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Journal of Zhejiang University SCIENCE A 2007 Vol.8 No.6 P.949-956

http://doi.org/10.1631/jzus.2007.A0949


Some stationary weak solutions to inhomogeneous Landau-Lifshitz equations in three dimensions


Author(s):  FANG Dao-yuan, LI Tai-long, XUE Ru-ying

Affiliation(s):  Department of Mathematics, Zhejiang University, Hangzhou 310027, China

Corresponding email(s):   m9845@163.com

Key Words:  Landau-Lifshitz equation, Stationary condition, Hausdorff measure


FANG Dao-yuan, LI Tai-long, XUE Ru-ying. Some stationary weak solutions to inhomogeneous Landau-Lifshitz equations in three dimensions[J]. Journal of Zhejiang University Science A, 2007, 8(6): 949-956.

@article{title="Some stationary weak solutions to inhomogeneous Landau-Lifshitz equations in three dimensions",
author="FANG Dao-yuan, LI Tai-long, XUE Ru-ying",
journal="Journal of Zhejiang University Science A",
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pages="949-956",
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publisher="Zhejiang University Press & Springer",
doi="10.1631/jzus.2007.A0949"
}

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%T Some stationary weak solutions to inhomogeneous Landau-Lifshitz equations in three dimensions
%A FANG Dao-yuan
%A LI Tai-long
%A XUE Ru-ying
%J Journal of Zhejiang University SCIENCE A
%V 8
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%@ 1673-565X
%D 2007
%I Zhejiang University Press & Springer
%DOI 10.1631/jzus.2007.A0949

TY - JOUR
T1 - Some stationary weak solutions to inhomogeneous Landau-Lifshitz equations in three dimensions
A1 - FANG Dao-yuan
A1 - LI Tai-long
A1 - XUE Ru-ying
J0 - Journal of Zhejiang University Science A
VL - 8
IS - 6
SP - 949
EP - 956
%@ 1673-565X
Y1 - 2007
PB - Zhejiang University Press & Springer
ER -
DOI - 10.1631/jzus.2007.A0949


Abstract: 
In this paper, we describe several stationary conditions on weak solutions to the inhomogeneous landau-Lifshitz equation, which ensure the partial regularity. For certain class of proper stationary weak solutions, a compactness result of the solutions, a finite hausdorff measure result of the t-slice energy concentration sets and an asymptotic limit result of the Radon measures are proved. We also present a subtle rectifiability result for the energy concentration set of certain sequence of strong stationary weak solutions.

Darkslateblue:Affiliate; Royal Blue:Author; Turquoise:Article

Reference

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[3] Daniel, M., Porsezian, K., Lakshmanan, M., 1994. On the integrability of the inhomogeneous spherically symmetric Heisenberg ferromagnet in arbitrary dimensions. J. Math. Phys., 35:6498-6510.

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[5] Ilmanen, T., 1994. Elliptic regularization and partial regularity for motion by mean curvature. Mem. Amer. Math. Soc., 108(520).

[6] Landau, L., Lifshitz, E.M., 1935. On the theory of the dispersion of magnetic permeability in ferromagnetic bodies. Phys. Z. Sowjetunion, 8:153. (Reprinted in Collected Papers of L.D. Landau, Pergamon Press, New York, 1965, p.101-114).

[7] Li, J., Tian, G., 2000. The blow-up locus of heat flows for harmonic maps. Acta Mathematica Sinica, 16:29-62.

[8] Li, Y., Wang, Y., 2006. Bubbling location for F-harmonic maps and inhomogeneous Landau-Lifshitz equations. Comment. Math. Helv., 81(2):433-448 (in Chinese).

[9] Lin, F.H., 1999. Gradient estimates and blow-up analysis for stationary harmonic maps. Ann. Math., 149:785-829.

[10] Liu, X., 2007. Blow up sets of the Landau-Lifshitz system and quasi-mean curvature flows. Manuscript.

[11] Moser, R., 2002. Partial Regularity for the Landau-Lifshitz Equation in Small Dimensions. MPI Preprint, 26. Http://www.mis.mpg.de/preprints/2002/

[12] Preiss, D., 1987. Geometry of measure in Rn: distribution, rectifiability, and densities. Ann. Math., 125:537-643.

[13] Struwe, M., 1988. On the evolution of harmonic maps in higher dimensions. J. Differ. Geom., 28:485-502.

[14] Tang, H., 2001. Inhomogeneous Landau-Lifshitz systems from a Riemann surface into S2. Chin. J. Contemp. Math. 22:215-230.

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