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Large norm inflation of the current in the viscous, non-resistive magnetohydrodynamics equations

Michele Dolce, Niklas Knobel, Christian Zillinger

Abstract

We consider the ideally conducting, viscous magnetohydrodynamics (MHD) equations in two dimensions. Specifically, we study the nonlinear dynamics near a combination of Couette flow and a constant magnetic field in a periodic infinite channel. In contrast to the Navier-Stokes equations this system is shown to exhibit algebraic instability and large norm inflation of the magnetic current on non-perturbative time scales.

Large norm inflation of the current in the viscous, non-resistive magnetohydrodynamics equations

Abstract

We consider the ideally conducting, viscous magnetohydrodynamics (MHD) equations in two dimensions. Specifically, we study the nonlinear dynamics near a combination of Couette flow and a constant magnetic field in a periodic infinite channel. In contrast to the Navier-Stokes equations this system is shown to exhibit algebraic instability and large norm inflation of the magnetic current on non-perturbative time scales.

Paper Structure

This paper contains 21 sections, 12 theorems, 256 equations.

Table of Contents

  1. Introduction
  2. Linear Stability and Adapted Unknowns
  3. Nonlinear Resonances and Short Time Stability
  4. Nonlinear Resonances and Paraproducts
  5. Main Resonance Mechanisms
  6. Nonlinear Stability: Energy Functionals and Bootstrap Approach
  7. Weights and Frequency Sets
  8. Useful sets
  9. Definition of weights
  10. Properties of the weights
  11. Nonlinear estimates
  12. Bound on the nonlinear action of $\phi$ on $G$
  13. Bound on the nonlinear action of $G$ on $\phi$
  14. The $NL_\phi$ estimate
  15. The $NL_G$ Estimate
  16. ...and 6 more sections

Key Result

Proposition 1

Let $\alpha=1=\nu$. Then for every $N\geq 5$ there exists $\epsilon_0>0$ such that for any $0<\epsilon<\epsilon_0$ and any mean free initial data with the corresponding solution satisfies the bound for all times $0<t \ll \epsilon^{-1/2}$.

Theorems & Definitions (24)

  • Proposition 1: Perturbative time-scale
  • Remark 1
  • Theorem 1
  • Proposition 2
  • Remark 2
  • proof : Proof of Proposition \ref{['prop:linear']}
  • Lemma 3
  • proof : Proof of Lemma \ref{['lem:perturbative']}
  • Remark 4: On the zero-mode energy
  • Lemma 5
  • ...and 14 more