The Landau–Lifshitz–Maxwell equation in dimension three

Abstract

In dimension three, we show the existence of weak solutions (u, H, E) to the Landau-Lifshitz equation coupled with the time-dependent Maxwell equation such that u is Holder continuous away from a closed set Σ that has locally finite 3-dimensional parabolic Hausdorff measure. For two reduced Maxwell equations, Holder continuity of ∇u away from Σ is also established.