Global Well-Posedness of the Landau-Lifshitz-Gilbert equation for initial data in Morrey space

Abstract

We establish the global well-posedness of the Landau-Lifshitz-Gilbert equation in ${\mathbb{R}}^n$ for any initial data ${\mathbf{m}}_0\in H_{\ast }^1({\mathbb{R}}^n,{\mathbb{S}}^2)$ whose gradient belongs to the Morrey space $M^{2,2}({\mathbb{R}}^n)$ with small norm $\Vert \nabla {\mathbf{m}}_0{\Vert }_{M^{2,2}({\mathbb{R}}^n)}$. The method is based on priori estimates of a dissipative Schrödinger equation of Ginzburg-Landau types obtained from the Landau-Lifshitz-Gilbert equation by the moving frame technique.

21 pages