Quasi-Stationary Limit and a Degenerate Landau–Lifshitz Equation of Ferromagnetism

Abstract

In this paper, we study a model of Landau–Lifshitz equations of ferromagnetism that does not contain the regularizing term of exchange energy. Without the exchange energy, due to the lack of certain derivative estimates and compactness, such an equation becomes degenerate and cannot be studied by the usual Galerkin method based on the elliptic equation theory. For such a degenerate model, it is known that the weak solutions can be obtained through the quasi-stationary limits of certain coupled Landau– Lifshitz–Maxwell systems as the dielectric permittivity tends to zero. In this paper, we use a simplified Landau–Lifshitz–Maxwell system with constant permittivity to present a different but more direct proof of this quasi-stationary limit result. We also establish a finite-time local L2-stability result for weak solutions of the degenerate Landau–Lifshitz equation, which yields the new uniqueness result on weak solution with bounded initial data.