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arXiv · 2410.02436

Invariant measures and their limiting behavior of the Landau-Lifshitz-Bloch equation in unbounded domains

Daiwen Huang·Zhaoyang Qiu·Bixiang Wang
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Abstract

This paper deals with the existence and limiting behavior of invariant measures of the stochastic Landau-Lifshitz-Bloch equation driven by linear multiplicative noise and additive noise defined in the entire space $\mathbb{R}^d$ for $d=1,2$, which describes the phase spins in ferromagnetic materials around the Curie temperature. We first establish the existence and uniqueness of solutions by a domain expansion method. We then prove the existence of invariant measures by the weak Feller argument. In the case $d=1$, we show the uniform tightness of the set of all invariant measures of the stochastic equation, and prove any limit of a sequence of invariant measures of the perturbed equation must be an invariant measure of the limiting system. The cut-off arguments, stopping time techniques and uniform tail-ends estimates of solutions are developed to overcome the difficulty caused by the high-order nonlinearity and the non-compactness of Sobolev embeddings in unbounded domains.

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Daiwen Huang, Zhaoyang Qiu, Bixiang Wang. 2024-10-03. Invariant measures and their limiting behavior of the Landau-Lifshitz-Bloch equation in unbounded domains. https://arxiv.org/abs/2410.02436

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