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Existence of a unique solution and invariant measures for the stochastic Landau-Lifshitz-Bloch equation. (English) Zbl 1448.35493

Summary: The Landau-Lifshitz-Bloch equation perturbed by a space-dependent noise was proposed in [10] as a model for evolution of spins in ferromagnetic materials at the full range of temperatures, including the temperatures higher than the Curie temperature. In the case of a ferromagnet filling a bounded domain \(D \subset \mathbb{R}^d, d = 1, 2, 3\), we show the existence of strong (in the sense of PDEs) martingale solutions. Furthermore, in cases \(d = 1, 2\) we prove uniqueness of pathwise solutions and the existence of invariant measures.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
35K59 Quasilinear parabolic equations
35R15 PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables)
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
35D35 Strong solutions to PDEs
78A25 Electromagnetic theory (general)
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References:

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