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 Abstract Acknowledgements Table Table Bibliography Statement 1 Bibliography 2 Introduction 2.3 2.2 2.4 1.1 2.1 1.2 Global of Introduction Finite Proof Derived 1.1.2 1.1.1 Objectives Introduction 1.2.3 1.2.2 1.2.4 1.1.3 1.2.1 Contents existence of of time Co-Authorship of Well-posedness flow Model Landau-Lifshitz Nonlinear equations Solitary Schrodinger equations Instability Global nonlinear the and blow main and Contents equation well and main wave up Dirac main of theorem heat posedness blow maps standing solutions results and results equation for flow equations up Schrodinger scattering and for waves . for . blow Landau-Lifshitz . a class for of up maps a the for model of Landau-Lifshitz nonllnear nonlinear equation flow Dirac Dirac for . . 44 48 29 23 23 38 13 10 15 14 16 16 17 iii II’ vi ii v 2 9 1 1 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 67
4 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 88
5 Instability of solitary waves for nonlinear Dirac equations 90 5.1 Introduction 90 5.2 Spectral analysis for the linearized operator 95
Bibliography 107
6 Conclusions 109 6.1 Summary 109 6.2 Future research 111
6.2.1 Blow up for 1-equivariant harmonic map heat flow . 111 6.2.2 Construction of unique ground states for nonlinear Dirac equation when 0 < 8 < 1 112 6.2.3 Asymptotic stability for the model equation for Schrödinger maps 113 6.2.4 Nonlinear instability of the ground states 113
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 .4C Although this 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 dV C , there is a number N of magnetic moments then the 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 = —‘ymx 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
= —‘ymX 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 A = —‘ymx Heff — iFmx (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 3m = —7m x + X ._. (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” fcIVmIdx, , §2 and we can take Heff = —Lm. Let u = in : f C RTh x C 3R with = 1, then LL equation can be considered as a linear combination of heat I 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 IRand “ x “ denotes the usual vector product in .3R At first 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 = cu + 2u)IVuI + /3u x LU. (1.1.5) If we let U denote the orthogonal projection from 31R onto the tangent plane 2TS :={RIc.u=0} to g2 at n with 3
3 1.1. Introduction then an equivalent equation, written in a more geometric way, is 8tU = aPUZ.,u + ,BJUPUL\u where JU = ux is a rotation through 7r/2 on the tangent plane 2TS The borderline cases for Landau-Lifshitz equation include harmonic. map heat flow when 3 = 0, a = 1: ãu = iu + 2u,IVuI (1.1.6) and Schrödinger maps if a = 0, j3 = 1: 8u=uxzu. (1.1.7) 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 + 2uIVuI and integrating over 2 x [0,t), E(u(t)) + a f f Iu + 2uIdxdtIVuI = 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 2JR problem is the relationship between the energy and the topology, as expressed by a lower bound for the energy:
E(u) IVuI 4Ideg(u)I = 2 2R 2dx —> 2• where deg(u) is the topological degree for the map u : 2JR 2S So space dimension 2 is particularly interesting mathematically.
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 2,to the case where the domain is the unit disk in ,2R the target manifold is and for a special class of solutions:
u(., t) : (r, 0) — (cos mOsin (r, t), sin mOsin (r, t), cos (r, t)) e 2S 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
5 1.1. Introduction sin (0, t) = 0 is necessary for solutions to have finite energy + Sm E() = f (r )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= ,r0 which is a contradiction to Struwe’s result. Using the standard notation for Lebesgue norms: we say solutions blow up at time T < oo, if limsup IIVu(.,t)II= 00, or limt-Tsup IIu(,t)IIL2 = 00, 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 (r) qo 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 > ir. A result I I 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)) = m2arctanr . 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 E .31R Their approach was based on the 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, )2S 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 2Dn. For M with smooth 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 2Du, L — L and 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 ,2R Harpes [48] analyzed the geometric description of the flow at isolated 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, map”. For maps u : R x R+ local well-p osedness is established for 0Vu e 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 0U E H’(R’) (1.1.7) has a unique global solution. While for n = 2, small energy implies global existence and uniqueness for radial and equivariant —, maps. For u : 2R C ,3R by an m-equivariant map, we mean u(r,8) = emOv(r) (1.1.10) where m e Z a non-zero integer,(r, 8) are polar coordinates on ,2R and R is the matrix generating rotations around the u3-axis. The argument in [19] is based on a generalized version of the “Hasimoto transformation”. The key
8 1.1. Introduction observation involves the vector field w(x, t) = — !JiJuRu e which lies in the tangent plane to 52 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 2R 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
Let u : 2R x —* .2S For geometric evolution equations such as harmonic 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.
Let 7(x, t) : x R —* C be a radial scalar function and m > 0 be an integer. By analogy with (1.1.9) for the heat flow, we consider the nonlinear Schrodinger equation () =o(x). (1.1.11) The nonlinearity is designed to be gauge invariant (hence the 2L norm is preserved) and to admit a conserved energy defined by m E(c,b) irJ (IrI2 + —-sin2 hI)rdr. = 2r
9 1.1. Introduction
The nontrivial stationary solutions with finite energy are given by a 2- parameter class of functions := {eQ(r/s)IQ(r) = m),s2arctan(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 2E = c2p + c24m 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 3R is the momentum operator (h is Planck’s constant). For the wave function q5(x,t), a relativistic equation is obtained = m2c2, (V2 (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 + + + 7°8t)(7’O + + Dz + — 720y 73t), where ’(j‘y°, = 1,2,3) are to be determined. On multiplying out the right 7 yi side, one found out ,y0 must satisfy 773 737 (70)2 + = ,j126 j70 +7 = 0, = 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
07O, is f n k /c i 0 (L2x2 I — k_I ‘ 2 — iS\U \ 0 ix) ‘ o ‘ — where are Pauli matrices:
(0 1” — (0 — (‘1 0 a 2 — o)’ o)’ ko —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 7O_ichkOk + 27°.mc for ,b : x IR —* .4C Without loss of generality, we always assume c = h = 1. 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— Dm’b + 70VF(b) = 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 24C(C,R) and )8F(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 — 70VF() 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 75 = 70717273 F() := 2a() + 52b)b(y (1.1.15) where ‘z,b = (7°,?) and (.,.), is the Hermitian inner product in .4C This 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:
(1.1.16) if(r) cos’ sin ’1We 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) = 0. (1.1.17) the existence of infinity many stationary2 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
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, c2 > 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 < < 2, is continuous 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)’° y8 + 0m is not bounded from below. However there are some partial resultsj 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 : 21R 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 u with equivariant symmetry u = emRv(r), the energy E(u) has a minimal energy 4irImI,which is attainable 8by a two-parameter family of harmonic maps: = e(mOh(r/s), ’5H E IR, s> 0 / hi(r) \ / 2 ‘ \ I r+r h(r)=( 0 1=1 0 \ h3(r) J \ 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 tangent vector field u,. — JuRu satisfy a nonlinear heat-Schrödinger type equation. It leaves us with an equation with small 2L 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 2D in .2R 1.2.2 Well-posedness and scattering of a model equation for Schrödinger maps Recall the model equation (1.1.11) sin2I4 2m 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 = {o: [0,oc) — C, 0E( 15 1.2. Objectives 1.2.3 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— 2IvIv=0. (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 lxiand decays exponentially at infinity. We prove that when 1 < 8 < 2, there exists e0 > 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. Then we use the “Lyapounov-Schmidt reduction” method from bifurcation theory to verify it rigorously. 16 Bibliography [1] S. ANGENENT AND J. HuLsH0F, Singularities at t = oo in Equivari ant Harmonic Map Flow, Geometric evolution equations, 1—15,Contemp. Math., 367, Amer. Math. Soc., Providence, RI, 2005. [2] F. AL0uGEs AND A. S0YEuR, On global weak solutions for Landau Lifshitz equations: Existence and nonuniqueness, Nonlinear Anal., 18 (1992), pp. 1071C1084. [3] M. BALABANE, T. CAzENAvE, A. D0uADY, & F. MERLE, Existence of excited states for a nonlinear Dirac field, Commun. Math. Phys. 119, 153-176 (1988). [4] I. BEJENARu, On Schrödinger maps, arXiv:math/0604255. [5] I. BEJENARu, A. D. IoNEscu, C. E. KENIG, D. TATARu, Global Schrödinger maps, arXiv:0807.0265. [6] I. L. B0G0LuBsKY, 1979, Phys. Lett. A 73, pp. 87-90. [7] M. BERT5cH, P. PoDlo-GurnuaLl, AND V. VALENTE, On the dynam ics of deformable ferromagnets. I. Global weak solutions for soft ferromag nets at rest, Ann. Mat. Pura Appi. (4), 179 (2001), pp. 331C360. [8] J. D. BJ0RKEN, S. D. DRELL, Relativistic quantum mechanics, McGraw-Hill, 1964. [9] M. BALABANE, T. CAzENAvE, & L. VAzQuEz, Existence of Stand ing Waves for Dirac Fields with Singular Nonlinearites, Commun. Math. Phys. 133, 53-74(1990). [10] J.B. BERG, J. HuL5H0F, & J.R.KING, Formal asymptotics of Bub bling in the harmonic map heat flow, SIAM J. APP1. MATH. Vol. 63, No. 5, pp. 1682—1717. [11] 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. 17 Chapter 1. Bibliography [12] F.BETHuEL, J.-M. CoRoN, J.-M. GrnDAGLIA AND A. SoYEua, Heat flows and relaxed energies for harmoinc maps. Nonlinear Diffusion Equa tions and their Equilibrium States, Gregynog, Birkhäuser (1990). [131J.-M. C0R0N, Nonuniqueness for the heat flow of harmonic maps, Ann. Inst. Henri Poincare 7 (1990), 335-344. [14] Y. CHEN, W.-Y. DING, Blow up and global existence for the heat flows of harmonic maps, Invent. Math 99. 567—578(1990). [151Y. CHEN, Dirichlet boundary value problems of Landau-Lifshitz equa tions, Comm. Partial Differntial Equations 25(1/2) (2000) 101-124, [16] Y. CHEN,Existence and singularities for Dirichlet boundary value prob lems of Landau-Lifshitz equations, Nonliner Analysis 48(2002) 411-426, [171K.-C. CHANG, W.-Y. DING, A result on the global existence for heat flows of harmonic maps from 2D into ,2S Nemantics, J.-M. Coron et al. de., Kiuwer Academic Publishers, 1990, 37 — 48. [18] 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. [19] N.-H. CHANG, J. SHATAH, & K. UHLENBEcK, Schrodinger maps, Comm. Pure Appl. Math. 53 (2000), no. 5, 590—602. [20] J.-M. C0R0N, J.-M. GHIDAGLIA Explosion en temps fini pour le flot des applications harmonique, C.R. Acad. Sci Paris Ser. I 308 (1989) 339- 344. [211 T. CAzENAvE, L. VAzQuEz, Existence of localized solutions for a clas sical nonlinear Dirac filed, Commun. Math. Phys. 105, 35-47 (1986). [22] M. CHuGuN0vA, D. PELIN0vsKY, Block-Diagonalizatiorz. of the Sym metric First-Order Coupled System, SIAM J. APPLIED DYNAMICAL SYSTEMS, Vol. 5, No. 1, pp. 66-83. [23] W. Y. DING, Y. D. WANG, Local Schrödinger flow into Kãhler man ifold, Sci. China Ser. A 44 (2001), no. 11, 1446-1464. [24] A. DESIM0NE, AND P. PoDlo-GuIDuGLI, Inertial and self interac tions in structured continua: liquid crystals and magnetostrictive solids. Meccanica 30 (1995), 629-640. 