The Landau–Lifshitz–Maxwell equation in dimension three

Shijin Ding, Xiangao Liu and Changyou Wang

Vol. 243 (2009), No. 2, 243–276

DOI: 10.2140/pjm.2009.243.243

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 Hölder continuous away from a closed set Σ that has locally finite 3-dimensional parabolic Hausdorff measure. For two reduced Maxwell equations, Hölder continuity of ∇u away from Σ is also established.

Keywords

Landau–Lifshitz–Gilbert equation, time-dependent Maxwell equation

Mathematical Subject Classification 2000

Primary: 35K55

Secondary: 35B65

Milestones

Received: 4 December 2008

Revised: 9 February 2009

Accepted: 24 May 2009

Published: 1 December 2009