PARTIAL COMPACTNESS FOR LANDAU-LIFSHITZ MAXWELL EQUATION IN TWO-DIMENSION

doi:10.1016/S0252-9602(11)60272-5

Abstract

Abstract:

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.

Key words: Landau-Lifshitz-Maxwell equations, ginzburg-landau approximations, Hausdorff measure, partial regularity

CLC Number:

35J55

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.

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References

[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