Acta mathematica scientia, Series B >
PARTIAL COMPACTNESS FOR LANDAU-LIFSHITZ MAXWELL EQUATION IN TWO-DIMENSION
Received date: 2010-02-01
Online published: 2011-03-20
Supported by
The second author is partially supported by the National Natural Science Foundation of China (10471050), Guangdong Provincial
Natural Science Foundation (031495), and National 973 Project (2006CB805902)
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∞ 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.
KOU Yan-Lei , DING Shi-Jin . PARTIAL COMPACTNESS FOR LANDAU-LIFSHITZ MAXWELL EQUATION IN TWO-DIMENSION[J]. Acta mathematica scientia, Series B, 2011 , 31(2) : 727 -748 . DOI: 10.1016/S0252-9602(11)60272-5
[1] Carbou G, Fabrie P. Time average in micromagnetism. J Diff Eqs, 1998, 2(147): 383--409
[2] Chen Yunmei, Ding Shijin, Guo Boling. Partial regularity for the weak solution to Landau-Lifshitz System. Acta Math Sinica, New Ser,
1998, 14: 423--432
[3]Chen Y, Struwe M. Existence and partial regularity for the heat flow for Harmonic maps. Math Z, 1989, 201: 83--103
[4] Ding Shijin, Liu Xiangao, Wang Changyou. Landau-Lifshitz-Maxwell equation in dimension three. Pacific Journal of Mathematics, 2009, 2(243): 243--276
[5] Ding Shijin, Guo Boling. Hausdorff Measure of the Singular Set of Landau-Lifshitz Equations with a Nonlocal Term. Communications in
Math Phys, 2004, 250(1): 95--117
[6] Ding Shijin, Guo Boling. Initial-Boundary value Problem for higher dimensional Landau-Lifshitz systems. Appl Anal, 2004, 83(7): 673--697
[7] Ding Shijin, Guo Boling, Lin Junyu, et al. Global Existence of Weak Solutions for Landau-Lifshitz-Maxwell Equations. Discrete and Continuous Dynamical System, Ser A, 2007, 17(4): 867--890
[8] Ding Shijin, Guo Boling. Existence of Partially Regular Weak Solutions to Landau-Lifshitz-Maxwell Equations. Journal of Differential Equations, 2008, 244(10): 2448--2472
[9] Guo Boling, Hong M C. The Landau-Lifshitz equations of the ferromagnetic spin chain and harmonic maps. Calc Var PDE, 1993, 1(3): 311--334
[10]Guo B, Su F. Global weak solution for the Landau-Lifshitz-Maxwell equation in three space dimensions. J Math Anal Appl, 1997, 211(1): 326--346
[11] Harpes P. Partial compactness for the 2-D Landau-Lifshitz flow. EJDE, 2004, 2004(90): 1--24
[12] Ladyzenskaja O A, Solonnikov V A, Uralceva N N. Linear and quasi-linear equations of parabolic type. Providence: American Mathematical Society, 1968
[13]Landau L, Lifshitz D. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys Z Sovietunion, 1935, 8: 153--169
[14] Ye Yunhua, Ding Shijin. Partial regularity for the 2-dimensional weighted Landau-Lifshitz flow. Journal of Partial Differential
Equations, 2007, 20(1): 11--29
[15] Wang C Y. On Landau-Lifshitz equation in dimensions at most four. Indiana University Mathematics Journal, 2006, 55: 1615--1644
[16] Zhou Y L, Guo B L. Weak solution of systems of ferromagnetic chain with several variables. Science in China, Series A, 1987, 30: 1251--1266
[17]Zhou Y L, Guo B L, Tan S. Existence and uniqueness of smooth solution for system of ferromagetic chain. Scientia Sinica, 1991, 34A(2): 157--166
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