PARTIAL REGULARITY FOR THE 2-DIMENSIONAL WEIGHTED LANDAU-LIFSHITZ FLOW

摘要

We consider the partial regularity of weak solutions to the weighted Landau-Lifshitz flow on a 2-dimensional bounded smooth domain by Ginzburg-Landau type approximation. Under the energy smallness condition, we prove the uniform local C∞ bounds for the approaching solutions. This shows that the approximating solutions are locally uniformly bounded in C∞(Reg({uε}) ∩((Ω)×R+)) which guarantee the smooth convergence in these points. Energy estimates for the approximating equations are used to prove that the singularity set has locally finite two-dimensional parabolic Hausdorff measure and has at most finite points at each fixed time. From the uniform boundedness of approximating solutions in C∞ (Reg({uε }) ∩((Ω)×R+)), we then extract a subsequence converging to a global weak solution to the weighted Landau-Lifshitz flow which is in fact regular away from finitely many points.