Weak Solutions of a Stochastic Landau-Lifshitz-Gilbert Equation

Abstract

The Landau-Lifshitz-Gilbert equation perturbed by a multiplicative spacedependent noise is considered for a ferromagnet filling a bounded three-dimensional domain. We show the existence of weak martingale solutions taking values in a sphere S 2 . The regularity of weak solutions is also discussed. Some of the regularity results are new even for the deterministic Landau-Lifshitz-Gilbert equation.

Weak solutions of the Stochastic Landau–Lifshitz–Gilbert Equations with nonzero anisotrophy energy

“…The proof of this result is based the a priori estimates (4.1)-(4.10). The proof of this result is omitted since it only a relatively simple modification of the proof of a corresponding result from [9].…”

“…A standard procedure is to approximate the full equation by some simpler problems. In the paper [9] we used Galerkin approximation, in a series of works with Banas, Prohl and Neklyudov culminating in a monograph [7], we used the finite element approximation. Here We follow the same method as used in Brzeźniak, Goldys and Jegaraj's paper [9] but with one important addition.…”

“…In the paper [9] we used Galerkin approximation, in a series of works with Banas, Prohl and Neklyudov culminating in a monograph [7], we used the finite element approximation. Here We follow the same method as used in Brzeźniak, Goldys and Jegaraj's paper [9] but with one important addition. We introduce, as in [9], finite dimensional subspaces H n of the Hilbert space H.…”

“…Here We follow the same method as used in Brzeźniak, Goldys and Jegaraj's paper [9] but with one important addition. We introduce, as in [9], finite dimensional subspaces H n of the Hilbert space H.…”

“…In Section 5 we state that the a priori estimates from the previous section are sufficient to prove that the laws of the solutions {u n } are tight on a suitable path space. The proof of this claim is omitted since it only a relatively simple modification of the proof of a corresponding result from [9].…”