The stochastic tamed MHD equations: existence, uniqueness and invariant measures

Abstract

We study the tamed magnetohydrodynamics equations, introduced recently in a paper by the author, perturbed by multiplicative Wiener noise of transport type on the whole space $${\mathbb {R}}^{3}$$ R 3 and on the torus $${\mathbb {T}}^{3}$$ T 3 . In a first step, existence of a unique strong solution are established by constructing a weak solution, proving that pathwise uniqueness holds and using the Yamada–Watanabe theorem. We then study the associated Markov semigroup and prove that it has the Feller property. Finally, existence of an invariant measure of the equation is shown for the case of the torus.

Martingale solutions of the stochastic Hall-magnetohydrodynamics equations on $\mathbb{R}^3$

“…Stochastic MHD equations with the Gaussian noise were considered, e.g. in [5], [13], [20], [21], [31], [32], [34], [41].…”

Three-dimensional magnetohydrodynamics system forced by space-time white noise

“…Let us emphasize that we just saw that the first and third nonlinear terms Based on the energy bound, mathematical analysis of the MHD system was pioneered by Duvaut and Lions [20] and fundamental results such as the global existence of a Leray-Hopf weak solution in both cases d ∈ {2, 3} and its uniqueness in case d = 2 can be found in [50,Theorem 3.1]. In the stochastic case, various results were obtained by many researchers in case noise is white only in time: the existence of a global-in-time weak solution in cases of additive and multiplicative noise in the three-dimensional (3D) case, along with path-wise uniqueness in the 2D case as long as the noise is Lipschitz [43,48,52]; ergodicity in case of an additive noise in the 2D case [6]; large deviation principle in the 2D case [17]; Markov selection, irreducibility, and strong Feller property in the 3D case [63,65]; tamed stochastic MHD system [49].…”