for solutions with the form u(r, t) = Q(r) + ri(r, t) where Q is the stationary solution. The linearized operator acting on the real and imaginary parts of 77 is non-self-adjoint, and the continuous spectrum is the whole imaginary axis. When one studies matrix operators, one usually as sumes that zero is neither an eigenvalue nor a resonance. But unfortunately, sin Q is the unique ground state if m> 2 and is a resonance if m = 1. The dispersive estimate for the linearized operator is difficult to obtain. We are only able to deal with local existence for finite energy, and global existence for small energy by the usual Strichartz estimates. In Chapter 4, the uniqueness and continuity of the solitary wave solutions for a class of nonlinear Dirac equations are proved. Although there are many researches on the existence of solitons for Dirac equation, none of them have proved that the solution is a continuous curve. It is Quanies [17] who first uses the perturbation method to relate the ground states to those of nonlinear Schrödinger equations. One can find the application of this approach in [16]. By a symmetry argument, the nonlinear Dirac equation is reduced to a simpler system. In order to show that this system is solvable and admit a unique solution, the linear operator is proved to be isomorphism, and the nonlinear terms are Lipschitz continuous. The properties of the ground states Q of the nonlinear Schrodinger equations play an very important role in the analysis. Moreover, we show that the solutions have exponential decay. So they are classical solutions. There is one discrepancy between our result and that in 1171. But it leaves an interesting problem for the future research. Can we construct a unique 110  6.2. Future research solution for 0 < 0 < 1? In Chapter 5, we continue to study the nonlinear Dirac equations. Since we prove that the map from the parameter w to the solitary waves is con tinuous in Chapter 4, it is natural to consider the stability. The current research either is devoted to a numerical study of stability, or uses energy minimization as a stability criterion. However, these criteria are not nec essary rigorous. Mathematically, the question of stability is related to the eigenvalues of the linearized operator. The goal of this chapter is to give a rigorous proof of the instability. The linearized operator is a four-by-four matrix with a parameter Most of the spectrum lies on the imaginary axis. We want to show that the linearized operator has an eigenvalue with posi tive real part. To do that, the first step is to formally expand eigenvalues and eigenfunctions as a power series of e. It turns out that the eigenvalue is related to that of the linearized operator of nonlinear Schrodinger equa tions. The latter is known to be unstable. We then verify the result by the “Lyapunov-Schmidt reduction” method. Indeed our result is called linear instability. Although linear instability always implies nonlinear instability, it needs to be verified. To prove the nonlinear instability, according to [6], Theorem 6.1 in [6] must be satisfied. This will be discussed in details in our future research. .  6.2 6.2.1  Future research Blow up for 1-equivariant harmonic map heat flow  In Chapter 2, we proved that if the degree of the map m = 1, then the solution with initial energy close to the harmonic map energy may blow up in finite time. The approaches we used are maximum principle and the comparison principle. From [1], we know blow up is generic for m = 1 with the symmetry class The maximum principle is widely used to study the harmonic map heat flow both for global existence [7] and blow up [4]. The finite-time blow-up for the energy-space critical (m = 2) wave maps was studied in [2, 13, 15, 18] in the subclass (3.1.1). Blow-up for higher degree (m 4) turns out to be generic [18]. When the degree m = 1, the nongeneric blow up behavior was given in [15]. This is consistent with the harmonic map heat flow [8], but the methodologies are totally different. Our proof of finite time blow up greatly relies on maximum principle. The possible alternative technique to study 1-equivariant harmonic map heat flow is provided in [15]. Suppose that u(r,t) = Q(r/s(t)) +v(r,t), where u is the polar angle on the sphere, Q(r) = 2 arctan r is the ground state harmonic 111  6.2. Future research maps, s(t) = t ,, and v(r, t) is the error with local energy going to zero as 1 t —* 0. The scheme is to first find an approximate solution Q(r/s(t))+ue(r, t). e (r, 11 t) can be obtained by a finite sequence of approximation near the origin (r = tb’). But this process does not lead to an actual solution, since we keep losing time derivatives which leads to worse and worse implicit constants. So we turn to construct a parametrix for the heat equation by passing to coordinates (R, r) where R r= Then it gives us an equation with zero initial value. We want to solve the equation by the contraction mapping theorem.  