Strong solutions of the Landau-Lifshitz-Bloch equation in Besov space

TL;DR

This paper proves local and global existence and uniqueness of strong solutions to the 3D Landau-Lifshitz-Bloch equation in Besov spaces, introducing new estimates and criteria for blow-up and global existence.

Contribution

It establishes the first local and global well-posedness results for the Landau-Lifshitz-Bloch equation in Besov spaces using a novel commutator estimate.

Findings

Local existence and uniqueness in Besov space for initial data in $ ext{dot}B_{2,1}^{3/2}$.

Global existence for small initial data in the same Besov space.

A blow-up criterion and conditions for global solutions in Sobolev space.

Abstract

We focus on the existence and uniqueness of the three-dimensional Landau-Lifshitz-Bloch equation supplemented with the initial data in Besov space $\dot{B}_{2,1}^{\frac{3}{2}}$. Utilizing a new commutator estimate, we establish the local existence and uniqueness of strong solutions for any initial data in $\dot{B}_{2,1}^{\frac{3}{2}}$. When the initial data is small enough in $\dot{B}_{2,1}^{\frac{3}{2}}$, we obtain the global existence and uniqueness. Furthermore, we also establish a blow-up criterion of the solution to the Landau-Lifshitz-Bloch equation and then we prove the global existence of strong solutions in Sobolev space under a new condition based on the blow-up criterion.