Existence of Partially Regular Solutions for Landau–Lifshitz Equations in ℝ3

Abstract We establish the existence of partially regular weak solutions for the Landau–Lifshitz equation in three space dimensions for smooth initial data of finite Dirichlet energy. The construction is based on Ginzburg–Landau approximation. The new key ingredient is a nonlocal representation formula for the penalty term that permits us to take advantage of the special trilinear structure of the limiting nonlinearity.

[1]  Lev Davidovich Landau,et al.  ON THE THEORY OF THE DISPERSION OF MAGNETIC PERMEABILITY IN FERROMAGNETIC BODIES , 1935 .

[2]  S. Campanato.  Equazioni paraboliche del secondo ordine e spaziL2, θ(ω, δ)(ω, δ) , 1966 .

[3]  M. Giaquinta,et al.  Partial regularity for the solutions to nonlinear parabolic systems , 1973 .

[4]  A. Friedman.  Partial Differential Equations of Parabolic Type , 1983 .

[5]  Michael Struwe,et al.  On the evolution of harmonic mappings of Riemannian surfaces , 1985 .

[6]  David Kinderlehrer,et al.  Existence and partial regularity of static liquid crystal configurations , 1986 .

[7]  Michael Struwe,et al.  On the evolution of harmonic maps in higher dimensions , 1988 .

[8]  Michael Struwe,et al.  Existence and partial regularity results for the heat flow for harmonic maps , 1989 .

[9]  François Alouges,et al.  On global weak solutions for Landau-Lifshitz equations: existence and nonuniqueness , 1992 .

[10]  F. Lin,et al.  Evolution of harmonic maps with Dirichlet boundary conditions , 1993 .

[11]  Mikhail Feldman,et al.  Partial regularity for harmonic maps of evolution into spheres , 1994 .

[12]  Fanghua Lin,et al.  Remarks on approximate harmonic maps , 1995 .

[13]  M. Marschark,et al.  Everywhere discontinuous harmonic maps into spheres , 1995 .

[14]  A. Freire,et al.  Uniqueness for the harmonic map flow in two dimensions , 1995 .

[15]  Chen Yunmei,et al.  Two Dimensional Landau－Lifshitz Equation , 1996 .

[16]  Ding Shi-jin,et al.  Partial regularity for two dimensional Landau-Lifshitz equations , 1998 .

[17]  G. Bo-ling,et al.  Initial-boundary value problem for the Landau-Lifshitz system (I) ——Existence and partial regularity , 1998 .

[18]  正人 木村.  Max-Planck-Institute for Mathematics in the Sciences(海外,ラボラトリーズ) , 2001 .

[19]  Giandomenico Orlandi,et al.  Asymptotics for the Ginzburg–Landau Equation in Arbitrary Dimensions , 2001 .

[20]  R. Moser.  Regularity for the approximated harmonic map equation and application to the heat flow for harmonic maps , 2003 .

[21]  P. Harpes,et al.  Uniqueness and bubbling of the 2-dimensional Landau-Lifshitz flow , 2004 .