| We study regularity theory for Landau Lifshitz equation (LLE). It describes the magnetization distribution in certain ferromagnetic materials. It can be regarded as a good example of supercritical nonlinear system of partial differential equations, for which regularity theory is not known in general. Many basic questions (such as uniqueness, regularity and singularity formations) are unknown.;In this work we study some of these questions. We study the regularity of continuous weak solutions, a priori gradient estimates for continuous weak solutions and well-posedness of LLE.;We use many techniques from the regularity theory of a quasilinear system of differential equations of parabolic type, a harmonic map equation, a harmonic map heat flow and Navier Stokes equation. |
