Stochastic generalized magnetohydrodynamics equations with not regular multiplicative noise: Well-posedness and invariant measure

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:474

Article

Idriss, Ali Zakaria¹  Razafimandimby, Paul Andre² 

[1] Univ South Africa, Dept Math Sci, ZA-0005 Florida, South Africa

[2] Univ Pretoria, Dept Math & Appl Math, Lynnwood Rd, ZA-0002 Pretoria, South Africa

关键词: Martingale solution;    Invariant measure;    Magnetohydrodynamics;    Yosida approximation;   

DOI  :  10.1016/j.jmaa.2019.02.026

来源: Elsevier

【 摘 要 】

In this paper we study a stochastic magnetohydrodynamics (MHD) system with fractional diffusion and resistivity (-Delta)(alpha), alpha > 0, in R-d, d = 2, 3. Our main goal is to identify the conditions on alpha under which we can prove the existence of a martingale solution, the pathwise uniqueness of solution and the existence of invariant measure when the noises are multiplicative and take values in functional space bigger than the space of square integrable functions. Roughly speaking, we prove that if alpha >= 1, theta is an element of (0, alpha) and the driving noises take values in H-theta, then the stochastic system has at least a weak martingale solution. We also establish the pathwise uniqueness of solution whenever alpha >= d/2. Finally, under the latter condition and under the addition of linear damping to the equations we are able to establish the existence of an invariant measure. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free