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Global Questions for Evolution Equations Landau-Lifshitz Flow and Dirac Equation

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Global Questions for Evolution Equations Landau-Lifshitz Flow and Dirac Equation Global questions for evolution equations Landau-Lifshitz flow and Dirac equation by Meijiao Guan M.Sc., Huazhong University of Science and Technology, 2003 Ph.D., The University of British Columbia, 2009 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in The Faculty of Graduate Studies (Mathematics) THE UNIVERSITY OF BRITISH COLUMBIA (Vancouver) July 2009 © Meijiao Guan 2009 Abstract This thesis concerns the stationary solutions and their stability for some evolution equations from physics. For these equations, the basic questions regarding the solutions concern existence, uniqueness, stability and singular ity formation. In this thesis, we consider two different classes of equations: the Landau-Lifshitz equations, and nonlinear Dirac equations. There are two different definitions of stationary solutions. For the Landau-Lifshitz equation, the stationary solution is time-independent, while for the Dirac equation, the stationary solution, also called solitary wave solution or ground state solution, is a solution which propagates without changing its shape. The class of Landau-Lifshitz equations (including harmonic map heat flow and Schrödinger map equations) arises in the study of ferromagnets (and anti-ferromagnets), liquid crystals, and is also very natural from a geometric standpoint. Harmonic maps are the stationary solutions to these equations. My thesis concerns the problems of singularity formation vs. global regu larity and long time asymptotics when the target space is a 2-sphere. We consider maps with some symmetry. I show that for m-equivariant maps with energy close to the harmonic map energy, the solutions to Landau Lifshitz equations are global in time and converge to a specific family of harmonic maps for big m, while for m = 1, a finite time blow up solution is constructed for harmonic map heat flow. A model equation for Schrödinger map equations is also studied in my thesis. Global existence and scattering for small solutions and local well-posedness for solutions with finite energy are proved. The existence of standing wave solutions for the nonlinear Dirac equation is studied in my thesis. I construct a branch of solutions which is a con tinuous curve by a perturbation method. It refines the existing results that infinitely many stationary solutions exist, but with uniqueness and continu ity unknown. The ground state solutions of nonlinear Schrodinger equations yield solutions to nonlinear Dirac equations. We also show that this branch of solutions is unstable. This leads to a rigorous proof of the instability of the ground states, confirming non-rigorous results in the physical literature. 11 Table of Contents Abstract ii Table of Contents iii Acknowledgements v Statement of Co-Authorship vi 1 Introduction and main results . 1 1.1 Introduction 1 1.1.1 Landau-Lifshitz flow 2 1.1.2 Model equation for Schrodinger maps 9 1.1.3 Nonlinear Dirac equations 10 1.2 Objectives 13 1.2.1 Global well posedness and blow up for Landau-Lifshitz flow 14 1.2.2 Well-posedness and scattering of a model equation for Schrodinger maps 15 1.2.3 Solitary wave solutions for a class of nonlinear Dirac equations 16 1.2.4 Instability of standing waves for the nonllnear Dirac equations 16 Bibliography 17 2 Global existence and blow up for Landau-Lifshitz flow . 23 2.1 Introduction and main results 23 2.2 Derived nonlinear heat equation 29 2.3 Proof of the main theorem 38 2.4 Finite time blow up 44 Bibliography 48 II’ Table of Contents 3 Well-posedness and scattering for a model equation for Schrödinger maps 50 3.1 Introduction and main results 50 3.2 Local weilposedness 56 3.3 Global small solutions and scattering states 62 Bibliography 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 C4. 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 = —‘ym x H. In 1935 Landau and Lifshitz proposed the Landau-Lifshitz equation [51] as a model for the precessional motion of the magnetization, in which H is replaced by the effective field Heff.
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