Global well-posedness of the Landau–Lifshitz–Gilbert equation for initial data in Morrey spaces

We establish the global well-posedness of the Landau–Lifshitz–Gilbert equation in ℝ^n for any initial data 𝐦_0∈ H^1_*(ℝ^n,𝕊^2) whose gradient belongs to the scaling critical Morrey space M^2,2(ℝ^n) with small norm ‖∇𝐦_0‖ _M^2,2(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.