@Article{JPDE-20-11, author = {}, title = {Partial Regularity for the 2-dimensional Weighted Landau-Lifshitz Flow}, journal = {Journal of Partial Differential Equations}, year = {2007}, volume = {20}, number = {1}, pages = {11--29}, abstract = {

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({u_∈})∩(\overline{\Omega}×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({u_∈})∩(\overline{\Omega}×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.

}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5290.html} }