Existence of partially regular weak solutions to Landau–Lifshitz–Maxwell equations

Abstract

In this paper, the authors establish the existence of partially regular weak solutions to the Landau–Lifshitz equations coupling with static Maxwell systems in 3 dimensions by Ginzburg–Landau approximation. It is proved that the Hausdorff measure of the singular set is locally finite. This extends the similar results of Ding and Guo [S. Ding, B. Guo, Hausdorff measure of the singular set of Landau–Lifshitz equations with a nonlocal term, Comm. Math. Phys. 250 (1) (2004) 95–117] from the stationary solutions to weak solutions and the results of Wang [C. Wang, On Landau–Lifshitz equations in dimensions at most four, Indiana Univ. Math. J. 55 (5) (2006) 1615–1644] from Landau–Lifshitz equations to Landau–Lifshitz–Maxwell equations.