6.2.2  Construction of unique ground states for nonlinear Dirac equation when 0 < 8 < 1  In Chapter 4, we constructed unique ground states for Dirac equation with 1 <0 <2, but our method does not work for 0 < 8 < 1 since an important technical lemma is used  Ia + b + cl 9  —  a+  —  Ia + cl° + laI <C(lcI 9  +  ibI ) 8 lbl,  for any a, b, c e R. This lemma is true only for 8 1. However we didn’t exclude the possibility of existence and uniqueness of ground states at least for 0 <8 < 1. For any qS ,2 E C 1 , we know 4  C(° + l2l29)l1  -  -  2l  So the nonlinear term of the Dirac equation is Lipschitz continuous for all 8>0. To solve equation (4.3.4) for 0 < 0 < 1, one can consider to study this equation in some weighted function space. Using the positivity of Q and the boundedness of Q’/Q, we define new functions such as = , f2 = —4. Then a first order differential system for (fi, f2) is obtained. The idea to solve this system is the contraction mapping theorem. One advantage chang ing functions from (er, C2) to (fi, f2) is that the K ,K 1 2 can be simplified such as:  2 1v  —  l 2 eu  =  2 e(—Q’ + ) 1 I(Q + e  =  2 Q  —  (1 + fi) 2  -  e() ( 2 l+  f2)2.  For sufficiently small e and f, f2 E X (X is a function space to be deter mined), (1 + fl) 2 r()2(1 + f2) 2 is pointwise positive. Hence we can avoid —  112  6.2. Future research  using the technical lemma. Then we left to show that linear operator is an isomorphism. Through studying this problem, a long term goal is to prove the asymptotic stability of ground state solutions.  6.2.3  Asymptotic stability for the model equation for Schrödinger maps  For the model equation (3.1.3), we have obtained the linear operator as an matrix £ =  (— -)  2 rn L=-Z+---cos2Q,  (6.2.1)  2 sin2Q m L.=-L+--2Q  To prove the asymptotic stability of the stationary solution, the first thing is to prove the space-time estimates (Strichartz estimates) for eta. This oper ator is non-self-adjoint and the continuous spectrum is the whole imaginary axis. When studying this kind of operator, one always assumes that zero is neither an eigenvalue nor a resonance. But this assumption does not apply to £. We know that L has sin Q as the unique ground state if m 2 and as a resonance if m = 1. Also notice that L_ has Q as a resonance LQ= -LQ+sin2Q =0.  but for m> 1 ItQIIL2=o0.  It may be a good starting point to study the resolvent (n—A)’ for A The dispersive estimates through resolvent analysis was also given in [141.  6.2.4  Nonlinear instability of the ground states  In Chapter 5, we prove that zero is linearly unstable, i.e. AE has an eigen value with positive real part. Although linearly instability always implies nonlinear instability, we still need to verify it. In [6], the theorem on unstable zero solution is given. =  Ax + f(x),  x(t)  X  (6.2.2)  where X is an Hilbert space. let f: X —* X be a locally Lipschitz mapping such that If (x)Ii < kIIxII 2 for ixil 1, for some constants k > 0 and 1 > 0. 113  6.2. Future research Let A be a linear operator which generates a strong continuous semigroup exp(tA) on X. Assume that A has an eigenvalue A with Re A> 0 and that  Ie” I 1 I <bet  for some0<  t<  2ReA  (6.2.3)  Then the zero solutions is (nonlinearly) unstable for (6.2.2). Thus we need to check equation (6.2.3) is satisfied for A . 6  114  Bibliography [1] J.B. BERG, J. HuLsH0F, & J.R.KING, Formal asymptotics of Bubbling in the harmonic map heat flow, SIAM J. APP1. MATH. Vol. 63, No. 5, pp. 1682—1717. [2) P. BizoN, Z. TAB0R, Formation of singularities for equivariant 2+1 -dimensional wave maps into the 2-sphere, Nonlinearity 14(2001), no. 5, 1041-1053.  [3] K.-C. CHANG, W.-Y. DING, A result on the global existence for heat flows of harmonic maps from D 2 into 52, Nemantics, J.-M. Coron et al. de., Kiuwer Academic Publishers, 1990, 37 48. —  [4] K.-C. CHANG, W.Y. DING, & R. YE, Finite time blow-up of the heat flow of harmonic maps from surfaces, J. Differential Geom. 36 (1992), no. 2, 507—515. [5] N.-H. CI-iANG, J. SHATAH, & K. UHLENBEcK, Schrödinger maps, Comm. Pure Appl. Math. 53 (2000), no. 5, 590—602. [6] M. GRILLAKIs, J. SH.k’rA, W. 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