至多四维加权Landau-Lifshitz方程的部分正则性
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摘要
在本文，我们得到了维数n≤4的加权Landau-Lifshitz方程弱解的整体存在性和部分正则性。其主要技巧是运用了Ginzburg-Landau逼近和能量衰减估计的方法。
In this paper, we establish the existence of partially regular weak solu-tions to the weighted Landau-Lifshitz equations for n≤4. The approach isbased on Ginzburg-Landau approximation and energy decay estimates.
In this paper, we establish the existence of partially regular weak solu-tions to the weighted Landau-Lifshitz equations for n≤4. The approach isbased on Ginzburg-Landau approximation and energy decay estimates.
引文
[1] 戴道生，钱昆明，铁磁学(上册)．科学出版社，2000．
[2] 钟文定，铁磁学(中册)．科学出版社，2000．
[3] 廖绍彬，铁磁学(下册)．科学出版社，2000．
[4] 郭柏灵，丁时进，自旋波与铁磁链方程．杭州，浙江科学技术出版社，2000．
[5] 肖勇，微磁方程数值解及其物理意义．中山大学，2002．
[6] L.Landau and E. Lifshitz, on the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Physikalische Zeitschrift der Sowjetunion, 1935, 8, 153-169.
[7] F.Alouges, A. Soyeur, On global weak solutions for Landau-Lifshitz equations: existence and nonuniqueness. Nonlinear Anal. TMA 18. 1071-1084(1992).
[8] Y. Chen, S. Ding, B. Guo, Partial regularity for 2-D Landau-Lifshitz equations. Acta Mathmatics Sinica, New S, 1998 July, Volume 14, No.13,423-432(1998).
[9] Y. Chen, F. Lin, Evolution of harmonic maps with Dirichlet boundary conditions. Comm. Anal. Geom. 1(1993), no. 3-4, 327-346.
[10] Y. Chen, M. Struwe, Existence and Partial Regularity for the heat flow for Harmonic maps. Math. Z. 201, 83-103(1989).
[11] Y. Chen, C. Wang, Partial regularity for weak heat flows into Riemannian homogeneous spaces. Comm. PDE 21(1996), no 5-6, 735-761.
[12] S. Ding, B. Guo, Hausdorff measure of the singular set of Landau-Lifshitz equations with a nonlocal term. Comm. Math. Phys. 250(2004), no. 1,95-117.
[13] S. Ding, B. Guo and F. Su, Measure-valued solutions to the strongly degenerate compressible Heisenberg chain equation. Jounal of Phys. Math. 40(1999), 1153-1162.
[14] S. Ding, B. Guo and F. Su, Smooth solution for one-dimensional inhomogeneous Heisenberg chain equation, Proceeding of the Royal Socienty of Edinbrgh,(29A(1999)), 1171-1184.
[15] C. Fefferman, E. Stein, H~P spaces of several variables. Acta. Math. 129(1972) 137-193.
[16] B. Guo, M. Hong, The Landau-Lifshitz equations of the ferromagnetic spin chain and harmonic maps, Calc. Var. PDE. 1: 311-334(1993).
[17] P. Harpes, Partial compactness for the 2-D Landau-Lifshitz flow, EJDE, No.90,1-24(2004).
[18] X. Liu, Partial regularity for the Landau-Lifshitz system. Calc. Var. 20(2004), no. 2,153-173.
[19] J. Lin, S. Ding, Smooth solution to the one dimensional Inhomogeneous Non-automorphic Landau-Lifshitz equation. Proc. R. Soc. A. , doi:10. 1098/rspa. 2006. 1689.
[20] O. Ladyzenskaja, V. Solonnikov, N. Uralceva, Linear and quasilinear equations of parabolic types. AMS Trans. Math. Monograph, 23 (1968).
[21] C.Melcher, Existence of partially regular solutions for Landau-Lifshitz equations in R~3, Comm. PDE. 30:567-587, 2005.
[22] S. Semmes, A primer on Hardy spaces, and some remarks on a theorem of Evans and Mller. Comm. PDE 19(1994), no. 1-2, 277-319.
[23] C. Wang, On Landau-Lifshitz equation in dimensions at most four. Indiana Univ. Math. J. Vol. 55, no.6 (2006) 1615-1644.
[24] Y. Ye, S. Ding, Partial Regularity for the 2-D weighted Landau-Lifshitz flow. To appear in PDE.
[25] Y. Zhou, B. Guo, Weak solutions systems of ferromagnetic chain with several variables, Science in China, A30:1251-1266(1987).
