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Transition and Stability of 3D MHD Around Couette Flow

Niklas Knobel

TL;DR

The paper analyzes the 3D incompressible MHD equations around Couette flow with a vertical magnetic field, identifying a sharp nonlinear stability threshold at 5/6 ≤ γ ≤ 1 for Sobolev-data sized μ^{γ}. It develops a moving-frame formulation, decomposes dynamics into z-average and non-average parts, and proves linear damping and enhanced dissipation for non-average modes while revealing nonlinear transient growth in z-averaged components. The authors establish a precise energy-functional framework that remains bounded up to times of order μ^{−1/3}, proving global stability within the regime and demonstrating optimality of the threshold via explicit constructions showing nonlinear growth at γ=5/6 and lower bounds below it. They also show that, in 3d, the presence of a constant magnetic field stabilizes z-dependent modes but does not fully suppress the nonlinear coupling that drives z-averaged growth, yielding sharp Sobolev thresholds and insights into 2d reductions. This work sharpens the understanding of nonlinear stability thresholds in MHD shear flows and highlights the nuanced role of magnetic tension and shear in high-dimensional regimes.

Abstract

We study the three-dimensional incompressible magnetohydrodynamic (MHD) equations near Couette flow with a constant magnetic field perpendicular to the shear plane. Couette flow induces mixing and generates magnetic induction, while the constant magnetic field stabilizes $z$-dependent modes. In contrast, the $z$-averaged magnetic field exhibits algebraic growth. Letting $μ$ denote the inverse fluid and magnetic Reynolds numbers, we analyze how $μ$ governs stability thresholds in Sobolev spaces. We identify a nonlinear transient-growth regime characterized by the sharp threshold $5/6\le γ\le 1 $. For $x$-average-free initial data of size $μ^γ$, solutions are nonlinearly stable; however, for certain initial data, the solution departs from the linear dynamics at rate $μ^{γ-1}$ due to a first-order nonlinear instability. The exponent $γ= 5/6$ is optimal for the associated energy functional and cannot be improved in Sobolev spaces without secondary transient-growth mechanisms. Below this threshold, solutions necessarily transition away from the linear dynamics at a minimal rate. As a consequence, the $3d$ results yield sharp Sobolev stability thresholds for the $2d$ MHD equations around Couette flow without a constant magnetic field. In particular, the threshold is strictly larger than in prior $2d$ results with a constant field, revealing destabilizing effects normally suppressed by a constant magnetic field. Crucially, this stabilization is restricted to the direction of the magnetic field.

Transition and Stability of 3D MHD Around Couette Flow

TL;DR

The paper analyzes the 3D incompressible MHD equations around Couette flow with a vertical magnetic field, identifying a sharp nonlinear stability threshold at 5/6 ≤ γ ≤ 1 for Sobolev-data sized μ^{γ}. It develops a moving-frame formulation, decomposes dynamics into z-average and non-average parts, and proves linear damping and enhanced dissipation for non-average modes while revealing nonlinear transient growth in z-averaged components. The authors establish a precise energy-functional framework that remains bounded up to times of order μ^{−1/3}, proving global stability within the regime and demonstrating optimality of the threshold via explicit constructions showing nonlinear growth at γ=5/6 and lower bounds below it. They also show that, in 3d, the presence of a constant magnetic field stabilizes z-dependent modes but does not fully suppress the nonlinear coupling that drives z-averaged growth, yielding sharp Sobolev thresholds and insights into 2d reductions. This work sharpens the understanding of nonlinear stability thresholds in MHD shear flows and highlights the nuanced role of magnetic tension and shear in high-dimensional regimes.

Abstract

We study the three-dimensional incompressible magnetohydrodynamic (MHD) equations near Couette flow with a constant magnetic field perpendicular to the shear plane. Couette flow induces mixing and generates magnetic induction, while the constant magnetic field stabilizes -dependent modes. In contrast, the -averaged magnetic field exhibits algebraic growth. Letting denote the inverse fluid and magnetic Reynolds numbers, we analyze how governs stability thresholds in Sobolev spaces. We identify a nonlinear transient-growth regime characterized by the sharp threshold . For -average-free initial data of size , solutions are nonlinearly stable; however, for certain initial data, the solution departs from the linear dynamics at rate due to a first-order nonlinear instability. The exponent is optimal for the associated energy functional and cannot be improved in Sobolev spaces without secondary transient-growth mechanisms. Below this threshold, solutions necessarily transition away from the linear dynamics at a minimal rate. As a consequence, the results yield sharp Sobolev stability thresholds for the MHD equations around Couette flow without a constant magnetic field. In particular, the threshold is strictly larger than in prior results with a constant field, revealing destabilizing effects normally suppressed by a constant magnetic field. Crucially, this stabilization is restricted to the direction of the magnetic field.

Paper Structure

This paper contains 26 sections, 29 theorems, 397 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Change of Coordinates
  4. Literature Overview
  5. Notations and Conventions
  6. Effects and Heuristics
  7. The $2d$ MHD and the $z$-Average
  8. Heuristics on the threshold of the $z$-average
  9. Linear dynamics and Proof of Proposition \ref{['prop:lin']}
  10. Main Theorems and Energies
  11. Notation for nonlinear interaction
  12. Non-Average Energy Estimates
  13. Linear Estimates
  14. Bound on $NL_{\sim}$,
  15. Bound on $NL_{\diamond\to \sim }$,
  16. ...and 11 more sections

Key Result

Proposition 1

Let $(V, B) \in C([0,\infty); L^2(\mathbb{T} \times \mathbb{R} \times \mathbb{T}))$ solve MHD2lin with $x$-average free initial data $(V_{\mathrm{in}}, B_{\mathrm{in}}) \in H^2$. Then, for all $t > 0$: $\bullet$ Doubly nonzero modes undergo inviscid damping and enhanced dissipation: $\bullet$ The $z$-averaged, $x$-nonzero modes of $V$ undergo inviscid damping and enhanced dissipation: $\bullet$

Theorems & Definitions (37)

  • Proposition 1: Linearized stability
  • Theorem 1: Stable nonlinear transient growth
  • Theorem 2
  • Proposition 2
  • Remark 1.1
  • Corollary 1: Stability of $f$ and $g$
  • Corollary 2: Optimality the energy functional
  • Corollary 3: Below the threshold $5/6$
  • Remark 2.1
  • Theorem 3: Stable nonlinear transient growth region
  • ...and 27 more