18 Chapter 1. Bibliography [25] A. DESIM0NE, AND P. PoDlo-GuIDuGLI, On the continuum theory of deformable ferromagnetic solids, Archive Rational Mech. Anal. 136 (1996), 201-233. [26] A. DESIM0NE, AND P. PoDlo-GurnuGLi, Pointwise balances and the construction of stress fields in dielectrics. Math. Models Methods Appl. Sci. 7 (1997), 447-485. [27] R. VAN DER H0uT, Flow alignment in nematic liquid crystals in flows with cylindrical symmetry. Duff. mt.Eqns. 14(2001), 189-211. [28] M. J. EsTEBAN, E. Stationary solutions of the nonlinear Dirac Equations: A Variational Approach, Commnu. Math. Phys. 171, 323-350 (1995). [29] J. EELLs, J. SAMPSON, Harmonic maps of Reimannian manifolds, Amer. J. Math. 86 (1964), 109-160. [30] M. J. ESTEBAN, M. LEwIN, E, Variational methods in rela tivistic quantum mechanics, Bulletin (New Series) of the American Math ematical Society, Vol. 45 No. 4, Oct. 2008, pp. 535-593. [31] A. FREIRE, Uniqueness of the harmonic map flow from surfaces to general targets, Comment math. Helv., 70(1995), pp. 310—338. [32] R. FINKELsTEIN, R. LELEvIER AND M. RuDERMAN, 1951, Phys. Rev. 83. pp. 326-332. [33] R. FINKELSTEIN, C.F. FR0N5DAL, AND P. KAus, 1956, Phys. Rev. 103, pp. 1571-1579. [34] T.L. GILBERT, “A Lagrangian formulation of the gyromagnetic equa tion of the magnetic field”, Physical Review 100: 1243 (1955). [35] M. GRILLAKI5, J. SHATAH, W. STRAuSS, Stability Theory of Solitary Waves in the Presence of Symmetry, 1*, Journal of Functional Analysis 74, 160-197(1987). [36] M. GRILLAKIs, J. SHATAH, W. STRAUSS, Stability Theory of Solitary Waves in the Presence of Symmetry, 11*,Journal of Functional Analysis 94, 308-348(1990). [37] J. F. GRoTowsKI, Heat flow for harmonic maps, J.- M. Coron et al.(eds.), Nematics, 129 -140. 19 Chapter 1. Bibliography [38] J.F. GR0TOwsru, J. SHATAH, Geometric evolution equations in criti cal dimension, Caic. Var. (2007) 30: 499-512. [39] M. GuAN, S. GusTAFs0N, T.-P. TsAI, Global Existence and Blow up for Harmonic Map Heat Flow, J. Duff. Equations 246 (2009) 1—20. [40] M. GuAN, S. GusTAFs0N, K. KANG, T.-P. 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. [47] S. GusTAFs0N, K. NAICANIsHI & T.-P. TsAI. Global Dispersive So lutions for the Gross-Pitaevskii Equation in Two and Three Dimensions. Ann. Henri Poincaré 8 (2007), 1303-1331 [48] P. HARPE5, Uniqueness and bubbling of the 2-dimensional Landau Lifshitz flow, Caic. Var. 20, 213-229 (2004). [49] A. D. JONESCU AND C. E. KENIG, Low-regularity Schrödinger maps, Differential Integral Equations 19 (2006), no. 11, 1271-1300, available at arXiv:math/0605210. [50] J. KATO AND H. KocH, Uniqueness of the modified Schrödinger map in (R Comm. Partial Differential Equations 32 (2007), no. 1-3, 415-427,/342),H available at arXiv:math/0508423. 20 Chapter 1. Bibliography [51] L. D. LANDAu, E. LIFsHITz, On the theory of the dispersion of magen tic permeability in ferromagnetic bodies, Phys Z. Sowj 8(1935) 153- 169 (Reproduced in Collected Papers of L. D. Landau, Pergamon Press, New York, 1965, pp. 101-114). [52] P. MATHIEu, Existence conditions for spnior solitions, Physical Review D, Vol. 30, no. 8, 1835-1836. [53] H. McGAHAGAN, An approximation scheme for Schrödinger maps, Comm. Partial Differential Equations 32 (2007), no. 1-3, 375-400. [54] F. MERLE, Existence of stationary states for nonlinear Dirac equations. J. Duff. Eq. 74(1), 50-68 (1988). [55] A. NAHM0D, A. STEFAN0v AND K. UHLENBEcK, On Schrödinger maps, Comm. Pure Appi. Math. 56 (2003), no. 1, 114-151; Erratum, 57 (2004), no. 6, 833-839. [56] H. OUNAIES, Perturbation method for a class of noninear Dirac equa tions, Differential and Integral Equations. Vol 13 (4-6), 707-720 (2000). [57] J. QING, On the singularities of the Heat Flow for Harmonic Maps from Surfaces into Sphere, Comm. Analysis Geom. 3, 297-315 (1995). [58] 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. [59] M. STRuwE, On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Rely. 28, 485—502(1988). [60] M. S0LER, Classical, Stable, Nonlinear Spinor Field with Positive En ergy. 1970, Phys. Rev. Dl. pp. 2766-2769. [61] M. STRuwE, Geometric evolution problems. In: Nonlinear Partial Dif ferential Equations in Differential Geometry, lAS Park City Mathematics Series, Vol. 2, Amer. Math. Soc., Providence, RI, 1996, pp. 257-339. [62] W. STRAuss, L. VázQUEz, Stability under dilations of nonlinear spinor fields, Phys. Rev. D 34(2) pp. 641-643, 1986. [63] P.-L. SULEM, C. SuLEM, C. BARD05, On the continuous limit for a system of classical spins, Comm. Math. Phys. 107(1986), No. 3, 431-454. 21 Chapter 1. Bibliography [64] B. THALLER, The Dirac Equation, Springer-Verlag, 1992. MR1219537 (94k:81056). [65] P. ToPPING, Repulsion and quantization in almos-harrnonic maps, and asymptotics of the harmonic map flow. Ann. of Math. (2), 159 (2004), pp. 465-534. [66] p. 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 = 1u + 2u,IVuI u(x, 0) = Uo(x) (2.1.1) where u(., t) : C R” §, 2S is the 2-sphere {u = 1} = 3(‘ul,u2,u I lul ,3CR denotes the Laplace operator) in R”, 2IVuI= Z— This equa tion written in a more geometric way is 12() = puU Ut = D8u . where U denotes the orthogonal projection from 3R onto the tangent plane 2TS :={ e 3R Iu =0} to 2 at u, 8 = -. is the usual partial derivative, and 3D is the covariant. derivative, acting on vector fields (x) e 2T()S •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 .2S 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 space- time 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 ,2R and for a special class of solutions: u(., t) : (r, 8) —÷ (cosm8 sin (r, t), sinm8sin (r, t), cos (r, t)) (2.1.3) for a positive integer m, usually m = 1. (r, 0) are polar coordinates. Then satisfies the equation: trr+r_m2S1, 0 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 x .• 2S 2 2 with m E Z a non-zero integer. An m-equivariant map u : —‘ is of the form u(r, ) = emOR v(r) 52, where (r,9) are polar coordinates on ,2R v : [0,co) — and R is the matrix generating rotations around the u3-axis: /0 —1 0\ /cos c — sin c 0 R 1 0 0 , sin c cos c 0 = ) e° = \o Do) \o o 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 2IVuI = 2IUrI+ 1u = 2Ivr + IRvI2 and so 201 p00/ 2 r E(u)=lrJ rdr. (IvrI+—-(v+v))r / 2 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) = —kand denote by 2m the class of m-equivariant maps with v(oo) = k: emGRv(r), = {u : u = (u) v(0) = —, v(oo) <, = We measure distances between maps in m in the energy norm lu — ulI = IIV(u— 2ü)11L The class contains (3.1.1) as a special case. (up to a trivial reflection m ‘U3 —# —‘U3, and ignoring boundary conditions). For u m-equivariant, the energy E(u) can be rewritten as follows: 00 2 E(u) = irf (IvTl2 + IJ0RvI2) rdr = irf lVr_JvRhl2rth+Emin where J0 := vx is a ir/2 rotation on 2TS and p00 , p00 Emin = 27rJ J’Rvrdr 1] r = 27r1m1J3(v)dr = 27r1m1[v3(oo)+ (using v + v + V 1). The number Emin, which depends only on the boundary conditions, is in fact 47r times the absolute value of the degree of the map u, considered as a map from 2S to itself by compactifying the domain 2R (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 VT = HJVRV (2.1.5) 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) [0,27r)} Is>0, a where h8’°(r) := e00h(r/s), 26 2.1. Introduction and main results and hi(r) / \ 2 1rIm — r_ImI h(r) = I 0 1, h,(r) = , ha(r) = rIm rImI rim rHmi \ It3T)/ \ /1 1 + 1 + We record for later use that h(r) satisfying (2.1.5) means m m (hi)r = ——h,h )r(h = —h,.2 3 r 3 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) = ( h3(r/s) 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 U0 C Dm and (uo) = 4-,rm+ S for some 5o 5, then the following hold: (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 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 u0 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 2).H Statement (c) can be viewed as a characterization of 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 ö1 e (0, ó) and C> 0 such that if 8o < 6, then the existence time T = T(uo) in Theorem 2.1.1 can be taken to be T = cc. One also has IIV(u(x,t)— emSRh(5(t) t)))IIL2Lc,flLoOL2 < .0Co (2.1.8) Moreover there exist c, and positive s such that (s(t), c(t)) — (s, ) as t —* cc. (2.1.9) Remark 2.1.4 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. 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 r2hi(r) e 2L(rdr), which requires m > 3. 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 < E(uo) — < 62 such that the corresponding solution of the harmonic map 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 2l/1x1 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 — tmJVRV) where,2 recall, D is the covariant derivative, acting on vector fields tangent to at v. Let e E 2TS be a unit tangent vector field parallel transported along the curve v(., t) De=O. Then {e,J’-’e} is an orthonormal frame on 2TS Let q(r,t) = qj(r,t) + — !!2JvRv iq2(r, t) be the complex coordinates of the vector. field ‘vj,. 2TS in this basis m v. — JVRV = qie + Jve. r q2 29 2.2. Derived nonlinear heat equation JU We sometimes write qe qj.e + q e for convenience. Define JveJvRv=vie+v v=v+izi Now is 12 it a straightforward2 matter to show that the complex function q(r, t) solves the following nonlinear heat equation. with a non-local nonlin earity — (1 —mV3) — m(v3)rq = 2 q — qN(q) (2.2.1) where 1 —my N(q) Q(r’)dr’, Q := i Tm (rT + mP + 3 q]) = f°° [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) — 4irm < we would like to write the solution u = eme?v(r,t) with: v(r,t) = e t)R(h(r/s(t)) +(r/s(t),t)) (2.2.2) where (r/s(t), t) is a perturbation. Using the explicit orthonormal basis of ,2TS 1 and 3jh 0 3 = ( = ( \oJ \ h1 J it is convenient to decompose the perturbation term into components tan gential and normal to 2S at h(p): (p,t) = z(p,t)3-{- (p,t)h(p,t)Jl3+ (2.2.3) for p = r/s(t). This decomposition defines the complex-valued function z := Using v3(p,t) 27 we find +iz = (p,t), z (v = 12 (p)+ 3)r which.z we substitute into h3(2.2.1) to obtain (p) — (r) qt=—Hq—2m h3 h3 q (2.2.4) — 332m +m m3q — m(3)rq 2h 2— 2 — qN(q), where H is the operator = 1 + 2m — 2mh H = — + V(r), V(r) 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 as r —* 0 and as r — oo (with different constants). To some extent,2 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, (2.2.5) I e(t—s)t f \ 3 fY S j S)uS LL ‘—‘ Jo t where (r,p) is an “admissible pair” — 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 ) Let V(r) be a continuous radial potential on R\{0} satisfying V(r)+ Then —L V(r) is essentially seif-adjoint on. °°(R\{0}). + 3 C On the basis of Lemma 2.2.1, we have the following:0 Lemma 2.2.2 The operator H extends to a positive self-adjoint operator on a domain D(H) with \{0})C8°(R C D(H) C ),L(R hence —H is the 2 {e_t}t>o infinitesimal generator of2 a contraction semigroup on 2).L(R Furthermore, D(H) C 2).L°°(R Proof. We first consider m 2. Since V(r) (1 — /rm) by Lemma 2.2.1, is essentially self-adjoint 2 H on C8°(R\{•0}), and so its closure (which we still denote by H) can be uniquely2extended to a self-adjoint, operator on a dense domain D(H) C 2).L(1R When m = 1, the operator H is simplified as 4 4 Hr+ r2 21+r 31 2.2. Derived nonlinear heat equation By the above argument H = — + extends to a seif-adjoint operator on 2),L(R and under the bounded perturbation it remains seif-adjoint by the Kato perturbation theory ([17]). So, H generates a semigroup e_tH on ).L(R The non-negativity of e_tH2 2).L(RV immediately implies H 0, and so is a contraction semigroup on The final part of the lemma, the L°° estimate, is more delicate. Let e 2\{0}).C8°(R We have f {lV + }2V1y1 = (,H) and since for a fixed disk DR centered at the origin 2lxIV(lxl) 1 on DR, and V(lxl) bounded on D, (2.2.6) we conclude + C(ltii2 + IlHyll (2.2.7) lxi L2 2). Now multiply Hço = — f2f + Vçoby /jxl and integrate by partsto obtain llHil — — + )drd8 lxi 2L 0 0 2V p27r p00 1 + /r + 2(rV — )drd8 —)2 0 0 ( = J J 2/r and so by (2.2.6) again, and (2.2.7), VlilllxI” + lIlxl_3/2li2 2C(Jlll, + 2).ilHll (2.2.8) Fix p e 2(1,4/3), and set q := (2,4). Using (2.2.8), we have llliLP(DR) llHllLP(DR)+ llVIlLP(DR) + C(H II1x1312VIILQ(DR)Iiixi312col1L2(DR)) llsoilL2(D) llHcoIlL2+ llVllL(D) ISOIIL2C(liHiiL2 + IlOilL2) and a Sobolev inequality, yields the required estimate IkolILco 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 e_tH satisfied by the well-known semigroup eta. Lemma 2.2.4 Let {e_tH}t>o bethe semigroupgenerated by the operator H in 2).L(R Let 1 0 6 IIYIILa (here e’ can be defined by density). Proof. If a = b = 2, the statement just follows from the fact that {e_tH}to is a contraction semigroup on ).L(R Now let C°(R\{0}) C D(H). 2 = He_tHço = e_tHHy, Then u(t) e_tp D(H). Thus Hu(t) 2 and so IHu(t)IIL2 IIHsoI.Using (2.2.9), we find IIu(t)IL°° C(JIH’u(t) 11L2+ IU(t)1IL2) 2C(IIHoIIL+ IIS2IIL2), so there exists M> 0 such that sup, Iu(x,t)I Ut = — V(x)u, and V(x) > 0, given any 0 < o E °°(R\{0}), we have Ut < Lu, which means is a subsolution to the heat equation. Since t) e1ço 02C sup Iu(x, I M, it follows by the maximum principle that 0 < e_tIcp eço for all t> 0, and hence e_tHLb( < c t_(h/a_1/b) I!ILa(R2). 