Global weak solutions to Landau-Lifshitz equations into compact Lie algebras

doi:10.1007/s11464-019-0803-7

Front. Math. China

2019, Vol. 14

Issue (6) : 1163-1196 https://doi.org/10.1007/s11464-019-0803-7

RESEARCH ARTICLE

Global weak solutions to Landau-Lifshitz equations into compact Lie algebras

Zonglin JIA¹, Youde WANG^2,^3,⁴(

)

¹. Institute of Applied Physics and Computational Mathematics, China Academy of Engineering Physics, Beijing 100088, China
². College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China
³. Hua Loo-Keng Key Laboratory of Mathematics, Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China
⁴. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

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Abstract

We consider a parabolic system from a bounded domain in a Euclidean space or a closed Riemannian manifold into a unit sphere in a compact Lie algebra g; which can be viewed as the extension of Landau-Lifshitz (LL) equation and was proposed by V. Arnold. We follow the ideas taken from the work by the second author to show the existence of global weak solutions to the Cauchy problems of such LL equations from an n-dimensional closed Riemannian manifold $T$ or a bounded domain in $ℝ n$ into a unit sphere $S g (1)$ in g. In particular, we consider the Hamiltonian system associated with the nonlocal energy-micromagnetic energy defined on a bounded domain of $ℝ 3$ and show the initial-boundary value problem to such LL equation without damping terms admits a global weak solution. The key ingredient of this article consists of the choices of test functions and approximate equations.

Keywords Landau-Lifshitz (LL) equations Lie algebra test functions

Corresponding Author(s): Youde WANG

Issue Date: 07 January 2020

Cite this article:

Zonglin JIA,Youde WANG. Global weak solutions to Landau-Lifshitz equations into compact Lie algebras[J]. Front. Math. China, 2019, 14(6): 1163-1196.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-019-0803-7
https://academic.hep.com.cn/fmc/EN/Y2019/V14/I6/1163

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