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Well-posedness of the electron MHD with partial resistivity

Mimi Dai, Hassan Babaei

TL;DR

The paper addresses local well-posedness for the 2.5D electron MHD with a Hall nonlinearity under partial resistivity, showing existence and uniqueness in Sobolev spaces for either horizontal ($ ext{μ}>0$, $ ext ν=0$) or vertical ($ ext μ=0$, $ ext ν>0$) dissipation. Using a dyadic Littlewood–Paley framework, paraproduct decompositions, and commutator estimates, the authors derive sharp a priori bounds in $H^{s+1} imes H^{s}$ for $s>2$, exploiting cancellations of the Hall term to control nonlinear interactions. The resulting differential inequalities, solved by Grönwall, yield a local-in-time solution whose lifespan depends on the initial data and the dissipativity parameter. This work extends the mathematical understanding of EMHD with Hall nonlinearity by proving local well-posedness under partial resistivity and clarifying the role of horizontal vs. vertical dissipation in the 2.5D setting.

Abstract

Due to the singular nonlinear Hall term, the non-resistive electron magnetohydrodynamics (MHD) is not known to be locally well-posed in general. In this paper we consider the $2\frac12$D electron MHD with either horizontal or vertical resistivity and show local well-posedness in Sobolev spaces.

Well-posedness of the electron MHD with partial resistivity

TL;DR

The paper addresses local well-posedness for the 2.5D electron MHD with a Hall nonlinearity under partial resistivity, showing existence and uniqueness in Sobolev spaces for either horizontal (, ) or vertical (, ) dissipation. Using a dyadic Littlewood–Paley framework, paraproduct decompositions, and commutator estimates, the authors derive sharp a priori bounds in for , exploiting cancellations of the Hall term to control nonlinear interactions. The resulting differential inequalities, solved by Grönwall, yield a local-in-time solution whose lifespan depends on the initial data and the dissipativity parameter. This work extends the mathematical understanding of EMHD with Hall nonlinearity by proving local well-posedness under partial resistivity and clarifying the role of horizontal vs. vertical dissipation in the 2.5D setting.

Abstract

Due to the singular nonlinear Hall term, the non-resistive electron magnetohydrodynamics (MHD) is not known to be locally well-posed in general. In this paper we consider the D electron MHD with either horizontal or vertical resistivity and show local well-posedness in Sobolev spaces.

Paper Structure

This paper contains 10 sections, 2 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. Basic energy estimate
  5. Littlewood-Paley projection and commutator estimate
  6. Proof of Theorem \ref{['thm']} for $\mu>0$ and $\nu=0$
  7. Estimate of $I_1+K_1$
  8. Estimate of $I_2+K_2$
  9. The a priori estimate
  10. Proof of Theorem \ref{['thm']} for $\mu=0$ and $\nu>0$

Key Result

Theorem 1.1

Let $(a_0, b_0)\in H^{s+1}(\mathbb R^2)\times H^{s}(\mathbb R^2)$ be the initial data with $s>2$. Assume either $\{\mu>0, \nu=0\}$ or $\{\mu=0, \nu>0\}$. There exists $T>0$ depending on $\|a_0\|_{H^{s+1}}$, $\|b_0\|_{H^{s}}$ and $\mu$ (or $\nu$) such that system eq-ab has a unique solution $(a(t), b

Theorems & Definitions (2)

  • Theorem 1.1
  • Lemma 2.1