2 ço = — p+ For general) 0, we can rewrite p where = 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 C°°(R\{o}) in La. 02 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+, IfIILL(R2xI) := f2(LIf(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 IIeIILL + e_(t_8)Hf(s)dsIILrL < C(IiIIL2 + IifiIL’Ls’) for all functions ,,f(.,t) on .2R If, in addition, m > 2, and ,f(.,t) are radialt functions, then the estimate holds also in the “endpoint” case r = = 2, and we have weighted estimates Iie_tIiLL + Ii(e)TIiLL2 tf e 22t_s)Hf(s)ds11 + li(f e f(5)d5)rIIL2L2t So F is of weak type (1, ---), 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 ‘ p oo, we have 1ft e_(t_8)Hf(s)dsiiL f( s)h/’h/P)iif(s)IiLwds, so by the Hardy-Littlewood inequality, tif e_(t_8)Hf(s)dsipLrL IiUiILoo IIUTIIL2L2+ IiIILL CIIVvIIL2L2 (2.2.12) since 2IVvi = 2irl + li2. Now v(x,t) solves the equation: 2m — 2mh Vt = LV 3 V (2.2.13) — r2 with initial data eiecp. Multiplying equation (2.2.13) by V, taking the real part and using 2m — 32mh > 0 (since m 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 OIL1 CIIp!1L2 (which in any case is covered by the above argument). Similarly4 we get: = II ILLl+IIUrIILL Now let 13)(i, = (2,oo). If w(r,t) := f e_(t_8)Hf(s)ds, then w(r,O) = 0. By the above arguments we get, for v = e8w, IIWIJL CIIVvIIL 2cff Ifvldxds 6IIvIILo + LiC(E)IIfII — 2 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 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 qt = (1 + i)( — V(r))q. (2.2.14) We can space-time estimates for the linear evolution operator e_(1i)tH with out using maximum principle. Theorem 2.2.8 Let the exponent pairs (r,p) and (i,) be “admissible” (i.e. 1/r + 1/p = 1/2), includingf the case r = = 2. Then we have lie_ti e_(t_s) LL + II f(S)d5IILrLP C( + 36 2.2. Derived nonlinear heat equation for m 2, and radial functions , f(., t). Moreover the weighted estimates hold 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 IIUIIL2L°° + 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+ IHJIL2L2 CI +eIIuIILL +C(e)IfIIL2Li. If e is small, we arrive at the endpoint estimate IIfIILLo + IIUrIILL + IIIILL C(IIIIL2+ 1).IIfIILL 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< Lemma 2.3.1 /13J If m> 3 and 6 is sufficiently small, then for any map ‘U E m with (u) < 4nm + ö, there exists s > 0, a R, and a complex function z = z1 + iz, such that u(r, 8) = )R[(1 7(p))h(p) zi(p)3 p := 8e(m + + + z2(p)J], r/s, with z satisfying (z,hl)L2(pdp) = 0 and IzIIc Here X := {z: [0,oo) —* C E with the norm Iz1 2L(pdp), 2L(pdp)} 2 co Z , 2 1 ,‘ 2 zx.— I i)pup Jo 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) + z1(p) + 2(p)J =: h(p) + z (p). 7)2 The pointwise constraint lvi= 1 gives 1 = 21z1+ (1 + and since (as we shall prove) remains pointwise small, 7piCIZpZl. 7= —1o,V1—izi 1712Ciz1 Making the decomposition2 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: 100 iIzIl := / (izI + p p so that X = .2X Lemma 2.3.2 /13J Let 2 3, then lIzIlx CIILOzIILP IlIzI/p+2IiL2izi/p CiILoz/pIiL2 where 0L is the operator Lo:=(a+h(p))=h,(p)8_1 (2.3.1) p h,(p) Since, modulo nonlinear terms, (Loz)(p) sq(sp), we also have: Lemma 2.3.3 /13J Under the assumptions of Lemma 2.3.2, if lizlix<< 1, then 3 1—2/p Z Xp(pdp) — ,— q LP(rdr) IIIIL2(pdp)+ Il-11L2(pdp)< 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 R2. Vt = MrV + 2(IvrI+ IRvI2)v, Mr : ã + + (2.3.2) Inserting the decomposition v(r,t) = et[h(p) + (p,t)J, p = r/s(t) into (2.3.2), wearriveat (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 + Jz2 + 7h, we find, after routine (though somewhat involved) computations, that the tangential components of (2.3.3) yield our desired equation for z: 2SZt = —Nz+ (ims — s26)hi + 1F + 2F (2.3.4) where N denotes the differential operator N := —8 + -(1 — 2h) = (2.3.5) — 0LL (here L5 is the adjoint of 0L in L(pdp)). We will not write the nonlinear terms 1F and 2F explicitly, but 2only give the necessary estimates (we are omitting many of these details since very similar calculations are presented in [13]): Co>OsuchthatIIzIIx 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) we see that ker(N) = span{hi}, it is natural2 to impose (z, hl)L2 z(p)hj(p)pdp 0. (2.3.8) = f 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 .2L But we have (z,hl)L21 IIIIL2IPhlIL2 1zxph and so to make sense of the condition (2.3.8), we require z/p E ,2L which 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<<1, and (z,hl)L2 0, then J • 11z11L00 41 2.3. Proof of the main theorem • 2liphliiLoo Lemma 2.3.5 For cr > 0, set I := [0,u), Q := x I and define the spacetime norm iill + + 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) — 4irm) <(E(uo) - 4rm) = can be taken small. We start by rewriting equation (2.2.4) in integral form: — (p) q(t) = e_tHqo e_(t_8)H(F(q(s)) h3 q(s))ds (2.3.10) + f + for 0 < t < u where — 2m + m — m3q — m( : 33 F(q) = 2h 2 2 3)r q — qN(q) =: I +11+111. (2.3.11) Due to Lemma 2.2.5 (including the endpoint case) we have + - (p) C(ilqoilL2 (r) h3 ilF(q) 1h3 + (2.3.12) liqily IiLL 1IL4I3L4/3)• 42 2.3. Proof of the main theorem Note ha(r) — h3(p) h3(r) — (r/s) II IILL CIIqI2I h3 r2 (2.3.13) < CIIqI2 1IsIILi’oIIs — lIILi’° where the last estimate comes from expressing (r)—ha(r/s) as the integral of its derivative with respect to s. Next we use Lemma 2.3.3 to estimate IIF(q)IJ4/34/3(Q)h3 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) = h3(p,t) +3(p,t) and = z2h1 7h3+ for p = r/s(t), hence 2 IVIIL4’3C(s IqIILs+ (‘+ qIIL4) 34 PL8 PL So we obtain II’11L4/3L4/3C (iiqiii + IIILLf) C (IIqII+ IqII). (2.3.14) Nextweestimate := II(m(3)T/r)qM4/34/3.Compute — mh m7h] = [12 z1h32 + h3 + Again using Lemma 3.3, we arrive at III’HIL4/3L4/3C 4IIqIILL(1+ C(q + IIqI) (2.3.15) Finally, using the Hardy inequality ‘1 i/p (L(Lfd) XP_ldx) (f(Yf(u))PP_’dY) for f 0, inthecasep=p=2, wefind IIN(q)IILfCQI[L2 4C(Iq + and since i-’ <1, we obtain II”UL4I3L/3 IIqULLfHN(q)WLL + IIq/rII 2 (2.3.16) C (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. From Lemma 2.3.1, we have = e(me+a[(1 +7(r/so))h(r/so) + (zo)i(r/so)+ (zo)(r/so)J’3] with (zo,hl)L2 = 0 and0 lizolix C5o.Let (r,t) u(sor,st).2 Then ü is also a solution to the heat flow equation (2.1.1) with initial data (r,0) = e(mO 1o)R[( 0+7(r))h(r) + (zo)i(r)3+ 2(zo)(r)J], and time-dependent decomposition = e(me(t))R[(1 + 7(r/s(t)))h(r/s(t)) + zi(r/s(t),t) + (r/s(t),t)Jj with s(t) E C([0,T);Rj, (t) E C([0,T);R), s(0) = 1, z2and c(0) = cEO. If 5o is sufficiently small, the estimates of Lemma 2.3.4 and Lemma 2.3.5 together yield 118sIlL + IIlIL Cllql G8, and in particular that s(t) Co > 0. Hence the solution extends to T = and as t —* 00, s(t) — > 0, and a(t) —> a. Furthermore, for any pair (r,p)with+=,2roo, s (t IIV(u— 9emRh8 t))IILrLP CS’IIZlILXp CIIqIIr CIIqy Coo. Finally, undoing the rescaling u(r, t) = ü(r/so, t/s) completes the proof of Theorem 2.1.3. 2.4 Finite time blow up In this section we give the proof of Theorem 2.1.5 by constructing a 1- equivariant finite-time blow-up solution in 21R with near-harmonic energy. The proof is a variant of that of [5], adapted to the plane .2R One special subclass of 1-equivariarit solutions 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 3‘u-axis. If u solves the harmonic map heat flow, then, as is easily checked, (r, t) satisfies 1 sin2ql O 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 (O, t) = 0, lim t) = ir. (2.4.3) r—oo (r, We take C’ initial data: tlo e C’([O,oo)), limo(r) = 0, lim o(r) = r. (2.4.4) r—*O r—oo 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) e ([0,oo)))C([0,Tj;C flC°°((0,oo) x (0,T)) (one way to see this is to solve the full harmonic map heat-flow with the initial data corresponding1 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 2R 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). Lemma 2.4.1 Let q1,12 E BC([0,oo) x [0,T]) fl C((0,oo) x (0,T)) be solutions of the problem (2.4.2)— (2.4.4) with initial 2data If ç5j then bi(r,t) b2(r,t), (r,t) [0,oo) x [0,T]. Proof. Let o := — i. Then satisfies = rr + + p(r, t), (2.4.5) 45 2.4. Finite time blow up where sin 22 — sin 1 sin( — p(r,t) := = ——cos(h +2) 2 — 2r 2 @2—&) r Fix 1T 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 = Vrr + + (p(r,t) — K — 1)v. And setting we(r, t) := v(r, t) + 2f(T — t)_1eT/(4(T2_t)) for some 2T > ,1T we find 2 — 1)(we — w = w + + (p(r, t) — K — 2f(T t)_1eT/(4(T2_t))). (2.4.6) Now suppose inf[ ’i][0, 6w <0. By the boundary2 conditions and bound 0 7 wE(r,t) eness of , this) implies = inf[O)[OT w < 0 for some r > 0, t (0,Tij. Hence w(r,t) 0, w(r,t) = 0, and wr(r,t) 0. This contra wE 1 dicts (2.4.6). So we have 0, and] sending e —* 0, we recover 0 in [0,oo) x [0,1].T Since 1T A(t) — 2 /,2 — r2(1+v)’\ f(r,t) := arccos 2 r )+arccos (r,t) e (A(t)2 2 2 2(1+v)) (0,1)x(0,TA) satisfies 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 ii) such that 1 sin2f ft TT+ (r,t) (0,1) x (0,TA). f — 22r for all a 46 2.4. Finite time blow up Proof of Theorem 2.1.5. Given any small 6> 0, let the initial data qlo(r) to equation (2.4.2) be of the form /s_r2 3 o(r) = arccos + + r) 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 It — arccos () It — 6, and choosing (1) > 10 (thus can be chosen independent of 6). It is then easy to check that e(qSo) 4ir+ 062. Moreover, C(r) and s0 can be chosen such that e(O)(7/8 1 sin2g gt=grr+—gr— 2r 0 47 [10] [9] [8] [4] [5] Bibliography [7] [6] 12] [3] [1] folds, flow flows ularity rio. de., 186 pp. Comm. applications eral Heat 161(2002) in of A. J. Y. J.-M. N.-H. K.-C. K.-C. J.B. M. harmonic Y. the 2, —212. 1682—1717. BELLS, Kluwer targets, FREIRE, CHEN, of BERTsCH, Flow Am. of GIGA, 507—515. BERG, harmonic of CoRoN, Pure harmonic CHANG, CHANG, harmonic CHANG, Weak 93-112. J. of W.-Y. Academic Comment J.H. harmonique, maps, Solutions Uniqueness Math AppI. J. Harmonic Solutions R. J.-M. HuLsH0F, W.Y. map W.-Y. SAMPSON, J. maps maps Invent. DING, 86 D. Math. SHATAH, heat for Publishers, math. GHIDAGLIA PAsso (1964), DING, from from of Maps DING, C.R. of Semilinear Blow 53 Math flow, & the the Helv., Harmonic (2000), surfaces, D 2 & J.R.KING, Acad. 109-160. & harmonic on & up A Navier- R. SIAM R. into 1990, 99. the Explosion result K. and V. 70(1995), YE, Sci Parabolic no. 567—578 Disk, S 2 , UHLENBEcK, D. J. J. global 37 mappings Paris Stokes on 5, Formal map Finite-time APP1. Differential Nemantics, HouT, — 590—602. Arch. the 48. en pp. existence flow Ser. (1990). Equations system, temps global MATH. asymptotics 310—338. Nonuniqueness of Rational I from 308 Riemannian Schrödinger blow-up fini J.-M. Geom. existence for J. surfaces (1989) Vol. in pour Duff. the Mech. L’ Corori of 36 of 63, heat le Eqns and Bubbling 339-344. for the flot (1992), to for No. mani maps, Anal. et flows Reg gen heat heat 61, des the 48 al. 5, Chapter 2. Bibliography [11] M. GuAN, S. GusTAFs0N, K. KANG, T.-P. TsAI, Global Questions for Map Evolution Equations, Centre de Recherches Mathé matiques, CRM Proceedings and Lecture Notes,Vol. 44, 2008(61-73). [12] S. GusTAFs0N, K. KANG, T.-P. TsAI, Schrödinger flow near har monic maps. Comm. Pure Appi. Math. 60 (2007), 463-499. [13] S. GusTAFs0N, K. KANG, T.-P. TsAI, Asymptotic stability of har monic maps under the Schrodinger flow, Duke Math. J. 145, No. 3 (2008), 537-583. [14] A. KosEvicH, B. IVANOV, & A. KOvALEv, Magnetic Solitons, Phys. Rep. 194 (1990) 117-238. [15] J. KRIEGER, W. ScHLAG, D. TATARu, Renormalization and blow up for charge one equivari ant critical wave maps, preprint, arxiv:math.AP/0610248. [16] L. D. LANDAU & E. M. LIFsHITz, On the theory of the dispersion of magnetic permeability in ferromagnetic bodies, Phys. Z. Sowj. 8 (1935), 153; reproduced in Collected Papers of L. D. Landau, Pergamon Press, New York, 1965, 101-114. [17] M. REED, B. SIMON, Method of Modern Mathematical Physics Vol. II: Fourier Analysis, Self-adjointness, New York: Springer-Verlag, 1990. [18] I. R0DNIANsKI, J. STERBENz, On the formation of singularities in the critical 0(3) u-model, preprint, arxiv:math.AP/0605023. [19] M. STRuwE, On the evolution of harmonic mappings of Riemannian surfaces, Comment. Math. Rely. 60, 558—581(1985). [20] T. TAO, Spherically averaged endpoint Strichartz estimates for the two- dimensional Schrödinger equation, Comm. Partial Differential Equations 25 (2000), no. 7-8, 1471—1485. [21] J.L. VAzQuEz, E. 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 g2 (the unit 2-sphere): harmonic map heat flow (HMHF) 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 3R 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) : 2R x R —* 2S 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: 2m sin(21u1) jUt+U u=O, u(x,O)=uo(x). (3.1.3) 2 21u1 where u(x, t) : x IR — C is radial and m is a nonzero integer. Without loss 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 !