Global existence and partial regularity for the Landau-Lifshitz-Gilbert equation with helicity

Abstract

In this paper, the Landau-Lifshitz-Gilbert equation with helicity is considered. In R3 or a bounded regular domain Ω of R3, we establish the global existence of a weak solution. In Rn, a global existence criterion and uniqueness of the smooth solution are given. In R1, the local smooth solution is indeed global with large initial data. In R2, we prove the existence of a global weak solution, which is smooth with the exception of at most finite singular points.