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Non-uniform Continuity for the MHD equations with only Magnetic Diffusion

Quansen Jiu, Yaowei Xie

TL;DR

The paper proves that the data-to-solution map for the incompressible resistive MHD equations with only magnetic diffusion is not uniformly continuous in Sobolev spaces $H^s$ for any $s>0$ in dimensions $d=2,3$. It adopts a frequency-decomposition strategy, constructing low-frequency magnetic fields and high-frequency velocity perturbations, with diffusion-induced cancellations to control the error terms. It further analyzes the impact of a constant background magnetic field $\mathbf{B_0}$ by a coordinate transformation that cancels leading linear terms, showing that magnetic stabilization does not restore uniform continuity of the data-to-solution map. The results reveal refined continuity properties of the solution map and clarify how diffusion and background fields interact in the ill-posedness landscape of MHD.

Abstract

In this paper, we prove the non-uniform continuity of the data-to-solution map for the incompressible magnetohydrodynamic (MHD) equations with only magnetic diffusion in Sobolev spaces $H^s(\mathbb{R}^d)$ for all $s>0$ and $d=2,3$. Our results are first studies on the non-uniform continuity of the data-to-solution map for the resistive MHD equations. Moreover, our results permit the solution perturbation around an arbitrary constant background magnetic fields $\mathbf{B_0} \in \mathbb{R}^d$, which reveal that the strong magnetic background fields may provide the stabilization effect but still preserve the analytical feature of non-uniform continuity of the data-to-solution map.

Non-uniform Continuity for the MHD equations with only Magnetic Diffusion

TL;DR

The paper proves that the data-to-solution map for the incompressible resistive MHD equations with only magnetic diffusion is not uniformly continuous in Sobolev spaces for any in dimensions . It adopts a frequency-decomposition strategy, constructing low-frequency magnetic fields and high-frequency velocity perturbations, with diffusion-induced cancellations to control the error terms. It further analyzes the impact of a constant background magnetic field by a coordinate transformation that cancels leading linear terms, showing that magnetic stabilization does not restore uniform continuity of the data-to-solution map. The results reveal refined continuity properties of the solution map and clarify how diffusion and background fields interact in the ill-posedness landscape of MHD.

Abstract

In this paper, we prove the non-uniform continuity of the data-to-solution map for the incompressible magnetohydrodynamic (MHD) equations with only magnetic diffusion in Sobolev spaces for all and . Our results are first studies on the non-uniform continuity of the data-to-solution map for the resistive MHD equations. Moreover, our results permit the solution perturbation around an arbitrary constant background magnetic fields , which reveal that the strong magnetic background fields may provide the stabilization effect but still preserve the analytical feature of non-uniform continuity of the data-to-solution map.
Paper Structure (7 sections, 5 theorems, 51 equations)

This paper contains 7 sections, 5 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Auxiliary Analysis Tools
  4. Local well-posedness
  5. The Case ${\bf B_0}={\bf 0}$
  6. Approximate Solutions Scheme
  7. Error Estimates for the Approximate Solutions

Key Result

Theorem 1.1

Let $s>0, \lambda>0, d=2,3$ and $T>0$. Then the data-to-solution map $(u_0,b_0)\mapsto(u,b)$ for the equations mhd is non-uniformly continuous from a bounded subset in $H^{s}(\mathbb{R}^d)\times H^{s}(\mathbb{R}^d)$ into $C\left([0,T],H^{s}(\mathbb{R}^d)\right)\times C\left([0,T],H^{s}(\mathbb{R}^d)

Theorems & Definitions (9)

  • Definition 1.1
  • Theorem 1.1
  • Remark 1.1
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 3.1