_ 2 E(u) = 7t J 2(uJ + 2sin u)rdr. Finite energy solutions require sin Iuo(x)I= 0 both at lxi= 0 and xl= cc. Therefore this yields the following boundary conditions uo : = {uo: {0,oo) —* C,E(uo) 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 00 2m E(u) = f 2(Iui + —- 2sin Iuhrdr msinu 2 1°°sinluj =ir f u,. — — rdr + 2irm Re I üurdr Jo r ui I .‘o u where sin 2m Ref üurdr = mf lul(ittI2)dT o Iui o iul dr = —27rm — cos(iuhdr = 4irm. J dr Theref ore E(u) > 4irm. The minimal energy is attainable when — rnsiniul Ur = 0. (3.1.5) r ui The solution to equation (3.1.5) is obtained by the 2-parameter family of harmonic maps: := {eQ(r/s)iQ(r) = m),o2arctan(r 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 m m sin 2S(r) m = —77+ —-cos2S(r)Re,7+i----2 2 Im+ N(i),2 2S(r) (3.1.6) i(r,0) = iio(r), ?Jo(0)= ?70(oC)= 0. 52 3.1. Introduction and main results where N(ij) represents quadratic and higher order terms of ii = sin2lii±SI — sin2S — sin2S N(ii) (ii + S) — cos 2S Re ii Tmii. 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-) L=-+---cos2S,2m msin2S r L=-+-- 2r 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 )2L and as a resonance if m = 1(Isin QIIL2= oo). On the other hand, for the stationary solution eQ(r), -(er - sin 2Q(r)e) =0 gives a resonance Q for L_, 2m L_Q = -Q+ —sin2Q22r = 0 since Q 0 2L 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 m sin I2u 2 2 —1)u=0. (3.1.9) r 12u1 This equation resembles cubic NLS with inverse square potential. The boundedness (smallness) of m I (ILr + )rdr—-juI Jo 2r implies boundedness (smallness) of E(u). Hence the natural space for u is u e H’ and e .2L This is different from usual cubic NLS equations where u E or u E H’. Let us introduce some Banach spaces. Define X := {u: [0,cc) —* Cpu,- E 2L(rdr), 2L(rdr)}, with the norm := {1ur12+ m2} rdr. IIII f°° For 2 < p < cc, also define X’ := {u [0,cc) —* CIUrE L?(rdr), E LP(rdr)}, with the norm IIuIIcp:= [ {ir + mJ-} rdr. So X = .2X For radial complex-valued function u(r), v(r), define the follow ing inner product, (u, v)L2 ãvrdr = f and (u,V)x = f (ür(r)vr(r) + ü(r)v(r)) rdr. For an interval I 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 , we say that a pair of exponents (q,r) is admissible if + = r < oo. Then we have (i) lie < CilllL2 (ii) ei(t_T)g(.,r)dr One can find this Lemma in [8]. Here we provide a very elementary proof. For any radial function f e 02),C°°(R 00 00 f2(r) = _2f f(s)f’(s)ds = _2f -f’(s)sds. Thus by Holder inequality If112) 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; )TX where2 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 1, there exists Eo > 0 such that when uo e , k1 = k2 = 0 and Iluolx 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 Furthermore, there exist unique).T 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(t) is like — acting on function of the form v(x,t) = eimu(r,t). It suffices to prove that v is global and scatters in X. 6 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 S = 0 or S = Q = 2arctan(r Substitution u = S + i into equation (3.1.3) yields ).m 2m . 2m sin 2S 2m it=—+-cos2SReij+’t- 2S Imi+N(ij), (3.2.1) (r,0) = m = uo — S(r) 2 56 __ 3.2. Local weliposedness where N() = 775sin2frl±SI( sin2S Im—cos2SRe—i are nonlinear terms.) Let = + , and sin 2 21S + ui f(171,u2)= 2IS+uj (S+u), then using Taylor expansion for the function f u2), we have IN(ui)I (Ifxx(1,2)7??I + Ify(1,2)’I + Ifxy(1,2)u717?2I) I72C(Su + 5).Iu1 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 sin 2S(r) hmcos2S(r) = 1, lim = 1. r—O r—*O 2S(r) 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. 2 °° 2 2m 2 Lemma 3.2.1 Suppose ‘1I, = f0 (hr! +--H )rdr < co,i(0,t) = u(oo,t) = o and u(r,t) = 5(r) + u(r,t), then u E D. 7 Proof. If S = Q,rewriting the energy as roo 2 2 m .2 .2 E(u) = 4irm + I + 2QrReuir + —-(sm + — sin Q))rdr. (IuI r IQ iI Using integration by parts co 2 f 2QrRerkrdr = _f -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 2sin (Q+ ) , we get )2 n2 IQ+ — 2sin Q = sin 2Q Re ?7+ cos 2Q(Re sin2Q (3.23) + (Imp) 2 +R(?7). 2Q where R()I .3G Using (3.2.2), (3.2.3), the energy can be estimated as sm2Q E(u) < 4irm+f (177r12+-cos 2Q1Re?712+- 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 2 asrO T ( —,- asr—÷oo and msin2Q f asr—*O 2r2 2Q as r —* oo — From Lemma 3.1.2, HIL,co E(n) = = (IrI E() f 1 < 2 + 2sinI)rdr 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 eImOwhen acting on radial functions. By changing of variable from to with (x,t) = &meq(r,t), we get V’ IrIP+!cIIP,2 := {v : — CIVv e L, v/ri }2L (3.2.4) and (P = {v : CIVv E L”, v/ri E L,2 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 — 1)(Re)eme + — 1)(Im)eime + —N(rl)eim8, o = e”°r1o. 1 (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 = eto i ei(t_F((s))ds (t) + f (3.2.7) where .m (S1n2Q F(i) = -(cos 2Q —1)Re eimO + —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 4); D {(x, t) = (r, t)eimoi L°(I; ) fl L(I; iIiIL (3.2.8) 59 3.2. Local weilposedness for 6 > 0 to be specified later. Let I = (—T,T) E IRwith 0 e I. Suppose that for some 6> E ,1H L satisfies <6. (3.2.9) Then IiM()IILr(J.)flL(I,4) < 26 + CIIVFIIL4/3(IL4/3) (3.2.10) where we have used Strichartz estimate Lemma 3.1.1 and IIIIL CIIVIILP. Then we estimate IIVFIIL4/3L4/3term by term. Let (r) = (cos2Q— t S f1 —m sin2Q 1), —1). Both functions f2(r) ——-(--— are smooth. Since IV(fi(r)Reeimo)I CIfiIIVI+IfirUD, By using of Holder inequality in space and time, we have imO’ I HV(fiRee lIM()lILnL <26+ 4 lIL4C(T” S 3 + + IIIIL44) T’44IIII 2 t S <2ö+C(Ti6+T62+63)4 <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 )12d( = IIi — 2IILooflL44. We will use the two inequalities, II2 — I2I I2 I1C(I + I2I)I1 — 2I and IIiI2i — + I22)I1 22II2I 12C([ — 2I. Let T be small enough. Then by Strichartzj estimates ),Jv1(d(.A4( 1CV(F( — /aF())IJL 12)) ) — 243LV2IIL4L4 +T’(IIV + / - 2VIIL4L4 41+ 12II(IIV + IIVlII)LLfIIVl — 24VIL4L I< C(T’ + T1/46 + 12 2 ) 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 + r) — i ei(tt’)VF((r + t’))dt’. Strichartz estimate and 111,4IlVIILfyield it/,f \ r itLr7,-f \ e çiT ((O,Tmaxt),X e vc-r) ((O,Tmaxt),L 4L <2PlVIlL4((tTmax)L4)4L + ) + T lIViIL4((tTmax)L4)) + IIIl4((t,Tmax),L4))IIIIL((t,Tmax),L) 61 3.3. Global small solutions and scattering states Therefore for t sufficiently close to Tmax(r —> 0) such that + T’/IIVIIL4((t,Tmax),L4) + IVII4((t,Tmax),L4)) and 2 o HIL4((t,Tmax),L4) we obtain Ii X7 itL.,-i7 ye çi ) ((O,Tmax_t),L By the existence theorem, can4L be extended past Tmax which is a contrac tion. This shows ) “‘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 (sin2 E(u(t)) = 4m + f (ITI2+ 2QrRer + I(Q+ )I — 2sin Q))rdr. This integral is a conserved quantity which follows formally from multiplying equation (3.2.1) by integrating over ,2R and taking the real part. This can 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) m m sin 12u1 2 2 —1)u=O, u(x,O)=uo(x). I2uF There exists E such that if mlix EU, then u < ,0CE E(u) < 0CE and = 1G(u)l si2 2u - ii 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) = 8uoeim (3.3.1) By Duhamel’s formula, = etvo e(t_8)G(u)vds, v(t) + i L t I. For 0 g(t) = IVIIL((ot)) + IIVIL4((o,t),x4) It follows from Strichartz estimates 2 g(t) CIIVvOIIL2+ L/IIVG(u)vIIL/ 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+ f ei8G(u(s))v(s)ds. Therefore for 0 < t < T, = if w(t) — w(r) eG(u(s))v(s)ds. 63 3.3. Global small solutions and scattering states If follows from is unitaryf Hw(t)- w(r)II IeIv(s)I2v(s)IIids II II4((t,r);L4)IIHL4((t,r);L4) + II II4((t,r);L4) By the global existence for v in 4LX , —* IIw(t)— w(r)II — 0 as r,t 00. Therefore there exists v+ E X such that Ie_ttv(t)— vII —* 0 as t —* oo. One can show as well that there exists v E X such that IIetv(t) — vIIi — 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 E X such that the solution v e C(R, X) of equation (3.3.1) satisfies IIetv(t) — vIl. —* 0 as t 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 itL+ e V LX— 64 3.3. Global small solutions and scattering states So CT —* 0 as T —* cc. Define the set D = {v E 4L(IT,X flL°° (IT,X) : IIVIIL4(IT4) 2C} and T(v) by ) 00 2 T(v)(t) = if ei(t_TG(u(s))v(s)ds. Then IIT(v)IIL(IT) + IIT()IIL4(IT,5C4) C(IIv/rII4(ITL4) + IVv4(JT,L4)) C(2C. We see if T is sufficiently large, such that 3C(2CT) 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) for t > T maps D to itself if T is large enough. We can easily verify that 2d(Mvi,Mv 2),d(vi,v V1,V2 E D. So M has a fixed) point v e D satisfying equation (3.3.2) on [T,cc). Let = v(T) e X, and then ft v(t + T) = eit(T) + ei(t_G(u(s + T))v(s + T)ds. Therefore v is the solution of the equation (3.3.1) with v(T) = . By the global existence solution of equation (3.3.1), we know v(0) e X is well defined and e_2tv(t) e18G(u(s))v(s)ds. — = if Since v D, then je_itv(t) —* 0 —, — vII 3 00((t,IIVII as t cc. therefore, v(0) = satisfies the conclusion)4 of the lemma. 65 3.3. Global small solutions and scattering states Finally, we show uniqueness, let ,1i,b 2i,b E X and ,v1 v2 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 4L(R, 4).X By the routine argument we obtain vi(t) = v2(t) for t sufficiently large. The uniqueness of the Cauchy problem at finite time gives = .b2 The proof of Theorem 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. Vol. 63, No. 5, pp. 1682—1717 [2] 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] 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 [4] N.-H. CHANG, J. SHATAH & K. UHLENBE0K. Schrodinger maps. Comm. Pure Appi. Math. 53 (2000), no. 5, 590—602 [5] K.-C. CHANG,W.Y. DING & R. YE. Finite-time blow-up of the heat flow of harmonic maps from surfaces. J. Duff. Geom. 36 (1992), no. 2, 507—515 [6] K.-C. CHANG, W.-Y. DING. A result on the global existence for heat flows of harmonic maps from 2D into .2S Nemantics, J.-M. Coron et al. de., Kiuwer Academic Publishers, 1990, 37—48 [7] M. GuAN, S. Gu5TAFs0N & T.-P. TsAI. Global existence and blow up for harmonic map heat flow. J. Duff. Equations 246 (2009) 1—20 [8] 5. Gu5TAF50N, K. KANG & T.-P. TsAI. Schrödinger flow near har monic maps. Comm. Pure Appl. Math. 60 no. 4 (2007), 463-499 [9] S. GusTAFs0N, K. KANG & T.-P. TsAI. Asymptotic stability of har monic maps under the Schrodinger flow. Duke Math. J. Volume 145, Num ber 3 (2008), 537-583 [10] J. F. GRoTowsKI, J. SHATAH. Geometric evolution equations in cir itcal dimensions. Calc. Var. (2007) 30: 499-512 67 Chapter 3. Bibliography [11] S. GusTAFs0N, K. NAKANIsHI & T.-P. TsAI. Global Dispersive So lutions for the Gross-Pitaevskii Equation in Two and Three Dimensions. Ann. Henri Poincaré 8 (2007), 1303-1331 [12] J. KRIECER, W. SCHLAG, D. TATARu. Renormalization and Blow Up for Charge One Equivariant Critical Wave Maps, Invent. Math. 171 (2008), no. 3, 543—615 [13] I. R0DNIANsKI, J. STERBENz. 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() = 0. (4.1.1) Here F : R —‘ R models the nonlinear interaction, : 41R — 4C is a four- component wave function, and m is a positive number. 8, = 8/ax, and ‘y are the 4 x 4 Dirac matrices: k) 7k = (_k k = 1,2,3 = ( ) where k are Pauli matrices: 1 — (0 1” 2 — (0 i’\ — (1 0 i o)’ o)’ —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) = e_t(x) Cz where x0 = t, x = (x,, X2, x3). It follows that : 3R —+ solves the equation — m + + F() = 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() = + 7bIS2), 75 = 70717273 for 0 < ai, c2 < . 