Partial compactness for Landau-Lifshitz maxwell equation in two-dimension

Abstract

We study the partial regularity of weak solutions to the 2-dimensional Landau-Lifshitz equations coupled with time dependent Maxwell equations by Ginzburg-Landau type approximation. Outside an energy concentration set of locally finite 2-dimensional parabolic Hausdorff measure, we prove the uniform local C infin bounds for the approaching solutions and then extract a subsequence converging to a global weak solution of the Landau-Lifshitz-Maxwell equations which are smooth away from finitely many points.