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

Ruimeng Hu, Qirui Peng, Xu Yang

TL;DR

We address the ill-posedness of the 3D Electron MHD system without resistivity by introducing stochastic regularization: resistivity is replaced by multiplicative noise and the nonlinear advection is regularized with a fractional gradient $\nabla^\alpha$ for $0<\alpha\le 1$. After a stochastic change of variables $\Gamma = e^{-\mu W_t \Lambda^s}$, the system becomes a random PDE with damping $-\frac{1}{2}\mu^2 \Lambda^{2s}U$ and a nonlinear term $Q(U,V)$; we prove almost sure local well-posedness in Gevrey spaces for $\alpha\in(\tfrac{3}{4},1)$ and, for small initial data and large enough $\mu$, global well-posedness with high probability. The results demonstrate that stochastic perturbations can restore well-posedness for quasilinear magnetic models relevant to plasma dynamics and turbulence. The analysis relies on stochastic Gevrey estimates and sharp Fourier-space bilinear bounds that exploit the regularizing effect of random diffusion, offering a framework for noise-induced stabilization in complex PDE systems.

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 $(B \cdot \nabla^α)J$ with $0 < α\leq 1$. We show that for $α< 1$, 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.

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

TL;DR

We address the ill-posedness of the 3D Electron MHD system without resistivity by introducing stochastic regularization: resistivity is replaced by multiplicative noise and the nonlinear advection is regularized with a fractional gradient for . After a stochastic change of variables , the system becomes a random PDE with damping and a nonlinear term ; we prove almost sure local well-posedness in Gevrey spaces for and, for small initial data and large enough , global well-posedness with high probability. The results demonstrate that stochastic perturbations can restore well-posedness for quasilinear magnetic models relevant to plasma dynamics and turbulence. The analysis relies on stochastic Gevrey estimates and sharp Fourier-space bilinear bounds that exploit the regularizing effect of random diffusion, offering a framework for noise-induced stabilization in complex PDE systems.

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 is replaced by its fractional version with . We show that for , 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.

Paper Structure

This paper contains 18 sections, 7 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Main theorem and relevant previous works
  3. Main results
  4. Related work in the literature
  5. Well-posedness of the Hall and Electron-MHD systems
  6. Ill-posedness of the Hall and Electron-MHD systems
  7. Gevrey regularity and stochastic regularization
  8. Preliminaries
  9. Notations
  10. Sobolev spaces and Gevrey class
  11. Probability space
  12. Proof of Theorem \ref{['Thm:main_theorem_1']}
  13. Proof of Theorem \ref{['Thm:main_theorem_1']}
  14. Proof of Theorem \ref{['Thm:main_theorem_2']}
  15. Energy estimates
  16. ...and 3 more sections

Key Result

Theorem 2.1

Let $\frac{3}{4} < \alpha<1$, $\alpha_0, \gamma > 0$, $s \in [\frac{1+\alpha}{2},1]$ and $\sigma \in \left[\frac{1+\alpha}{s},2 \right)$. The equations eq:EMHD_random_1 are locally well-posed in the space $\dot{{\mathcal{G}}}^{\sigma,s}_{\gamma}$ for arbitrary initial data $B_0 \in \dot{{\mathcal{G}

Theorems & Definitions (13)

  • Theorem 2.1
  • Remark 2.1
  • Theorem 2.2
  • Remark 2.2
  • Lemma 3.1: R92, Proposition 6.8.1
  • Proposition 4.1
  • proof
  • Proposition 5.1
  • Lemma 5.2
  • proof
  • ...and 3 more