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} and on the torus {mathbb {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.
Highlights
In this paper, we consider a randomly perturbed version of the tamed MHD (TMHD) equations introduced recently in [42]
We consider a randomly perturbed version of the tamed MHD (TMHD) equations introduced recently in [42]. This aims at modelling the turbulent behaviour of a flow of electrically conducting fluids through porous media
We study existence and uniqueness of strong solutions, as well as existence of invariant
Summary
We consider a randomly perturbed version of the tamed MHD (TMHD) equations introduced recently in [42] This aims at modelling the turbulent behaviour of a flow of electrically conducting fluids through porous media. Yamazaki [51] proved ergodicity in the case of random forcing by finitely many modes In three dimensions, he proved ergodicity of a Faedo–Galerkin approximation of the MHD equations for degenerate noise in [50]. From a physical point of view, the fact that our model is most appropriate for low to moderate Reynolds numbers, cf [42, Section 1.2.1], raises the question of whether or not a stochastic model for turbulence (which is commonly associated with high Reynolds numbers) is appropriate in this setting It is, an interesting mathematical problem that we want to address in this work. The randomness can be seen as a model for other features of our system, including uncertainty
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