Weak solutions of the Landau–Lifshitz–Bloch equation

Abstract

The Landau-Lifshitz-Bloch (LLB) equation is a formulation of dynamic micromagnetics valid at all temperatures, treating both the transverse and longitudinal relaxation components important for high-temperature applications. We study LLB equation in case the temperature raised higher than the Curie temperature. The existence of weak solution is showed and its regularity properties are also discussed. In this way, we lay foundations for the rigorous theory of LLB equation that is currently not available.

Global weak solution and smooth solution of the periodic initial value problem for the generalized Landau-Lifshitz-Bloch equation in high dimensions

“…where λ > 0 is a constant. In [6], the author asserted that if L 1 = L 2 , (2) can be reduced as follows Z t = ∆Z + Z × ∆Z − k(1 + µ|Z| 2 )Z, (k, µ > 0) (3) and the existence of the weak solution for the equation 3has been obtained. The purpose of this paper is to discuss the following initial value problem for the GLLB in high dimensions.…”

“…The structure of the paper is as follows. In section 2, we obtain the approximate solution to the periodic initial value problem (4)- (6), and the estimation of the approximate solution is derived in section 3. In section 4, we get the weak solution of (4)- (6).…”

“…In section 2, we obtain the approximate solution to the periodic initial value problem (4)- (6), and the estimation of the approximate solution is derived in section 3. In section 4, we get the weak solution of (4)- (6). Finally, we prove the existence and uniqueness of the smooth solution for the problem (4)-(6) in high dimensions.…”

“…2. Approximate solution to the periodic initial value problem (4)- (6). In order to prove the existence of global weak solution to the problem (4)-(6), we use the Galërkin method.…”

Existence of a unique solution and invariant measures for the stochastic Landau--Lifshitz--Bloch equation

“…The aim of this paper is to initiate the analysis of stochastic Landau-Lifschitz-Bloch equation (1.3). For the reader's convenience we recall here some background material introduced in [17].…”