On the Landau-Lifshitz equation in dimensions at most four

Abstract

For n < 4 and any bounded smooth domain Q c R n , we establish the existence of a global weak solution for the Landau-Lifshitz equation on Q with respect to smooth initial-boundary data, which is smooth off a closed set with locally finite n-dimensional parabolic Hausdorff measure. The approach is based on the Ginzburg-Landau approximation, a time slice energy monotonicity inequality, and an energy decay estimate under the smallness of renormalized Ginzburg-Landau energies.