Landau-Lifshitz-Bloch equation on Riemannian manifold

Abstract

We bring in Landau-Lifshitz-Bloch equation on m-dimensional closed Riemannian manifold and prove that it admits a unique local solution. When m\ge \mathrm{3} and the initial data in L^{\infty }-norm is suciently small, the solution can be extended globally. Moreover, for m\ge \mathrm{2}, we can prove that the unique solution is global without assuming small initial data.

Key words： Orientable vector bundle; Riemannian curvature tensor on vector bundle; Sobolev space on vector bundle

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