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Ideal Magnetohydrodynamics Around Couette Flow: Long Time Stability and Vorticity-Current Instability

Niklas Knobel

TL;DR

The paper analyzes the ideal 2D MHD equations on $\mathbb{T}\times\mathbb{R}$ near Couette flow with a constant magnetic field and proves existence up to $T \ge c_0\varepsilon^{-1}$ and nonlinear stability for Gevrey perturbations of size $\varepsilon$. It introduces tailored unknowns and a change of coordinates that align with the shear, coupling the linearized dynamics to the nonlinear terms, and develops a bootstrap energy method with time-dependent Gevrey weights to control resonances. A key finding is that vorticity and current grow like $O(t)$ while the velocity and magnetic field remain Gevrey-stable on the same time scale, suggesting a dynamo-like mechanism that can lead to breakdown without dissipation. The results extend to dissipative MHD when dissipation is small relative to the initial data, highlighting the delicate balance between nonlinear growth and damping and providing a framework for understanding long-time dynamics and potential turbulence in high-Reynolds-number regimes.

Abstract

This article considers the ideal 2D magnetohydrodynamic equations on an infinite periodic channel close to a combination of an affine shear flow, called Couette flow, and a constant magnetic field. This setting combines important physical effects of mixing and coupling of velocity and magnetic field. We establish the existence and stability of the velocity and magnetic field for Gevrey-class perturbations of size $\varepsilon$, valid up to times $t \sim \varepsilon^{-1}$. Additionally, the vorticity and current grow as $O(t)$ and there is no inviscid damping of the velocity and magnetic field. This has parallels to the above threshold case for the $3D$ Navier-Stokes \cite{bedrossian2022dynamics} where growth in `streaks' leads to time scales of $t\sim \varepsilon^{-1}$. In particular, for the ideal MHD equations, our article suggests that for a wide range of initial data, the scenario ``dynamo effect $\Rightarrow $ vorticity and current growth $\Rightarrow $ vorticity and current breakdown'' leads to instability and possible turbulences.

Ideal Magnetohydrodynamics Around Couette Flow: Long Time Stability and Vorticity-Current Instability

TL;DR

The paper analyzes the ideal 2D MHD equations on near Couette flow with a constant magnetic field and proves existence up to and nonlinear stability for Gevrey perturbations of size . It introduces tailored unknowns and a change of coordinates that align with the shear, coupling the linearized dynamics to the nonlinear terms, and develops a bootstrap energy method with time-dependent Gevrey weights to control resonances. A key finding is that vorticity and current grow like while the velocity and magnetic field remain Gevrey-stable on the same time scale, suggesting a dynamo-like mechanism that can lead to breakdown without dissipation. The results extend to dissipative MHD when dissipation is small relative to the initial data, highlighting the delicate balance between nonlinear growth and damping and providing a framework for understanding long-time dynamics and potential turbulence in high-Reynolds-number regimes.

Abstract

This article considers the ideal 2D magnetohydrodynamic equations on an infinite periodic channel close to a combination of an affine shear flow, called Couette flow, and a constant magnetic field. This setting combines important physical effects of mixing and coupling of velocity and magnetic field. We establish the existence and stability of the velocity and magnetic field for Gevrey-class perturbations of size , valid up to times . Additionally, the vorticity and current grow as and there is no inviscid damping of the velocity and magnetic field. This has parallels to the above threshold case for the Navier-Stokes \cite{bedrossian2022dynamics} where growth in `streaks' leads to time scales of . In particular, for the ideal MHD equations, our article suggests that for a wide range of initial data, the scenario ``dynamo effect vorticity and current growth vorticity and current breakdown'' leads to instability and possible turbulences.

Paper Structure

This paper contains 22 sections, 20 theorems, 188 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Ideas and Challenges of the Proof
  3. Change of coordinates
  4. Adapted Unknowns for the Linearized Dynamics
  5. Nonlinear Equations for the Adapted Unknowns
  6. Nonlinear Growth Mechanism
  7. Stability of the Nonlinear Equations
  8. Linear Terms
  9. Nonlinear Estimates
  10. Reaction Term
  11. Transport Term
  12. Remainder Term
  13. Average Term
  14. Other Nonlinear Terms
  15. The Low Frequency Average Estimates
  16. ...and 7 more sections

Key Result

Theorem 1

Let $\alpha \neq 0$ and $\lambda_1 > \lambda_2 >0$. Then for all $\tfrac{1}{2} <s\le 1$ and $N\ge 5$ there exists $c_0= c_0(\alpha,s, \lambda_1 ,\lambda_2 ) \in (0,1)$ such that the following holds true: Let $0<\varepsilon<c_0$ and consider mean and divergence-free initial data, which satisfies the Then the corresponding solution of MHD2 satisfies the following:

Figures (1)

  • Figure 1: Relation of different unknowns.

Theorems & Definitions (33)

  • Theorem 1
  • Corollary 1
  • Remark 1: On dissipation regimes
  • Remark 2: On time of stability and comparison to $3D$ Navier-Stokes equation
  • Remark 3: Choice of unknowns
  • Theorem 2
  • Lemma 1
  • proof
  • Proposition 1: Linear estimates
  • Proposition 2: Nonliner estimates
  • ...and 23 more