Weak–strong uniqueness for the Landau–Lifshitz–Gilbert equation in micromagnetics

Abstract

We consider the time-dependent Landau-Lifshitz-Gilbert equation. We prove that each weak solution coincides with the (unique) strong solution, as long as the latter exists in time. Unlike available results in the literature, our analysis also includes the physically relevant lower-order terms like Zeeman contribution, anisotropy, stray field, and the Dzyaloshinskii-Moriya interaction (which accounts for the emergence of magnetic Skyrmions). Moreover, our proof gives a template on how to approach weakstrong uniqueness for even more complicated problems, where LLG is (nonlinearly) coupled to other (nonlinear) PDE systems.

Homogenization of the Landau-Lifshitz equation

“…In contrast, our goal is to study how classical solutions to (1.1) can be approximated by the homogenized solution and associated correction terms. We note that while existence of weak solutions to (1.1) is shown in [5], existence of classical solutions is only known for short times and/or for small initial data gradients, see for example [10,11,14,16,24]. In particular, in [10,11], the authors prove local existence and global existence given that the gradient of the initial data is sufficiently small.…”

Vanishing Gilbert damping limit problem of Landau–Lifshitz–Gilbert equation

“…However, the solution of LL equation still can not be got by taking the vanishing Gilbert limit of LLG equation, due to the relationship between solutions of the two equations is not clear, and even the explicit dynamic solution has not been seen so far. It can be seen from recent articles, for example, Guo and Huang [27] and Fratta and Innerberger [28], that the relationship between solutions of LL equation ( 2) and LLG equation (3) with large initial value or bounded domain is still an open problem. In this paper, we first reveal the fundamental reason why a solution of LL equation can not be generalized to a solution of LLG equation, and give a necessary and sufficient condition for the solutions of LL equation to be generalized to the solutions of LLG equation.…”

Strong solutions of the Landau-Lifshitz-Bloch equation in Besov space

“…Gutiérrez-De Laire [19] built the global existence of the self-similar solutions to LLG equation in any dimension under the hypothesis that the BMO semi-norm of the initial data is small. Di Fratta-Innerberger-Praetorius [13] proved the weak-strong uniqueness of the LLG equation on a bounded domain in R 3 . Very recently, Wang-Guo [37] deduced a blow-up criterion for the LLG equation on bounded domain with Neumann boundary condition in n-dimensional (n 2) space.…”