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. [Copyright &y& Elsevier]