Global weak solution of the Landau-Lifshitz-Baryakhtar equation

The Landau--Lifshitz--Baryakhtar (LLBar) equation is a generalisation of the Landau--Lifshitz--Gilbert (LLG) and the Landau--Lifshitz--Bloch (LLB) equations which takes into account contributions from nonlocal damping and is valid at moderate temperature below the Curie temperature. As such, it is able to explain some discrepancies between the experimental observations and the known theories in various problems on magnonics and magnetic domain-wall dynamics. In this paper, the existence and uniqueness of weak and regular solutions to LLBar equation are proven. H\"older continuity of the solution is also discussed.