On the Landau-Lifshitz equation in dimensions at most four Changyou Wang 35K5549N60Landau-Lifshitz equationGingburg-Landau approximationHardy spaceBMO space 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. Indiana University Mathematics Journal 2006 text pdf 10.1512/iumj.2006.55.2810 10.1512/iumj.2006.55.2810 en Indiana Univ. Math. J. 55 (2006) 1615 - 1644 state-of-the-art mathematics http://iumj.org/access/