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

Abstract

Abstract:We study a stochastic Landau-Lifschitz-Gilbert Equations with non-zero anisotrophy energy and multidimensional noise. We prove the existence and some regularities of weak solution proved. Our paper is motivated by finite-dimensional study of stochastic LLGEs or general stochasric differential equations with constraints studied by Kohn et al [17] and Lelièvre et al [19].

Large Deviations and Transitions Between Equilibria for Stochastic Landau–Lifshitz–Gilbert Equation

“…Here we only sketch the main arguments. Full details can be found in [12]. It is sufficient to prove the theorem for a bounded time interval [0, T ].…”

Stochastic Landau–Lifshitz–Gilbert equation with anisotropy energy driven by pure jump noise

“…We wish to use the notation u × ∆u for our weak martingale solution u even when we do not know that u has weak second order derivatives. First of all let us recall (see Appendix A in [8]) that for u ∈ H we say u × ∆u ∈ L 2 iff there exists…”

“…Brzeźniak et al [4] considered the above equation when φ = 0 and is perturbed by a Gaussian noise in the Stratonovich sense, and proved existence of weak martingale solutions taking values in a sphere S 2 . In a later paper Brzeźniak and Li [8], the result was generalised in the presence of anisotropy energy. In a very recent work [9], we have addressed existence of weak martingale solution of the stochastic LLGE in the absence of anisotropy energy.…”

Weak solutions of the Landau–Lifshitz–Bloch equation

“…Proof. The proof of this lemma can be found in [7]; for the reader's convenience we recall the proof as follows.…”