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

TL;DR

This paper establishes the existence of global martingale solutions for stochastic Hall-magnetohydrodynamics equations in three-dimensional space with multiplicative noise, overcoming challenges posed by the nonlinear Hall term.

Contribution

It introduces a novel approach using Fourier analysis and stochastic compactness methods to handle the nonlinear Hall term in stochastic MHD equations.

Findings

Existence of global martingale solutions proved

Construction of approximate solutions via Fourier analysis

Application of stochastic compactness and Skorokhod theorem

Abstract

We prove the existence of a global martingale solution of a stochastic Hall-magnetohydrodynamics equations on $\mathbb{R}^3$ with multiplicative noise. Using the Fourier analysis we construct a sequence of approximate solutions. The existence of a solution is proved via the stochastic compactness method and the Jakubowski generalization of the Skorokhod theorem for nonmetric spaces, in particular, the spaces with weak topologies. The main difficulty is caused by the Hall term which makes the equations strongly nonlinear.