Global well-posedness for the Landau-Lifshitz-Baryakhtar equation in $\mathbb{R}^3$

TL;DR

This paper proves the global existence and uniqueness of solutions for the Landau-Lifshitz-Baryakhtar equation in three-dimensional space, advancing understanding of its mathematical properties.

Contribution

It establishes the first comprehensive proof of global well-posedness for the LLBar equation in , including strong, classical, and smooth solutions, using energy methods and approximation techniques.

Findings

Existence and uniqueness of global strong solutions.

Existence and uniqueness of classical solutions.

Existence of arbitrary smooth solutions.

Abstract

This paper establishes the global well-posedness of the Landau-Lifshitz-Baryakhtar (LLBar) equation in the whole space $\mathbb{R}^3$. The study first demonstrates the existence and uniqueness of global strong solutions using the weak compactness approach. Furthermore, the existence and uniqueness of classical solutions, as well as arbitrary smooth solutions, are derived through a bootstrap argument. The proofs for the existence of these three types of global solutions are based on Friedrichs mollifier approximation and energy estimates, with the structure of the LLBar equation playing a crucial role in the derivation of the results.