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 u’ + — 1v1 + v — (iv — Eu — = 0, v28 2 2 I )v28iv1 (4.1.5) v’ + 2mu — E(1 + 2iv — 2eul°)u = 0 If = 0, (4.1.5) can be related to the nonlinear Schrodinger equation _+v_ivl20v=0, u=—--—. (4.1.6) 2m 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 0U = (Q,—Q’), then we want to continue Uo to yield a branch of bound states 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 21v — 2i°eu has a cancelation cone when v = Along this cone, the first derivative of — is unbounded for 0 < 8 1. But Ounaies’ argument may go Iv2 2Eu < through for 9 1, which gives us the motivation of the current research. However we can not work in the natural Sobolev space H’ 3(1R 2).1R Since ‘—* we lose regularity. To overcome , we )3H’(1R 63),L(1R these difficulties, 71 4.1. Introduction want to consider equation (4.1.5) in the Sobolev space W’P(IRS, 2II),p > 2 and 8> 1. To state the main result, we introduce the following notations. For any 1 < p < oo,L = 3L(1R denotes the Lebesgue space for radial functions on .3R = W,’’(R)) denotes the Sobolev space for radial functions on .3R Let X? = 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 ),W,’(1R,RC((0,e satisfy ing )) f(r) = 6(-Q’(/r) +042 e2(r)) g(r) = S(Q(vr) + ei(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 = u(e) > 0 such that Iej(r)I+ Iörej(r)I TCe j = 1,2. Moreover, the solutions (f,g) in Theorem 4.1.1 are classical solutions f,ge fl w,’P. 2 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. In fact, Theorem 4.1.1, Theorem 4.1.2 are both true for the Soler model. In fact, from (4.1.5) one can find out that 2(v — )2eu — v2 = 2—eu which is Lipschitz continuous. An adaption of the proofs of the above theorems will 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 —p R be defined by g(t) = tt,28 0 > 0, then g(a + g(a) — (26 + 1)IaI29u + C2uI201)2 ) — 29(CiIa where depends on 0 and = 0 if ’ 12C,0 1C 0<0 Proof. We may assume that a> 0 in our proof. It is trivial if a = 0. So we assume that a 0. If a < 21a1,then a + °i <31o1and g(a + a) — g(a)— (26 + 1)a29a 73 4.2. Preliminary lemmas where is between a+ and a. Since g”() = 28(28+1)29_1, if 28—1 <0, then g”()I ’.C28 If 28 — 1 > 0, we have < Cmax{(a + ’}u)28’,a C(a°’ + o1281). Hence we prove the lemma. Lemma 4.2.2 For any a, b E IR,8> 0, we have Ia— 8bI — IaI - 8b1 - aI C(Ia° + IbI°) On the other hand, if a 21b1,then a — bI a — IbI IbI.So by using the mean value theorem Ia- bI°- IaI = 8’IbIOItI where t is between a — b and a. If 8 — 1 > 0, then ItI°’ < C(a°’ + IbI°’), hence bI°- Ia- IaI IbI Lemma 4.2.3 For any a,b,c e R, ifl 8<2, then a + b+ 9c1 — a + 9b1 — 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, weassumethat bi Id in the following. Proof. For simplicity, let L = Ia+ b + 9Cl — Ia+ bI° — a + 8c1 + .9al It is trivial for 8 = 1, since if lal 5IbIthen L = 0. If la 51b1, ILl< C(IbI+ Id), So next we consider 8 > 1. If al 51b1,by triangle inequality and Lemma 4.2.2, we have LI If lal 5lbI,by using Taylor’s theorem ILl= c)I°lbcI.CI(a+tib+t where t12,t E (0,1) and + tib 2 — 2lbl — (a + t2c)l > lal id Id. Soifl<8<2,wehave ILl< 11b1.CIeI° The proof is complete. Lemma 4.2.4 Let 2 Proof. We begin with p = cc. Using integration by parts f(r) = (af + f)p2dp. (4.2.1) r2 fT0 P Hence f f2Ir I< iIf + fIIL s2ds = IIfr + fIIL 75 4.3. Proof of the main theorems which gives IIi I = 2ir + f)Lr2dr. By Holder inequality, I < 2(D)lIfrCIILIIL + fiIL2(D). On the other hand, we have 1 llil2(D) + 2(r2f(r2) - 2rif(ri)). Since rf)2(r > 0 we have Illi2(D) Let T2 —p cc, r1 —> 0, we obtain 2 lIliL2f C8,. + flIL2. The intermediate case 2 4.3 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— — 1v1 = 0, v28 2 2i°eu )v20 (4.3.1) ‘ + — e(1 + 21v — 2eu 8)u = 0. 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 — + v = 0 r vIv20 (4.3.2) VI + U = 0. This yields the elliptic equation —v + v = v,281v1 u = —v” (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) + e2(r) + e1 — (26+ 291)Q = Ki(e,ei,e e1 (4.3.4) = 2 e(r) + e2(r) ) where Ki(e,ei,e IQ+e(Q+ei)—(26+1)Q 1+28_Q )2 + 2(v2820 — 286U — v)v29 2K(e, ci, e2) e(1 + 21v — J28eu)u. Define L the first order e1linear differential operator L : X — Y,? by e1 = (1—(26+1)Q 8r+ (ei e2) 28 Or 1 )e2 Then we aim to solve the equation Le=K(E,e) (4.3.5) where e = (ei,e)T,K(e,e) = (Ki,K)T(c,e). Let 1= (O,),u>0. We say e(e) is a weak2 X-solution to equation2 (4.3.5) if e satisfies e = L’K(e, e) (4.3.6) for a.e. e e 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 L_(1(28+1 8r+ 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 e1 L ( = 0. \\ e2 J Then —rel + e1 —(28 + e1261)Q = 0, e = —e. (4.3.7) It is well known (see, eg. [14]) that e1 = 0 is the unique solution in .1H Next we prove that L is onto. Indeed L is a sum of an isomorphism and a relatively compact perturbation: L= ( 288r-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 e1 + (ôr + )e2 = 1 (4.3.8) ôrel + e2 = and show that ei, e2 e By eliminating e2 we know that ei satisfies (— + 1)ei = —(ãr + (4.3.9) Define G(x) = (47r)_hIxI_1e_khi. (4.3.9) has the solution ei = G(x) * - ( + = 1G(x)*q1 +8,G(x)*b .2 78 4.3. Proof of the main theorems Here we have used the property of convolution and the fact (ãr + )*f(r) = —8rf(r) in .3JR By Young’s inequality and G,0rG 3),L’(R we have IIelIILp IIGILLi11111LP+ IIDrGIIL1II2IIL which implies ei E L. Similarly e2 satisfies —8r&.(—r+1+)e2= Let H(x) = G(x), then e2 —H(x)*(8rcii)2 =H(x)* =H(x)*+(arH+H)*i eL since H, (ar+ )H e2L’(R ). To 3 improve the regularities2 of ei, e2, we go back to (4.3.8). Since = — e E L, we have e1 E Regarding the regularity of e2, we know that —e(aT+)e EL. By Lemma 4.2.4 21= II8re2ILp 79 For Thus then Let where and available and for Let Y,? in (Iv 20 ’ a Fix any if Ki(e,e) us IKflIL4 suppose IKjIjL4,leta=v 2 =(Q+el) 2 ,b=ru 2 small K(E, e C consider e 3, q + K 2 (,ei,e 2 ) Ki(e, for Xi?. if to K(e,e) IuI 20 ’) ball e) p> < is 11L’K(E, = be K 2 . e = Recall Cge(Q C(QI G e 1 , a K(e,e) in chosen K 1 , 3. ((Q real Q. From e2) X. We E IQ = We the Li’. IK 1 (e,e)I + that = = 4.3. + constant e) choose + First IIK1WL later. (v 2 Lemma E(1 ei)2 + know Q estimate By ei1 201 ) K(e,e) + Proof + = — + +6) W”P(1R 3 ) - Sobolev’s we e 1 1 28 (Q Consider {e p 2I9 v 2 that IIeH,’) (QI C(IKl(E, depending 4.3.1, = must E — of — for IIKIIL 4 X; where eu 2 1 8 )u. 6/4 Q 1 — the in + K 2 + v 28 )v we —÷ ensure + embedding the ei) IIeIIx the e2)29 main e212 is L(R 3 ) know e)11L4 + — following. on similar. set =E(—Q’+e 2 ) 2 IITIIL (28 + + that theorems - e, + that 6}, l(Q — + 8, 1K 2 (e, K(e, Q’ 1)Q 2 eei it Since + L’K The + suffices ei)126) e) e2I 28 IQ e)IL4). same is — inLemma4.2.2, well Xi?. Q 2 +l (Q to + argument + show eiIB defined ei). that 80 in is By For We Next Hence i.e. on if For IK(e,e) K(e)-K(f)I 6 6 6, Sobolev compute < L’K D, IIKI’11L4, 6) we -. 1 we to — and let want obtain K(e, is obtain A IDIIL a embedding e a the let similar = is to contraction f)I Q 11L’(K(6,e) a small r.h.s. +(2O+1)IQ+flI28(el_fI)_Q20(el_fl)=D+D. show IIKNIL that < + = < C(IQ IQ+e1I28(Q+e1)_tQ+f1I29(Q+fi)_(26+1)IQ+fi28(ei_fi) fi, Q(r),o argument 4.3. IK(e) Co(Hei Co(ö+6 2 Iei enough that term and g < < = + Ger(IIQII Proof mapping. for Holder Ce(IlQ 2 ’eIlL4 C 8 (e 1 fI 20 ’ e 1 — by = — WK2(e,e)IIL L’K(e,e) - such K(f)I — can f1II,,i,4 e 1 any term. K(E,f))IIy of fi in + IIi,4 be the inequality, + that e, and Lemma 620+1) — We f Iei + After applied f’IIw + main + E by + Kj(e) have — E IIei 6) Q, IIei ft <2C 0 c5 2 fi1 29 ’)Iei use rewriting + 4.2.1, — and theorems to II) we IIe II61tIL) — of K 2 IIei K(f)I ö,e have Lemma then - < (with as fItx, K(e) — — 6/4 above, + fiI 2 . flIIw’4 4.2.1, similar 1K 2 (e) — K(f), then — condition I(2(f)I. 81 where small), We 9 for Notice Then Hence (Q+ei) 2 —(Q+fi) 2 , JE(e, if K(e) = ö e discuss < 1 sufficiently f)I let E(e, separately. for that and (1Q - + + IE(e,f)IIL us the DI K(f) f) C(b the rewrite (Q I(Q the Using Ia+b+cI study + - = last IDIIL - ei)2 + + contractive first (I(Q ((Q small. + - = ei)2 e,)2 line, For + Kjs(e) Lemma (I(Q (J(Q E(e, - c (1Q ei 292 I + + line + 4.3. = e(-Q’ - - 0 C(e° f’) 2 we IcI)IbI e For — —e(Q+f2) 2 + + f) E(-Q’ E(-Q’ > + — Ce 1 in Ia+b1 9 e f) 2 Proof applied 4.2.2, ei)2 + 1, Kj(f) to - property the the - + — I2Qfi we e(-Q’ e(-Q’ get Q’ e2)210 - - + + + second r.h.s. - - we of e(-Q’ E(-Q’ use ö)IIe e2)28 e2)219 — e(-Q’ + Lemma fiIlw,p the + Ia+cI (notice find + + for - Lemma is f?1 8 ’)I2Qfi f2)218 e2)20 and main I(Q - - + + easy + + two - I(Q e2)219 f2) 2 1 9 (Q 4.2.2. e2)210 + that the + fIIw — - to - different theorems IIei ei)218) 4.2.3. + + Q’ IaIi I(Q third estimate ei)2 (Q b, ei)2 - - - + We c - I(Q I(Q + + e2 9 )(e1 + I(Q can Set - - fiII• (ei obtain line, f’) 2 1°)(Q ei) 28 )(Q Ie f?Uei situations + + r(-Q’ r(-Q’ + be a - ei) 2 1 8 )(Q fi) 2 1 8 )(Q since ei)216) - let = taken fi)WLp — fIIw — (Q us + + fiI. + + f2)2I8 f2)2I8 + define sufficiently (e 0 f’). fi) + + f,) 2 ,b > (4.3.10) - fi) f’). 1 f) and 82 Lemma The regularity. following sufficiently Note and a Similarly, there Therefore Thus Simply Hence then Next can for unique Let e, not its continuity we 6 we exists we contractibility. sufficiently us assume be can 4.3.2 prove IF(v) fixed standard only we if see used small. This a 8 satisfy WE(e, can why a iiE(e,fHIL need point — w.r.t. that that (see + since 0 1, is F(u)I prove IIKl(e,e)-Kl(E,f)ilL Then b such f)IiLP ia+b+ci smail. lemma: a done all 11K 2 (e) to + 11K 1 (e) /2]) E(e, e(e) solution ci 4.3. ib’ cl-Ia consider the This follows that the that Let by f) E Hence Ibi, condition Proof - — C 8 6 112 contraction = 1 completes using is C 8 Ee + F K 2 (f)Iy4 — Kl(f)ilL 4 we of IcI°’ which from E(e, contractive bi : Ia+bI— equation we C + have of - lie a Iui)Iv —* — f) the Ia the have above = standard - is fiiw” + C if 1. fIiw’,4 the main a mapping continuity ci lal ia+ci+ satisfy In for weak Iie Iie (4.3.5) 1 — + by for proof is iIe-fIiw. (4.3.10), uI 1 al small, theorems choosing - < bootstrap — 0 <8 solution Iie 1 F(0) = fii’. fliw’,4. for iIe theorem of which lal C(ibi w.r.t < 1 Theorem — i.e. all = if directly. - 2, = fII 1 ,”4. 6 IaI 0. fWwl4. if + is of 0, e = argument u,v implies of at Id). in equation C 8 e arid the max{51bI, < X’ 4 4.1.1. E Lemma 5 and assume C. map max{51b1, L’ has taking and LI (4.3.5). L’K K more 5IcI}, 4.2.3 that has the 83 e Slci}, This Next Lemma So Since and Recall and Proof It Let follows K(ei,e 2 ) by L gives from el,e2 Lemma is that of 4.3.1, an Theorem that that Lemma e el, isomorphism W,’ 4 (R) we 4.3.2, K(f 1 , if e2 u have satisfy 4.2.1, f2) L, VK 1 xVK 2 e 4.3. 4.1.2. 1K 2 1 (ei,e 2 ) -=+-, r 1 (el,e2) IK1I IVF(u)IILr Vu Lemma from e2) ( Proof L°°(R 3 ), el,e2 p First E e 1 (Iel—flI+1e2—f2D. e(IuI E C,(IvI 20 ’ E L, X 4 p<+oo q 4 p + of 4.2.2 E + we — fl < fl — then Q’1 28 4 p the 4 p then L vI 2 ’ fl can Y’. fl and main VF(u) 14’’ + ( K 1 We prove + Iei 28 LxL. Lemma + K 1 IuI 28 ’) x theorems x x know IuI 26 ) W,’’ K 2 that + L T 4.2.3 e2 2 ° that and x + L f 1 I for + p f2 28 ) 4. By 84 Thus with where We If if g Since find generality, where know theorem. Going and are = ii the We Moreover classical prove is e g(ro) g”(r) el, claim h’ small f uj,6 back — conclude = €2 e2 e 1 ” h, > Taking it = v(_e_(r_r0) suppose are then is E to < enough 0 by solutions. — — el(ro) we not W 2 ’ 0, is e2 e 1 equation classical g” an g’(ru) that g arbitrary show = = true, derivatives satisfies application e — and cri(r)e 2 Si(r)eu such el(r)I (1 4.3. there (ro) (1 = then that + solutions h(r) + + 1o’I, (4.3.5), that 0. 6 i)g = g(r)O /3e1(7’_T0)) Proof + 1 3) and + exist But + — g(r) 2€ je2p2P = ei, e2(r)I 6 < in 6 2 (r)eç u 2 (r)e + of (TO e2 this 0,g(oo) 0 we 52g (4.3.2) of obtains constants and + the < have is Ce(i the know < contradicts for < + + ii sufficiently + + maximum eu, Ce” T zih (1 main 6 3 (r)Q < and 3 (r)Q < + = exponential II maximum — rro, that 1e21 0. eT_T and 1,2,3) 1 r 0 , after Thus is for theorems < v(e), 0. el,e2 + Q to with C€ r principle. large). for for öu for tedious > be we C(e) by at + h, r r r 0 . E equation decay r claim r determined large large Sobolev’s then 5 3 1)h W’’ large. = Let positive computations r 1 Without at that C and (4.3.13) infinity. C 2 . embedding such g(r 1 ) later. So (4.3.13) (4 (4.3.12) loss (f,g) since that 3 > We 11) we 85 0. of If where We mined Then in if E(ei, not and Remark. Letting Thus Then for the claim E a r IE(ei(ro), want right valid, small e similar large Iei(ro) fi) under similarly is h later, E /3 true = H 2 . hand = to IE(ei(r),fi(r))I enough. —+ Let ,,/jQ’(ro)l. For enough. ( way. know this — we 0 if (Q+ei) 2 —E(Q’) 2 This fi(ro))I us fi(ro) side , 0 we z-’ E(ei(ro),fi(ro))I assume we ansatz, < < Once consider whether can completes Letting ei(r) of 8 have ei(r) /1 Q(ro) Q(ro) The ei(r) 4.3. < = = (4.3.13) Then show we — that 1, sh [(s 2 exponential C + + a Proof our have or r 0 —h(r) special ei(ro) fi(ro) that + h(r) Iei(r) and the not 10_ large is method 2s)° (5.2.5), positive of = IQ+ei proof = then the = = if e_T_ — the example. h28Iei(ro) enough if - < r decay ‘/IQ’(ro)I(1 i/Q’(ro)I. ((1 following r fi(r)t, Ce” T . is does main of 120_ it is + evaluated large Theorem is large s) 2 e estimate I(Q+fi) 2 —e(Q’) 2 1 8 +IQ+fi and not obvious + theorems Suppose enough. /3eu)(T_T0). inequality r enough. — work s E 1)]h 2 ’ + = - at (0,oo) 4.1.2. s), that for fi(ro)I. e, r since e2 = e2 o = Iãrej(r)I L is r 1 . > = can f2 Lemma true g(s)h 2 ’, 0 Therefore = to be 0, be obtained < then (4.3.14) 4.2.3 Ce deter 29 )(Q+fi) the 86 is not since In So and since We fact, claim hold 8 + we for cx(8 that have every — if > 1) cx> Cs 8 ’h 28 <0. r (1 4.3. (O,oo). + The s) 2 ° s Proof then = claim h 8 >> (s 2 CIQF(ro)j 28 E 8 + 0 _l) — g(s) + of is 2s)° the , 2 proved Cs 9 main > as + Cs 8 s 20 ) and theorems —* < consequently, 0. >> 1 as (4.3.14) e ‘. 