Global weak solutions of the fractional Landau–Lifshitz–Maxwell equation

The fractional Landau–Lifshitz–Maxwell equation is considered in this paper. We show the global existence of suitable weak solutions by Galerkin method and the vanishing viscosity method. The main difficulties in this study are due to the loss of compactness of this system and the fact that the nonlinear term is nonlocal and of the same order of the equation. To overcome these difficulties, we introduce the commutator estimates and some cancellation properties to this equation, which prove to be proper tools in the study of Landau–Lifshitz type equations.