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Sobolev Stability for the 2D MHD Equations in the Non-Resistive Limit

Niklas Knobel

Abstract

In this article, we consider the stability of the 2D magnetohydrodynamics (MHD) equations close to a combination of Couette flow and a constant magnetic field. We consider the ideal conductor limit for the case when viscosity $ν$ is larger than resistivity $κ$, $ν\ge κ>0$. For this regime, we establish a bound on the Sobolev stability threshold. Furthermore, for $κ\le ν^3$ this system exhibits instability, which leads to norm inflation of size $νκ^{-\frac 1 3 }$.

Sobolev Stability for the 2D MHD Equations in the Non-Resistive Limit

Abstract

In this article, we consider the stability of the 2D magnetohydrodynamics (MHD) equations close to a combination of Couette flow and a constant magnetic field. We consider the ideal conductor limit for the case when viscosity is larger than resistivity , . For this regime, we establish a bound on the Sobolev stability threshold. Furthermore, for this system exhibits instability, which leads to norm inflation of size .
Paper Structure (11 sections, 8 theorems, 146 equations)

This paper contains 11 sections, 8 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Linear Stability
  3. Sobolev Stability for the Nonlinear System
  4. Linear estimates
  5. Nonlinear terms without an $x$-average
  6. Nonlinear terms with an $x$-average in the second component
  7. Nonlinear terms with an $x$-average in the third component
  8. Nonlinear terms with an $x$-average in first component
  9. Other nonlinear terms
  10. Construction of the Weights
  11. Local Wellposedness

Key Result

Theorem 1.1

Consider $\alpha >\tfrac{1}{2}$, $N\ge 5$ and a small enough constant $c_0=c_0(\alpha )>0$. Let $0<\kappa \le \nu \le\tfrac{1}{ 40} (1-\tfrac{1}{2\alpha })^{\frac{6}{5}}$, then we obtain Sobolev stability for initial data which is sufficiently small in Sobolev spaces, where the estimates qualitativ In particular, we obtain Lipschitz stability for the Lipschitz constant $L\approx \max( 1, \nu \kap

Theorems & Definitions (14)

  • Theorem 1.1
  • Proposition 1
  • proof
  • Theorem 3.1
  • Proposition 3.1: Control of errors
  • proof : Proof of Theorem \ref{['thm:comp']}
  • Lemma 2: $M_L$ properties
  • proof
  • Lemma 3: Enhanced dissipation estimates
  • proof
  • ...and 4 more