Remarks on the two-dimensional magnetohydrodynamics system forced by space-time white noise

TL;DR

This paper establishes the global existence and uniqueness of solutions for the two-dimensional magnetohydrodynamics system driven by space-time white noise, using a novel approach based on the structure of Maxwell's equations and paracontrolled calculus.

Contribution

It introduces a new method for analyzing stochastic MHD systems without explicit invariant measures, leveraging Maxwell's structure and paracontrolled calculus.

Findings

Proved global-in-time existence of solutions.

Established uniqueness of solutions.

Developed a novel analytical approach for stochastic MHD systems.

Abstract

We study the two-dimensional magnetohydrodynamics system forced by space-time white noise. Due to a lack of an explicit invariant measure, the approach of Da Prato and Debussche (2002, J. Funct. Anal., \textbf{196}, pp. 180--210) on the Navier-Stokes equations does not seem to fit. We follow instead the approach of Hairer and Rosati (2023, arXiv:2301.11059 [math.PR]), take advantage of the structure of Maxwell's equation, such as anti-symmetry, to find an appropriate paracontrolled ansatz and many crucial cancellations, and prove the global-in-time existence and uniqueness of its solution.