The fractional Landau–Lifshitz–Gilbert equation and the heat flow of harmonic maps

Abstract

In this paper, we prove the existence of global weak solutions to the periodic fractional Landau–Lifshitz–Gilbert equation through the Ginzburg–Landau approximation and the Galerkin approximation. Since the nonlinear term is nonlocal and of full order of the equation, some special structures of the equation, the commutator estimate and some calculus inequalities of fractional order are exploited to get the convergence of the approximating solutions. The equation considered in this paper can also be regarded as a generalization of the heat flow of harmonic maps to the fractional order.