[26] Y. Zhou, B. Guo, S. Tan, Existence and uniqueness of smooth solution for system of ferromagnetic chain, Science in China, 34(A)(1991), 257-266.
[27] S. Ding, C. Y. Wang, Finite time singularity of the Landau-Lifshitz-Gilbert equation, 2006, Preprint.
[2] 钟文定，铁磁学(中册)．科学出版社，2000．
[3] 廖绍彬，铁磁学(下册)．科学出版社，2000．
[4] 郭柏灵，丁时进，自旋波与铁磁链方程．杭州，浙江科学技术出版社，2000．
[5] 肖勇，微磁方程数值解及其物理意义．中山大学，2002．
[6] L.Landau and E. Lifshitz, on the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Physikalische Zeitschrift der Sowjetunion, 1935, 8, 153-169.
[7] F.Alouges, A. Soyeur, On global weak solutions for Landau-Lifshitz equations: existence and nonuniqueness. Nonlinear Anal. TMA 18. 1071-1084(1992).
[8] Y. Chen, S. Ding, B. Guo, Partial regularity for 2-D Landau-Lifshitz equations. Acta Mathmatics Sinica, New S, 1998 July, Volume 14, No.13,423-432(1998).
[9] Y. Chen, F. Lin, Evolution of harmonic maps with Dirichlet boundary conditions. Comm. Anal. Geom. 1(1993), no. 3-4, 327-346.
[10] Y. Chen, M. Struwe, Existence and Partial Regularity for the heat flow for Harmonic maps. Math. Z. 201, 83-103(1989).
[11] Y. Chen, C. Wang, Partial regularity for weak heat flows into Riemannian homogeneous spaces. Comm. PDE 21(1996), no 5-6, 735-761.
[12] S. Ding, B. Guo, Hausdorff measure of the singular set of Landau-Lifshitz equations with a nonlocal term. Comm. Math. Phys. 250(2004), no. 1,95-117.
[13] S. Ding, B. Guo and F. Su, Measure-valued solutions to the strongly degenerate compressible Heisenberg chain equation. Jounal of Phys. Math. 40(1999), 1153-1162.
[14] S. Ding, B. Guo and F. Su, Smooth solution for one-dimensional inhomogeneous Heisenberg chain equation, Proceeding of the Royal Socienty of Edinbrgh,(29A(1999)), 1171-1184.
[15] C. Fefferman, E. Stein, H~P spaces of several variables. Acta. Math. 129(1972) 137-193.
[16] B. Guo, M. Hong, The Landau-Lifshitz equations of the ferromagnetic spin chain and harmonic maps, Calc. Var. PDE. 1: 311-334(1993).
[17] P. Harpes, Partial compactness for the 2-D Landau-Lifshitz flow, EJDE, No.90,1-24(2004).
[18] X. Liu, Partial regularity for the Landau-Lifshitz system. Calc. Var. 20(2004), no. 2,153-173.
[19] J. Lin, S. Ding, Smooth solution to the one dimensional Inhomogeneous Non-automorphic Landau-Lifshitz equation. Proc. R. Soc. A. , doi:10. 1098/rspa. 2006. 1689.
[20] O. Ladyzenskaja, V. Solonnikov, N. Uralceva, Linear and quasilinear equations of parabolic types. AMS Trans. Math. Monograph, 23 (1968).
[21] C.Melcher, Existence of partially regular solutions for Landau-Lifshitz equations in R~3, Comm. PDE. 30:567-587, 2005.
[22] S. Semmes, A primer on Hardy spaces, and some remarks on a theorem of Evans and Mller. Comm. PDE 19(1994), no. 1-2, 277-319.
[23] C. Wang, On Landau-Lifshitz equation in dimensions at most four. Indiana Univ. Math. J. Vol. 55, no.6 (2006) 1615-1644.
[24] Y. Ye, S. Ding, Partial Regularity for the 2-D weighted Landau-Lifshitz flow. To appear in PDE.
[25] Y. Zhou, B. Guo, Weak solutions systems of ferromagnetic chain with several variables, Science in China, A30:1251-1266(1987).
[26] Y. Zhou, B. Guo, S. Tan, Existence and uniqueness of smooth solution for system of ferromagnetic chain, Science in China, 34(A)(1991), 257-266.
[27] S. Ding, C. Y. Wang, Finite time singularity of the Landau-Lifshitz-Gilbert equation, 2006, Preprint.