0 does 87 [10] [9] [7] [8] [6] [5] Bibliography [3] [4] [2] [1] Amsterdam, Iberoamericana Phys. J. s> cles. matical tions. RI, in sical and Equations: (1995) 176 excited H. P. 5. F. T. M. M. T. M. A. Duff. Mathematics, 2003 1. MERLE. CAzENAvE. MAcHINHARA, (1988) MATHIEu, the CAzENAvE, OuNAJEs. In: ESc0BEDO, J. nonlinear BALABANE, F. Differential Rev. SIAM states Eq. Sciences, EsTEBAN, nonrelativistic RANADA. Quantum D A 74(1), Reidel: Existence J. for V.30, Variational Perturbation Dirac T. Math. 19 Semilinear L. New 10, a L. and T. 50-68 theory, Classical (2003), nonlinear F. E. No. VAzQuEz. K. 1982 VEGA. filed. New CAzENAvE, York; Integral Anal. MoRRIs. of limit 8, NAKANISHI, (1988) stationary group, Approach. York 1835-1836, 179 Commun. Schrödinger nonlinear American for 2 method Dirac Stationary A (1997), Equations. -194 Existence semilinear the Existence University, fields A. field. nonlinear states for Commnu. Math. T. Dirac D0uADY, 1984 338-362, Mathematical and equations, solutions Commun. a OzAwA. of Vol for condition particles class Dirac field localized Courant Phys. Dirac nonlinear 13 Math. models F. of (4-6), equation Courant of Small 105, Math. noninear MERLE. (editor for equation. Society, solutions the Institute Phys. 35-47 707-720 of spinor Dirac nonlinear global Phys. extended in Lecture A. 171, Existence Dirac Providence, H(R 3 ) (1986) 0. for Rev. of equations. solitions. solutions 119, (2000) 323-350 Barut). Mathe a Notes parti Dirac equa Mat. clas 153- for 88 of [15] [12] [14] [13] [11] Math. ematical Focusing tra turbative ergy. L. S.-M. M. M. C. of VAzQuEz. Phys. SuLEM, linearized S0LER. S0LER. Gen., CHANG, Analysis and and Rev. Vol. P.-L. exact Wave classical, classical Localized operators Dl, 10, S. 39 approaches. SuLEM. GusTAFs0N, No. 2766-2769 (2007), Collapse, Chapter electrodynamics stable solutions 8, of 1977(1361 NLS The no nonlinear Springer-Verlag, 4. Phys. ( 4. Nonlinear solitary 1970) K. 1070—1111. of Bibliography NAKANIsHI a Rev. -1368). nonlinear for spinor waves. Schröodinger 3424-3429 a nonlinear field Berlin, SIAM AND spinor with T.-P. (1973) Journal 1999. field. spinorfield: Equations: positive TsAI. J. on Phys. rest Math Spec Self- per en 89 A: S. linearities. with solutions We where where for In Instability 5.1 for Chapter Instability this are By the define A version (x,t) (.,) solitary Pauli a chapter nonlinear Introduction real nonlinear are While is of of matrices: parameter extensively = 1 solitary the : this R 3 wave, we Dirac Hermitian by /0 chapter x \o /12 show 5 waves shooting or o)’ iN equation and —12) standing — the studied will 0’\ for = C 4 ,m>0 (x,t) j=’ of (x) inner be instability nonlinear (Q) 3 ‘Y 2 method submitted =i k in wave = product decays /0 Dirac solitary = [2, Dirac e_t(x) and = 4, [2, we ZN of o)’ for 5, 4, as solitary mean equations. in publication. 9—12] 10] lxi C’. —* and 3 a for =o waves, solution co. equations k variational /1 a Guan, The large waves 1,2,3, () or —1 0 existence of M. standing variety the and method form Gustafson, of of (5.1.2) (5.1.1) waves non such [5], 90 suggests stability, But variation does the below. be that method tive metry where Shatah is ear w of in is E() and near sufficiently structed in form of the —, ities Gordon e unstable. solitary the R 3 . a positive However We PDEs second authors Strauss very this (0, eigenvalue, the with the not that be form that f, The m). and left equations to ground that g the method Dirac in have just general operator the such state waves are near variation Dirac and an Guan gives Strauss spectrum [9]. (5.1.2), [7, proved energy there This the form like a real open operator 8] as bounded d”(w) a Stability states local (at this ground setting, cannot the proved in equations. kernel Klein-Gordon, (5.1.4). and issue JI functions, are also there functional, L() least problem for [9] state of solitary minimum of <0 are = some nonlinear refined w came the Dm Dm has be states of a which E”(b) away of = up exists e unstable Werle gives general where 1 dimension applied f energy a (m 5.1. is been to and partial about wave up Bogolubsky IkIdx). solitary the in (Dm) —i from 8 and — can at rise symmetries) infinity with if(r) g(r) < if(r)cos — [161, and o, Schrödinger r, [9] Introduction d(w) extensively orbital the result if dynamics L”(q) 2, functional , be to L() the to are zero. rn) results they Schrodinger or this Strauss sin = ground waves Dirac one Their applied = a stability with unstable. y 0 yJt3j many (‘)&, are (—oo,m] solution in be E() ‘Pe stability as exist, Then in and [11] the standard operator to the about means assumptions for some (in [1] equations. states. and studied as to localized + the have to since there — charge sharp instability required later of the m 0 traveling a which fl ‘ Recall and wL(). and Vázquez show ej rest necessary the the {m,+oo). that only case The > the directly. spherical is is times. wave for instability instability related solitary of application 0. solutions exists a that not for any energy Grillakis, one of continuous the allow condition nonlinear the These waves criterion in e equations. solitary bounded for initial condition = simple Otherwise positivity and spectrum [14] Contrary to coordinates waves the m—> nonlinear functional of are condition condition for the remains claimed nonlin Shatah of in second datum (5.1.4) (5.1.4) (5.1.3) Klein- waves of nega curve every sym from con [13]. this Let the for 91 to to of 0, it eh/ 2 v(/gr,t), the In Here equations. But where to of Q and Since at the is a nonlinear infinity [9], For ground eigenvalues Q(x) smooth coupled this Introducing f(r, d’(w) we equation t), is = used (see states g(r, not and Q(IxI) II&11L2 = Dirac system then f —L() ( [6] rescaling of I t) positive, -wf a (x) (5.1.1), two d”(w)=—I(&Il 2 iaV and the are rigorous (u,v) satisfies equations = ) rQ+Q=Q 29 Q, new = the - - - Ce C-valued linearized + the = —(II1I)2 t) iOtf ( the ° and solve E(0U problem decreases functions = references (x) i(x) the proof. e_t 5.1. solitary + perturbation is the 0rg + nonlinear 0(e), functions. harder operator. ) + Introduction with coupled - monotonically, (u, The - - = ll2II,2), — if(r, waves if(r, g(r,t) therein). (lgI 2 v) (E(x) (IvI 2 nonlinearity than question such e(x) elliptic t) t) - Then system arguments. In have IvI 2 — II2II2 cos sin 1<6<2. ( If then that as EIuI 2 )v that general I 2 )f We re’ 1!et —eIuI 2 )u, equation the equation if of of and f(p, have + = I 0> form mf. nonlinear + stability 0(E). the decays t) Let v). ) I°&have , = (5.1.1) + stability we e Eu(/r, 0(e), = exponentially is have Schrodinger m the is related — reduced analysis t), ansatz (516) w, 517 515 g(p, p 92 to = t) = where with where the NLS can with eigenvalues Furthermore To When we linear A Where and (uo(p), prove — obtain The eigenvalue verify Q(x) with this Re Schrödinger A, e without vo(p)) —e(9r 1 idea is A the — expansion, = the the that A small > e(1 and Q(xI) instability, to solitary satisfy 0. + linearized following are A + loss A find IeMI eigenfunctions L=-L+1-Q 2 ,L+=—z+1-3Q 2 , Then enough, j f is 0 0 vg + two-by-two equation. satisfies is of 8vo 8uo+—(v—eu)vo+vo=O, related — 2uOvO) A(e) it the wave generality, vo=Q+e 2 , 3eu) we is linearized it + operator ground then useful with needs can 2mu 0 A=(02 11e11i, Theorem to can £= 5.1. Linearizing the —1+e(1+v—Eu) matrices. positive say as be to to we — state for (_ around Introduction eigenvalues —A+ operator a block-diagonalize (1 IIe2IIHl proved show that take E(8r+) some power 6.1 uo=—Q’+e 1 , of + real 0 0 The m solitary in that v the the °2x2 its —) A) to = [8] series — 0 part cubic, ground system formal A be eug)uo < . of The has is nonlinearly the wave of < focusing first states expansion 2ReA. (5.1.7) e. ground AE an cubic, = —E(1 is In 0. eigenvalue to into ôr linearly order — formally — around focusing, and states A 3vg unstable 2 euovo 0 0 implies with radial to + unstable. solutions A eu) proceed (u 0 , expand = (5.1.9) radial 518 if non that A(s) vo), we 93 0 0 we dimension where nentially Since Lemma satisfy of theorem ing satisfying which Lemma eigenvalue the and 8 used theory. the and with solution < nonlinear Now say An Before result eigenfunction formal e 1 ,e 2 let 2 to Q is or the important the conclude we (f,g)(E) in is known 5.1.2 at 5.1.1 < (b’)’, by = have n introducing i nonlinear expansion smooth [9]: are ground infinity, 0, ilejilW2 = Schrödinger _)2 are the the 3. the ready exponential [6] [9] to that “Lyapounov-Schmidt U admits Lemma /11 If first Suppose nonlinear Let and C((O,eo),W 4 (R3,R2)) be state then f(r) g(r) elliptic <0, 1 of to there the unstable e + positive, pair the state = = = the 112 equation is addressing main LrQ+Q=Q’Q, m decay e(-Q’(vr) e(Q(/r) ). exist < equation eigenvalue linearly 5.1. for = eigenvalues Dirac — solitary is the p w. result [15]. an some decreases < such as Introduction main = There equation is eigenvalue 1 lxi unstable. the + Notice as waves of an reduction” and — C + theorem , follows. this of exists instability eigenvalue. e1(/r)) = = monotonically, 00. satisfying + £, eigenfunction with of C(O) 0, that chapter, then e2(/r)) of (A 1 )± n+2 Eo the In in £ nonlinearities = Let > A method j this form this (5.1.9) eo( 9 ) of > 0,j = is Rather, is let Q 0. the real +/‘. chapter. chapter, not = (5.1.3) = us and > can with 1,2, from ground Q(lxi), recall 0 and seif-adjoint, we not decays and correspond bifurcation with the will Therefore the the easily states a x unique lowest verify expo space E (f,g) main 1 W 94 be so Q then the A tion operator are In theory carry In but 5.2 U, a Theorem in (u1,’u2,u3,u4) T . operator column convenient V Lemma this this To In C-valued real in (5.1.7) equation in out this study fact tell section, chapter, Spectral C 4 . and i-a-- A A. the vector 1.1 us chapter, 5.1.3 at this The has imaginary 5.2. the functions, way =th,e =8h same that (uo, Then (5.2.1) are E(IVo E((ivo we theorem we usual We an stability in For to VO). linearly Spectral present A + only argument eigenvalue + we C 4 . a + denote study analysis becomes + -— dU e the Let formal Hermitian hi 2 (f,g)L2 U — e\/E parts. use hi 2 then Sometimes consider is e(1 = = nonlinear u(p, of unstable — + equation a still — analysis the AU by (Ree,Ime,Reh,Imh) T , eiuo the block-diagonal Eh the + eiuo expansion to Let a t) = A + (U,V) ivo true following — system prove ground = + perturbation + with o(E) f product Dirac for + e(ivo forE + nonlinear uo(p)+e(p, e1 2 )vo Dirac for e1 2 ) we (5.2.1) for (f(z),g(x))d 3 x. hi 2 is positive the the it. + with the equations also sufficiently one — — state of + hi 2 notations. equation Hermitian in eiuo (vg is linearized eigenvalues e((vo) 2 representation of L 2 write linearized a to solution, — t), real (e, four-by-four the + EIUo (1R 3 , terms write 1 v(p, h) e1 2 )e— < with part. the (5.1.1), eigenvalue — small, C 4 ) + satisfy U product U E(Uo) 2 )vO. t) operator functions ei 2 )h— < column nonlinearity one and = = is 2. i.e. vo(p)+h(p, denoted the of The matrix the linearizes operator the of U U4 U2 Ui the of ground the vector system e, readers two bifurcation A. linearized h linearized operator by denotes in vectors (5.2.2) t), states equa U their can e, 95 as h For Theorem Moreover, is numerical where algorithm Because where and u(A), the such spectrum that A AE= more is dynamics, such block-diagonalizes that a especially —e(a++2euovo) first 1E(1+V—3EU) convenient of kind M’ of [3]. AE computations 5.2.1 its order, this 5.2. A +i\ and of We we = connection — operator of 0 0 — MT non-seif-adjoint There know adopt Spectral AE (—1 to the non-seif-adjoint (—1+e(1-i-v—eu) IV.IL ),fl e(8,,++2euovo) has study. = the eigenvalues + zero exists M, this of e(1 AE, the A operator to e(8,,+) M— analysis eigenvalues where is ?,f method + AE ‘‘—As the same the —1+E(1+v—eu) 0100 an an v — block-diagonalization — operator. eigenvalue I stability orthogonal I’ \ is 0001 0 1000 VO 0 operator. spectrum, A, for 3eu) hard to I’ 0 + the 0 0 reduce with 1 =0. because linearized problem, 0 E(1—3v+eu) e(1—v+eu) — 0 The and —0,, similarity / From u(A) the A + of analysis —9,, Chebyshev to the 2Euovo its = —e(1—3v+u) our a operator cr(AE). four-by-four new is phase Op2euovo transformation often interest of operator 0 the 0 interpolation invariance used spectrum is format. in in which fast the M, 96 e(1—v+eu) of —8,, 0 0 with Proof. with diagonalized the Theorem The almost is g(AE). Then is formation Proof. ( Since Thus, Zt equivalent a u Because localized sum ) following U 1 , , e we M For the then D(A’) M U 2 of The is conclude to 5.2.2 entire the is nonsingular, the of eigenvalue to 5.2. terms A nonsingular eigenvalue C 2 . Theorem are theorem continuous operator (A’) 2 U 1 to imaginary The Therefore the Spectral u(A 6 ) that yield VO, = eigenfunctions spectrum I o u(AE) problems A’— 5.2.1, identifies i[1 then = MAM’U AV=AV A i(1 —ir(Du A. problem = since since AeU=AU, — spectrum ?U 1 , analysis — u(AE) and — axis. it The , ‘\—iE(0p ( we = e)u — — suffices they IM co) of g(A). a i(1—e) change I (‘J2x2 + \Zi eigenvalue the (A’) 2 U 2 + relatively A’ AEU = U u) = for i8v A! have u(A), = with i(—oo, {ir, + continuous 1. V=M’U. to AeU the — = from UEC 4 , We ‘-‘ U2x2 n = r find iev exponential eigenvalue = 1 uU linearized e Au, —e], A! compact = problem can u 2 , by —E iãp IR, studying = the ,XU. is Weyl’s Av. 0 r apply spectrum equivalent spectrum > A perturbation, operator decay << to such the lemma studying 1. similarity that of of at to we A’. A infinity. the A’( can u(A) Suppose covering (5.2.4) (5.2.3) block trans drop ) AE 97 to — o seeking Hence For and then This We then we On Since Multiplying have On one claim any u, v we is v (A’ the the ( hand, f have satisfy has H’ ‘ that = — other first the AI)’ only by ( ) 5.2. = if E ‘ the the part second A L 2 hand, continuous ) Spectral spectrum is e x standard E, u= of invertible, L 2 + L 2 1v= (A+iE)(A—i(1 equation it (A+ie)(A-i(1-e)) A the if -‘ x satisfy A = — implies A L 2 (p 2 dp), = analysis theorem. (A’-AI)( i[1 i(1 e of spectrum (A+ie)(Ai(l6)) regularity i{1 il D, — A’ — which the by — 6 e, we that 2e) is e, + co) A for )(aPvf1) following on want let oo) Define ± — is u U —e)) (—oo, i(1 the iV’l argument. the U i(—oo, a ______ — to i(—oo, contradiction. - linearized imaginary + e) 0], (A = (-oo,0). show 4ev 2 and - —e]. let EL 2 , —e]. i(1 Since that using - operator axis A A the Thus e and o(A’). first i(1 u(A’) — equation, e), We ç then are a 98 value The Therefore Moreover, Hence Let Since and then Since e In c)( b)II(A a)II( v continuous H 2 with order A = V 3 Ui V 3 Ui e with / II( we —A)( )—*0weaklyas--oo. positive , )IL2xL2 to c 5.2. IR—/ 3 )jUL2-—O have IIjIL2 then prove u(A’) II( spectrum )tI2XL2 IIOpjIIL2 V 3 Ui Spectral real /i II(A—A)( 3 = )IIL2XL2—*OasJ--*oo, the and = )II2XL2 E 1, + part. 1 = instability, (—oo, the such of = = — (IIiIIL2 analysis E (, (,—qj) A 6 i[E, claim ( Ui that V 3 0] covers V 3 (- (1 co) = )II—>0 and )eD(A’) is and for we fl o-(). + proved. + + i(—oo, almost will the /3)) u A j—O IA - as = linearized By show i(1 —El. + A the By Weyl’s j—*oo. (j, - A 6 relation pure as E)j2 -/ 3 v) has operator j—oo. E II8PJIIL2). lemma, imaginary = at H’(A 1. (5.2.3), least there one i(1 we axis. eigen — know exists 99 of But a infinite that and where we To Proof. Theorem value where and more cubic, find can first A eigenfunction of of v dimensional subtle AE. be is the Recall proceed the focusing not written 5.2.3 /E 7 Hence eigenvalue form A 1 5.2. relation a standard = with is For and A 6 Spectral as and UE taken —1—Q 2 —O— a is e radial with a A A power = A 0 = the sufficiently a formal 0 0 AEU€ such perturbation (_i,mi,_ 2 ,m 2 )T+o(l). singular from = ( A— 1 analysis the essential A 0 NLS ) series — that = —A linearized —1+3Q 2 0 000 000 1 Lemma + expansion v) =eii+O(e) —L ( 2QQ’ EA 1 perturbation ( 0 small, 0 0 for of 0 spectrum 0 7o2 ‘io + e, —1 of the 5.1.2, 0 e 2 A 2 U L_ J operator A A 0 the 0 of 0 1-+-Q 2 O+ linearized = —9 e = eigenvalue 0 0 0 + 0 0 operator and is 0. C 4 , AEIo of ... dramatically ioi, A 0 . around 1—Q 2 operator 7o2 0 0 0 It A : the and turns the satisfy admits kernel eigenfunction. different ground out an of there (5.2.6) (5.2.5) eigen A 0 state from 100 is is Then for We concentrate following (5.2.11), Substituting Formally where exist It (U 01 , is some will known U 02 , v A 0 = let determine functions can sequences expanding U 03 , (A 0 on U 1 that into 5.2. be (1 U 04 )T, — = A 0 + with ,\ 0 )U 2 L=-z+1-Q 2 ,L=-z+1-3Q 2 . taken £ equation (U 11 ,U 12 , = Spectral Q 2 )8p , is of .\o, (1 ), 0. from i (A 0 Rev>0 unstable problems + + Then as A 1 tIE u —r Q 2 )7 (A 1 H’(p 2 dp) — u 01 +uç 2 —A 0 u 03 =o, A 0 u 01 +u 03 +uç 4 =o, + either equation from U 0 = (5.2.5), analysis U 13 , ?o)U 1 2QQ’ (A 0 U 01 — U0+EU1 + = (L + + and (see, ..\ 1 )U 1 (1 U 14 )T, are those Aiãp 0 — (1 = — + — and or + Ao)U 0 io 2 , — —U to for (5.2.10), obtained (A 1 3Q 2 ) e.g. Ai 0 pil + Q 2 )17 ±i. = = = be +E 2 U2+” sequences we = upon the (A 2 = — 0, 0. ( [6, U 13 determined = U 03 But obtain A 1 )Uo 0. = linearized iio = = 0, — 15]). we collecting + —Ai? 7 . = A 2 )U 0 we ) au 14 , U 11 have —U 4 . = of E From will 0, C 2 — equations. = operator later. ignore We such 0. the lemma take powers From that A 0 5.1.2, Let = equation of (5.2.12) (5.2.11) (5.2.10) ±i (5.2.8) (5.2.7) (5.2.9) e, U 0 there and 101 the = where The and tion nonzero nite bifurcation As equivalent We exists El) determined and For It From allows with we The will the the projections dimensional (5.2.13) the a mentioned proof small eigenvalue components. corresponding Rev not us second to theory. by to into push 5.2. solve > depends eigenvalue choose the / —L split 0, kernel and the before, 0 Spectral the with leading problem Let U equation range The fourth from = on eigenfunction perturbation L/ A and 0 leading Ojj (—,mi, A 0 = NA 0 U the idea A 1 =v, = order. N_/0 N_Il it analysis space (5.2.7) A 0 U NA 0 U (5.2.5), cokernel components, is A 0 ) — — “Lyapunov-Schrnidt (5.2.13) to a order \0 + ‘\0 = = very solve and Next i EY+O(E) EA, = EN)’, =i for 2’ the ()=m. theory 0 0 1 0 and eN(). complementary eigenfunction A degenerate into equation a the — leading 0 0 0 1 rigorous = 1702) A)U. we produces linearized 0 two 1 0 0 A)u, — A, further have A)U i =rAt + and equations: order (5.2.13) o(1). proof operator. reduction” only since AU (—i 1 , operator space term shows the is = the to 701 of A 6 U. of first It project method eigenvalue eigenvalue indeed N, has Then and where (5.2.15) (5.2.14) (5.2.13) an equa third there from infi it 102 is is is with Let U and is Now with then We and Now can J bounded is = will BE be the invertible it (NTo equation we B 0 is reduced first inverse use = ready = Zk=o (I + (I because this solve —8,. 5.2. NTI3) - — and (5.2.16) to to of eJN(A NU=(U:=a, eJN(A equation 6 k solve u1+4 11 3 +U 4 This equation Spectral (I (fl J 1 of (JN(A — N(A the (NT can eJN(A c = is - — ) in the —J, )N T ) 1 U to )N T )o It6JN(A exponential ) — = be + analysis (5.2.14). terms — = = write - NTB)I3 0 (NT solution written )NT)k then NTNU — ( ( .)NT) —u 3 —t4 of = uj, = i\1 (0 — we = for + 3 BL3, Let A)N T II (—8 ) Ek as U3 NTBE)/3 and decays for 00 (8 + get NU=(U2:= the —1 + into 0 exists, U NTNU, in eJN(A equation + , = linearized terms + ) (JN(A <1, equation eJN(A . (ul,u2,u 3 ,u4) T , of 6 JN(A \tL4J The u, = - N(A)U of 0. (5.2.14). v 0 . operator )U. - — u2, operator — (5.2.15), )NT)k. So tL4, A)NT), for e JN(A— Now and small then we we A. plugging have (5.2.17) (5.2.16) (5.2.14) have enough Since )NT 103 where projection Equation To hence Recall For Then Let (.,.) (),3) solve this us is equation we Q that identify a solve /3i equation, standard = (, can (5.2.19) (i onto I ,8), the 5.2. — - take the Q. e( (5.2.17) , we equation is inner Spectral Therefore after /31 leading — then - use solvable A (1 = v)’( ker(L_v)*= = (, becomes — AV+EA, computation, a product (L l.h.s.(5.2.18) A 1 solvability v)/i order, - analysis (] + if v)’(] = := (L 6 A, = EN in — + + L 2 (C 2 ). v) 0 )(77o for )) ((S condition. BE the 1 3 Th+E/ 3 1. (m2 = + the — = = (t. + )(o + = l.h.s. A)N T E ( 0, linearized Denote e)) — )(77o _ 8 p ) - =. + can = Since v)’Q(E + E/3). + 0, — by be E/3i) A(AT T operator Q simplified the + + L 2 -orthogonal NTB)) as (5.2.20) (5.2.18) (5.2.19) 104 and has Therefore then We theorem, Therefore, we We L_ computations, theorem. Next It A) yields is have is claim first an eigenfunction G(0, bounded, we nonnegative eigenvalue use a show for Let A 2 ) that function the if the e we 5.2. = eigenvalue that small, we (,o) A for solvability 0. choose G(e,A) of =Eli+E 2 A 2 , find Spectral Finally and sufficiently there U the G(O,A) there in = = > = Q A 2 terms 2Re(oi,o 2 ) = form problem 0 (jT (—,mi, exists —-(0,A 2 ) ÔG is = if condition since exists analysis• the - (0, = (L_io 2 ,i 7 o 2 ) (K + of small (1oY’(?7,]o?7o). A(?,17o) ground A 2 NTB)(o r, A 2 A — A 2 ) A, such = = for 102 AE)U e, = (5.2.20) and/3 1 5 (e) = (71o,7o) state the + that * 0, 2(Re = 02 )T > + (i,J 0 ). with we linearized = 0 0. )II + to [6], G(0, 3 1 (e, conclude E/ 3 i) obtain + A(0) 0 <1. L_Q o(1). A 2 ) .), = operator A = since A 2 . by by 0. 0, implicit implicit In After since (—v)’Q(N+ fact, io2 a function function series CQ, 105 of The The 5.2.1. formal linearly asymptotics instability 5.2. Spectral is expansion a analysis direct result is for verified. the from linearized Theorem operator 5.2.3 and Theorem 106 [11] [10] [9] [8] 16] [7] [5] [4] Bibliography [1] [2] [3] J. mitted, 94, 74, tions. Self-Focusing Equations: SYSTEMS, sical metric (1995). states (1988). Waves Waves M. M. M. T. M. C. M. I. M. H. F. Duff. 308-348(1990). 160-197(1987). L. CAzENAvE, Balabane, SuLEM GuAN, J. GRILLAKIS, GRILLAKI5, MERLE, nonlinear CI-iucuNovA, OUNAIBS, Differential for B000LuEsKy, in in First-Order Eq. http://arxiv.org/abs/0812.2273. ESTEBAN, the the a A 74(1), Vol. Soliatry AND nonlinear Existence Presence and Presence Variational T. Dirac Perturbation 5, L. J. J. and P.-L. 50-68 Cazenave, Wave E. No. D. Coupled VAzQuEz, SHATAi, SHATAH, Wave filed, Integral 1979, Dirac of PELIN0vsKY, of 1, of (1988). Collapse, SuLEM, Approach, Symmetry, pp. stationary Symmetry, Commun. Solutions Phys. System, A. Stationary method field, 66-83. W. W. Equations. Existence Douady, The Springer-Verlag, Lett. STRAUSS, STRAUSS, Commun. Commnu. states for SIAM for Math. 11*, J*, Block-Diagonalization Nonlinear A solutions Nonlinear F. a of Journal Journal Vol 73, for class Merle, localized J. Phys. Stability Stability pp. Math. 13 nonlinear APPLIED Math. of Schrodinger (4-6), of 87-90. of of 105, Berlin, noninear Dirac Existence Functional Functional solutions the Phys. Phys. Theory Theory 35-47 707-720 Dirac nonlinear Equation, DYNAMICAL 1999. 119, 171, of Dirac (1986). Equations: of of for equations, of the (2000). Analysis Analysis Solitary Solitary 323-350 153-176 excited a Dirac equa Sym sub clas 107 [16] [15] [13] [14] [12] 35Q20 spinor mun. Schrödinger energy, M. J. W. J. M. WERLE, SHATAH, Math. I. S0LER, fields, STRAuss, (78A45 Phys. WEINsTEIN, Phys. equations, Phys. Rev. 1981, W. 82A45). Classical, L. STRAuss, 100, Dl, Rev. Acta Modulational VázQUEz, Chapter SIAM 2766-2769 173-190 D stable Physica 34(2) Instability J. 5. (1985). nonlinear Math. pp. Stability Polonica ( Bibliography stability 1970). 