Global well-posedness of the Landau–Lifshitz–Gilbert equation for initial data in Morrey spaces

Abstract

We establish the global well-posedness of the Landau-Lifshitz-Gilbert equation in R n for any initial data m 0 ∈ H 1 * (R n , S 2 ) whose gradient belongs to the Morrey space M 2,2 (R n ) with small norm ∇m 0 M 2,2 (R n ) . The method is based on priori estimates of a dissipative Schrödinger equation of Ginzburg-Landau types obtained from the Landau-Lifshitz-Gilbert equation by the moving frame technique.

The Cauchy problem for the Landau–Lifshitz–Gilbert equation in BMO and self-similar solutions

“…for N ≥ 2, Theorem 1.1 can be seen as generalization of these results since it covers the case of less regular initial conditions. The arguments in [30,31] are based on the method of moving frames that produces a covariant complex Ginzburg-Landau equation. In Subsection 3.3 we give more details and discuss the corresponding equation in the one-dimensional case and provide some properties related to the self-similar solutions.…”

“…A few remarks about previously known results in this setting are in order. In the case α > 0, global well-posedness results for (LLG α ) have been established in N ≥ 2 by Melcher [31] and by Lin, Lai and Wang [30] for initial conditions with a smallness condition on the gradient in the L N (R N ) and the Morrey M 2,2 (R N ) norm 4 , respectively. Therefore these results do not apply to the initial condition m 0 c,α .…”

“…for some Q ∈ S 2 and ε > 0 small. Later, Lin, Lan and Wang [30] improved this result and proved global well-posedness under the conditions…”

“…for N ≥ 2, we deduce that Theorem 1.1 includes initial conditions with less regularity, as long as their essential range is not S 2 . The argument in [30,31] is based on the method of moving frames that produces a covariant complex Ginzburg-Landau equation. One of the aims of this subsection is to compare their approach in the context of the self-similar solutions m c,α , and in particular to draw attention to a possible difficulty in using it to study these solutions.…”

Energy decay rate of multidimensional inhomogeneous Landau–Lifshitz–Gilbert equation and Schrödinger map equation on the sphere

“…For a smallness initial condition on the gradient, the global well-posedness results for (1) have been established in n ≥ 2 by Melcher [17]. Some further works about the smallness condition for well-posedness of (1) were done by Lin, Lai, and Wang [15] in the Morrey space.…”