Global Weak Solutions to the Landau-Lifshitz System in 3D

Abstract

By considering a general form of the Landau-Lifshitz equation under the influence of a homogeneous external magnetic fields, we prove that for a ferromagnetic body which occupies a bounded domain â„¦ in â„³ there exists a global weak solution either for the Dirichlet problem or for the Neumann problem. Although there is, in general, non-uniqueness result for the Landau-Lifshitz equation, the uniqueness result for the dynamic equation with constant initial data, which connects with the ground state of the magnetization in physical meanings, is pointed out.