TY - JOUR
T1 - Partial Regularity for the 2-dimensional Weighted Landau-Lifshitz Flow
JO - Journal of Partial Differential Equations
VL - 1
SP - 11
EP - 29
PY - 2007
DA - 2007/02
SN - 20
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jpde/5290.html
KW - Landau-Lifshitz equations
KW - Ginzburg-Landau approximations
KW - Hausdorff measure
KW - partial regularity
AB - <p>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.</p>