Well-posedness of the relaxed Electron MHD equations with random diffusion

Abstract

We study the three-dimensional Electron Magnetohydrodynamics (EMHD) equations without resistivity, a regime known to be ill-posed in Sobolev and Gevrey spaces due to the quasilinear nature of the system. Motivated by recent work on stochastic regularization of the inviscid primitive equations [R. Hu, Q. Lin, and R. Liu, J. Nonlinear Sci. 35:84 (2025)], we introduce a modified EMHD model where resistivity is replaced by multiplicative noise and the nonlinear term is regularized by a fractional derivative. In particular, the classical advection term $(B\cdot \nabla )J$ is replaced by its fractional version $(B\cdot {\nabla }^{\hat{I}}\pm )J$ with $0<\hat{I}\pm \le 1$. We show that for $\hat{I}\pm <1$, the system is locally well-posed almost surely in suitable Gevrey spaces, and globally well-posed with high probability for small initial data. The results demonstrate that stochastic perturbations can restore well-posedness in a broader class of quasilinear magnetic models relevant to plasma dynamics and turbulence.