641-643, of Anal. Nonlinear spinor under B12, of 16 1986. ground pp.601-606. (1985), field dilations Bound states with no. States. of positive of 3, nonlinear nonlinear 472—491. Corn 108 rest decomposes estimates of system potential tion different try. solutions. harmonic the to given proof dinate function, dimensional butions Compared among Landau-Lifshitz the map Lifshitz around in maps. — This 6.1 Chapter Conclusions potential. R 2 that In flow “generalized on the The heat thesis to of is Chapter systems” My the (2.3.3) radial the Summary of to flow double global analyze proved coordinate equations but 1/r 2 map main are A flow, with the the few thesis ground focuses the linearized manifold, Usually and a functions. used can are linearized field component, 2, existence to more end map achievement most as the its flow hasimoto in we studies nonlinear be consider obtained, well. 6 states. can to on spectrum. systems the points of single into studied used complicated the of is replace Dirac study. the or not energy given, the For For for a double operator a to the transform” equation space-time stability nearby the and bigger be yield Dirac due equivariant current Schrodinger tracking operator Landau-Lifshitz the of For bounded It the uniqueness and described space this produce to turns end nonlinear more coupled Landau-Lifshitz equations. harmonic class double of the finite analysis for research chapter points the (see the of around estimates out complicated energy a domain, of near-minimal maps an maps time-varying scalar by Nonlinear time [5, endpoints system. an and maps Dirac equation estimates map of is a flow 8—10]), is the unstable There inequality to single continuity [9], localized blow-up to focused for function in and ground equations, plus study with use flow the In equations. H the Schrödinger which estimates. are (2.2.4). for equation my energy a do parameters with weighted two eigenvalue equivariant including whole perturbation. on the for solutions two states and not (2.1.4), a thesis, of will different the special a degree my flow the the major higher critical-decay hold In plane. One for is compact remove space-time research The stationary this a positivity these equations harmonic equation. explicitly is s(t), Landau even rigorous a symme class is m related contri degree “coor scalar equa other using Thus This c(t) = two 109 the for 2- of is 1 is properties solutions be interesting play is equation to many for dispersive sin whole We of them for only sumes imaginary compared tion is the to of the from adaptations In is m dinate ary of [17] (3.1.3) one the solvable a the the those this Chapter = make Q isomorphism, small In In a disc are condition subclass necessary who of able an 1, 0. is have class Chapter researches stationary imaginary Chapter that initial discrepancy nearest systems is singular approach looking we the very domain, of an For to is energy have first to to estimate parts of restricted and zero problem reduced proved 2, of nonlinear understand unique appropriate of deal equation m data (3.1.1) important the maximum q(1, nonlinear condition 4, uses harmonic disk exponential to 3, for solutions admit is = of by the the axis. with on solution. and in ground a show neither t) must 77 2, that solutions ground between for in the model to the for uniqueness [16]. is the = is borderline 3 ourselves Schrödinger the (3.1.2), R 2 local a a When the non-self-adjoint, r. we not usual more that be perturbation the choice principle the role simpler for Dirac unique maps. existence becomes states to an For nonlinear By equation conjecture linearized state made The decay. preserved existence finite our the solution time-varying future with eigenvalue in about one Strichartz this 6.1. the a to of equations for the and But linearized whole Q solution, symmetry system. result if to and studies study s(t), the domain harder. hyperbolic energy equations. m> of singularity of Summary So the research. for analysis. continuity construct terms the operator is for form comparison solitons that method the by a(t). plane. they and nor Schrodinger Schrodinger and a 2 estimates. (2.1.4). restriction finite In matrix solutions. Schrödinger are length-scale and continuous the The R 2 , nonlinear operator the u(r, a are argument, Finally order that the are resonance. equation One proved. Moreover, is is the linear finite formation for Can of to main energy, solution t) Lipschitz continuous As classical difficult a the operators, in relate theorem = to requirement Dirac resonance can we we maps we m acting maps time Q(r) Hence 1171. goal Schrodinger solitary show operator curve. s(t) is maps, and the find relate Although learn construct is But the more is equation, we to to 4 + blow-up solutions. continuous. when But is that has still are nonlinear that global on a spectrum must the ri(r, obtain. one study studied. show wave unfortunately, ground if from special the It the complicated. developed the is a it (oo, m global. this m application t) the usually is limit proved equations be leaves two there construc existence a = solutions = equation where solution. that real [3, Quanies none We bound system unique 1. placed t) 1. There choice states Dirac is Since 4], coor away The The But = and 110 For the the are are as an to as of in Q possible was polar degree is the the harmonic up the solution Our comparison In future instability. Theorem We 6.2 it tions. tive rigorous 6.2.1 is we matrix essary research minimization tinuous solution and eigenvalues “Lyapunov-Schmidt provided needs related Chapter in The prove In harmonic nongeneric want symmetry real studied proof eigenfunctions angle finite Chapter The research. rigorous. with (m Future finite-time alternative in Blow to for with proof either part. 6.1 to that map to of latter Chapter in be Although time. of on principle. 2, 0 show a in that in finite map as < [15]. 4) parameter initial verified. the the we heat of class 5, To the blow is up [2, [6] 0 Mathematically, a turns is we the of devoted The that < linearized proved research heat do map must time stability blow-up 13, Suppose sphere, technique as known for flow 4, the 1? The reduction” up continue energy linear instability. that, From a approaches 15, the it out flow To from behavior 1-equivariant blow be power linearized both . is maximum that to to 18] prove linearized Q(r) 6.2. Most satisfied. the for to that criterion. natural instability [1], [8], close operator. to the a be up to for in be the series if numerical first study we but method. Future study = u(r,t) unstable. greatly of the parameter was global the the The the generic to we operator energy-space 2 know the the to step principle arctan This nonlinear operator the of subclass question given degree used 1-equivariant However, linearized the The always spectrum consider = research methodologies e. existence relies harmonic blow is harmonic Q(r/s(t)) Indeed study will [18]. nonlinear We It are r goal to in of w of turns is is be implies on has to up then formally nonlinear [15]. (3.1.1). of maximum instability, the the When of the critical of these widely operator lies discussed our the stability maximum is [7] an stability, this harmonic out +v(r,t), ground verify This map generic map Dirac stability. on map and solitary result eigenvalue nonlinear criteria are the Blow-up that chapter expand the (m used Schrodinger is energy blow m principle the is totally according equations. is = in heat degree consistent state imaginary for is principle. or = the where a related map 2) waves details to result called are The four-by-four up m uses 1, eigenvalues is wave instability, eigenvalue for with may study harmonic flow different. heat = to then [4]. not m and u current is energy by 1 higher to to in give linear is equa maps Since = posi blow with with axis. flow con The nec 111 the our the the the the the [6], 1, a mined), For such Then equation ing boundedness solve So exclude 8>0. for with for In mapping technical coordinates losing 6.2.2 t 1 So maps, 11 e (r —* <0 (r, = Chapter the we To 0 any functions sufficiently 0. tb’). as: t) this zero <8 a <2, time The turn s(t) solve nonlinear Ia can first the a, (1 Dirac Construction in theorem. But system < lemma + b, initial + but scheme = be some 4, b 1. possibility derivatives to (R, c fl) 2 equation order + of from t 1 ,, this we e For obtained 1v 2 construct our small cl 9 r) Q’/Q, equation R. is — is value. weighted term and constructed — process where any is — r()2(1 the differential used method (er, This eu 2 l to e a v(r, - (4.3.4) contraction qS 1 , of of we first + C2) which by and We = = R lemma a existence the of t) 2 does define a function parametrix Q 2 to 6.2. + I(Q does when find f, finite want is — unique E for Dirac f2) 2 unique leads (fi, system the + Ia C 4 , f2 not (1 an Future is not 0 new e 1 ) 2 + + sequence r is to E f2) mapping < true error approximate and 0 we lead space. cl° equation = fi) 2 to pointwise X work solve < ground 0 functions — ground is for know + worse for C(° < uniqueness (X 8 research to only e(—Q’ - with that laI < 1, (fi, for Using e() 2 (l the an the of is theorem. 1 one states for is and approximation a the actual 0 local f2) positive. + heat equation Then states Lipschitz function such < + solution the can 8 is K 1 , worse 8 l2l29)l1 of energy for < equation + obtained. positivity solution, consider it as One ground 1. K 2 1 f2)2. Dirac gives for Hence However since Q(r/s(t))+ue(r, by implicit space + continuous can advantage = going ibI 8 )lbl, the nonlinear - us near states equation , by since an to be we 2l of to The an contraction f2 study passing Q important constants. to simplified we can the be equation we and = at idea zero chang for deter didn’t origin avoid —4. least with keep this 112 the all t). to to as where such zero value The In nonlinear 6.2.4 It but to neither To to as ator axis. matrix For 6.2.3 isomorphism. the using may a prove £. Chapter prove for asymptotic the dispersive solution resonance is that When We with the X be non-self-adjoint m> an model Nonlinear Asymptotic Schrödinger the the is instability, know a technical If eigenvalue positive an good 5, studying (x)Ii 1 space-time L=-Z+---cos2Q, asymptotic is we Through Hilbert equation estimates if given. that stability starting m < prove lemma. = real kIIxII 2 we L this instability nor LQ= 1. space. and still stability that studying estimates maps (3.1.3), part. has stability of Also through point rn 2 kind a for = ground 6.2. the resonance. Then need sin £ zero Ax let -LQ+sin2Q notice = Although to of ixil continuous ItQIIL2=o0. Q we Future f: + to this of study (— operator, resolvent is we (Strichartz as for of state f(x), have the verify X linearly that 1, the left problem, the —* L.=-L+--- the But for stationary the research solutions. obtained linearly unique to X L_ -) it. spectrum some x(t) ground resolvent analysis one this model show be unstable, In has estimates) a a always [6], assumption constants ground =0. long locally X Q instability that solution, the the m 2 was as equation states is (n—A)’ i.e. term linear a linear the theorem assumes sin2Q also state Lipschitz resonance for 2Q AE whole k the goal does eta. always > operator given operator has if for 0 first on that for m is This and imaginary not A an mapping unstable in to thing implies (6.2.2) zero (6.2.1) eigen 1 2 apply [141. prove oper as is > and 113 an an 0. is is to Then exp(tA) Let check A the be on equation zero a X. linear solutions Assume operator (6.2.3) Ie” 1 II that is is (nonlinearly) 6.2. which satisfied A has for Future generates an some0< for eigenvalue research A 6 . unstable a t< strong A for 2ReA with (6.2.2). continuous Re A> Thus 0 semigroup and we (6.2.3) that need 114 [10] [9] [8] [71 [6] [5] Bibliography [3] [4] [2) [1] for flow flows 537-583. 94, no. pp. monic Comm. de., monic dimensions. -dimensional 1041-1053. in Waves M. S. M. J. N.-H. K.-C. K.-C. P. J.B. S. the 308-348(1990). Harmonic F. 2, 1682—1717. GUsTAFs0N, Kiuwer BizoN, GuAN, GRILLAKIs, of GusTAFs0N, of 507—515. BERG, maps GRoTowsKI, maps. harmonic in Pure harmonic CHANG, CHANG, CI-iANG, harmonic the Academic Calc. under S. Z. Comm. Appl. wave J. Presence Map GUSTAFSON, TAB0R, HuLsH0F, W.Y. map W.-Y. J. K. Var. J. maps maps the maps K. Heat Math. SH.k’rA, J. Pure SHATAH, KANG heat Publishers, schrödinger KANG, (2007) SHATAH. of DING, from from Flow, DING, Formation into Symmetry, 53 Appl. flow, & & T.-P. (2000), surfaces, D 2 J.R.KING, the 30: W. & J. T.-P. T.-P. & A SIAM Geometric Math. Duff. R. into 2-sphere, 499-512. 1990, flow, STRAUSS, result K. TsAI, of YE, no. 11*, TsAI, TsAI, Equations 52, UHLENBEcK, J. singularities J. 60 Duke 37 on 5, Formal Finite Journal APP1. Differential Nemantics, Global — (2007), Nonlinearity evolution 590—602. the Asymptotic Schrödinger Stability 48. Math. time global MATH. asymptotics 246 Existence of 463-499. J. for Functional equations (2009) Schrödinger blow-up Theory J.-M. Geom. 145, existence equivariant 14(2001), stability flow Vol. No. and Coron 1—20. of 36 of of 63, near in 3 Analysis Bubbling Blow for Solitary the (2008), of (1992), ciritcal No. maps, no. et har har 2+1 heat heat 115 al. up 5, 5, [18] [15] [17] [16] [12] [14] [13] [11] J. arXiv:math/0610248. Ann. 754. no. tions, tial critical blow-up at of tions I. H. R. A. J. S. J. S. Math. Thresholds, linearized 4 equation, R0DNIANSKI, ISENBERG, GUSTAFSON, GusTAFs0N, Henri KRIEGER, J. for OuNAIES, JENSEN, (2007) Differential 0(3) MAGNuS, for Phys. the Poincaré8 1070-1111. u-model, operators charge Proc. Gross-Pitaevskii Reviews G. 43(2002), Perturbation S. W. and NENcTu, J. On K. S.-M. Royal. LiBuNG, one STERBENz, SciiLAG, preprint, Integral perturbation NAKANISHI, for in Chapter (2007), Mathematical no. equivariant CHANG, Soc. NLS A 1, method Equation Singularity 1303-1331. Equations. United Edinburgh., 6. AND 678-683. arxiv:math.AP/0605023. solitary On Bibliography K. of T.-P. the for D. a Approach cirtical NAKANISHI, Physics, translationaly waves, in formation TATARu, a Vol formation TsAI, class Two 110 13 wave SIAM Vol. (1998), of To and Global (4-6), Renormalization noninear of T.-P. Resolvent 13, for maps, Three singularities invariant J. 707-720 1 No. Dispersive 2+1 Math. -25. TsAI, 6 Dimensions, available Dirac wave (2001)717- Expansion Anal. (2000). differen Spectra in equa Solu maps 116 and the 39 at