Title: On the Landau-Lifshitz equation in dimensions at most four

Authors: Changyou Wang

Issue: Volume 55 (2006), Issue 5, 1615-1644

Abstract: For $n \le 4$ and any bounded smooth domain $\Omega \subset \mathbb{R}^n$, we establish the existence of a global weak solution for the Landau-Lifshitz equation on $\Omega$ 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.