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Volume 20, Issue 1
Partial Regularity for the 2-dimensional Weighted Landau-Lifshitz Flow

Yunhua Ye & Shijin Ding

J. Part. Diff. Eq., 20 (2007), pp. 11-29.

Published online: 2007-02

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  • 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.

  • Keywords

Landau-Lifshitz equations Ginzburg-Landau approximations Hausdorff measure partial regularity

  • AMS Subject Headings

35J55 35J60.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JPDE-20-11, author = {Yunhua Ye and Shijin Ding }, 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} }
TY - JOUR T1 - Partial Regularity for the 2-dimensional Weighted Landau-Lifshitz Flow AU - Yunhua Ye & Shijin Ding 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 -

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.

Yunhua Ye and Shijin Ding . (2007). Partial Regularity for the 2-dimensional Weighted Landau-Lifshitz Flow. Journal of Partial Differential Equations. 20 (1). 11-29. doi:
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