The Three-Dimensional Stochastic EMHD System: Local Well-Posedness and Maximal Pathwise Solutions

TL;DR

This paper establishes local well-posedness and maximal solutions for the 3D stochastic EMHD system with fractional dissipation, using advanced Sobolev energy methods and stochastic analysis techniques.

Contribution

It introduces a novel high-order Sobolev energy method combined with commutator estimates to prove well-posedness of the stochastic EMHD system.

Findings

Constructed martingale solutions for initial data in $L^2(\Omega; H^s)$.

Proved pathwise uniqueness using cancellations and stochastic Grönwall.

Established local pathwise well-posedness and maximal solutions.

Abstract

We study the three-dimensional stochastic electron magnetohydrodynamics (EMHD) system with fractional dissipation on the torus, driven by Stratonovich transport noise acting through divergence-free first-order operators. The noise generates an It\^o correction while preserving the transport structure of the Hall nonlinearity. Since the Hall term contains one more derivative, in the stochastic setting it must be controlled together with commutators arising from the transport operators. We develop a high-order Sobolev energy method based on Littlewood--Paley analysis and refined commutator estimates, which yields uniform bounds for Galerkin approximations in $H^s$ with $s > \tfrac{5}{2}$ together with suitable time regularity. Using stochastic compactness and identification of limits, we construct martingale solutions for initial data in $L^2(\Omega; H^s)$. Pathwise uniqueness follows from cancellations in the Hall term combined with a stochastic Gr\"onwall argument. An application of a Yamada--Watanabe type result then yields local pathwise well-posedness and the existence of maximal pathwise solutions.