Landau-Lifshitz-Bloch equation on Riemannian manifold

TL;DR

This paper studies the Landau-Lifshitz-Bloch equation on Riemannian manifolds, proving local existence of solutions and conditions for their global extension depending on the manifold's dimension and initial data size.

Contribution

It extends the Landau-Lifshitz-Bloch equation to Riemannian manifolds and establishes existence and global extension results under new geometric and initial data conditions.

Findings

Unique local solutions exist on closed Riemannian manifolds.

Global solutions are achievable for high-dimensional manifolds with small initial data.

In two dimensions, solutions are global without smallness assumptions.

Abstract

In this article, we bring in Landau-Lifshitz-Bloch(LLB) equation on $m$-dimensional closed Riemannian manifold and prove that it admits a unique local solution. In addition, if $m\geqslant3$ and $L^{\infty}-$norm of initial data is sufficiently small, the solution can be extended globally. Moreover, if $m=2$, we can prove that the unique solution is global without assuming small initial data.