Partial compactness for Landau-Lifshitz maxwell equation in two-dimension
Abstract We study the partial regularity of weak solutions to the 2-dimensional Landau-Lifshitz equations coupled with time dependent Maxwell equations by Ginzburg-Landau type approximation. Outside an energy concentration set of locally finite 2-dimensional parabolic Hausdorff measure, we prove the uniform local C infin bounds for the approaching solutions and then extract a subsequence converging to a global weak solution of the Landau-Lifshitz-Maxwell equations which are smooth away from finitely many points.
A weightedL2 norm is introduced in which Markov operators, e.g., associated with noisy maps, are contracting provided the kernel (i.e., the transitional distribution) is smooth enough. This results in strong relaxational properties of noisy maps. Similar to this norm, integral functionals appear useful when studying spatiotemporal chaos and random fields.