Global questions for evolution equations Landau-Lifshitz flow and Dirac equation

• Main Author: Guan, Meijiao
• Language: English
• Published: University of British Columbia 2010
• Online Access: http://hdl.handle.net/2429/22491

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 singularity 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 regularity 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 continuous curve by a perturbation method. It refines the existing results that infinitely many stationary solutions exist, but with uniqueness and continuity 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.