Well-posedness and invariant measures for the stochastically perturbed Landau-Lifshitz-Baryakhtar equation

2024/05/24 by Fan Xu, Lei Zhang, Xu, Fan +3
Economics, Econometrics and Finance · Mathematics · #Stochastic processes and financial applications #Differential Equations and Numerical Methods #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2405.15666

Abstract

In this paper, we study the initial-boundary value problem for the stochastic Landau-Lifshitz-Baryakhtar (SLLBar) equation with Stratonovich-type noise in bounded domains O⊂ℝ^d, d=1,2,3. Our main results can be briefly described as follows: (1) for d=1,2,3 and any u₀∈ℍ¹, the SLLBar equation admits a unique local-in-time pathwise weak solution; (2) for d=1 and small-data u₀∈ℍ¹, the SLLBar equation has a unique global-in-time pathwise weak solution and at least one invariant measure; (3) for d=1,2 and small-data u₀∈\mathbbL², the SLLBar equation possesses a unique global-in-time pathwise very weak solution and at least one invariant measure, while for d=3 only the existence of martingale solution is obtained due to the loss of pathwise uniqueness.

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