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Large amplitude traveling waves for the non-resistive MHD system

Gennaro Ciampa, Riccardo Montalto, Shulamit Terracina

TL;DR

The paper proves the existence of large-amplitude bi-periodic traveling waves for the two-dimensional non-resistive MHD system on the torus, driven by a traveling external force with large speed. The authors develop a Nash–Moser scheme tailored to a non-perturbative setting, combining micro-local analysis and normal-form transformations to invert the linearized operator despite large variable-coefficient perturbations and small divisors. A decoupling strategy reduces the linearized system to a scalar transport-type problem, enabling a quantitative inversion with tame estimates and high/low-frequency analysis in the large parameter $λ$. An approximate traveling-wave solution is constructed; together with the Nash–Moser iteration, this yields global-in-time, large-amplitude quasi-periodic solutions for a broad set of frequencies, with precise bounds on the velocity and magnetic-field components. The results extend the scope of KAM-type techniques to high-dimensional PDEs with large forcing and contribute new insights into non-resistive MHD dynamics under quasi-periodic forcing.

Abstract

We prove the existence of large amplitude bi-periodic traveling waves (stationary in a moving frame) of the two-dimensional non-resistive Magnetohydrodynamics (MHD) system with a traveling wave external force with large velocity speed $λ(ω_1, ω_2)$ and of amplitude of order $O(λ^{1^+})$ where $λ\gg 1$ is a large parameter. For most values of $ω= (ω_1, ω_2)$ and for $λ\gg 1$ large enough, we construct bi-periodic traveling wave solutions of arbitrarily large amplitude as $λ\to + \infty$. More precisely, we show that the velocity field is of order $O(λ^{0^+})$, whereas the magnetic field is close to a constant vector as $λ\to + \infty$. Due to the presence of small divisors, the proof is based on a nonlinear Nash-Moser scheme adapted to construct nonlinear waves of large amplitude. The main difficulty is that the linearized equation at any approximate solution is an unbounded perturbation of large size of a diagonal operator and hence the problem is not perturbative. The invertibility of the linearized operator is then performed by using tools from micro-local analysis and normal forms together with a sharp analysis of high and low frequency regimes w.r. to the large parameter $λ\gg 1$. To the best of our knowledge, this is the first result in which global in time, large amplitude solutions are constructed for the 2D non-resistive MHD system with periodic boundary conditions and also the first existence results of large amplitude quasi-periodic solutions for a nonlinear PDE in higher space dimension.

Large amplitude traveling waves for the non-resistive MHD system

TL;DR

The paper proves the existence of large-amplitude bi-periodic traveling waves for the two-dimensional non-resistive MHD system on the torus, driven by a traveling external force with large speed. The authors develop a Nash–Moser scheme tailored to a non-perturbative setting, combining micro-local analysis and normal-form transformations to invert the linearized operator despite large variable-coefficient perturbations and small divisors. A decoupling strategy reduces the linearized system to a scalar transport-type problem, enabling a quantitative inversion with tame estimates and high/low-frequency analysis in the large parameter . An approximate traveling-wave solution is constructed; together with the Nash–Moser iteration, this yields global-in-time, large-amplitude quasi-periodic solutions for a broad set of frequencies, with precise bounds on the velocity and magnetic-field components. The results extend the scope of KAM-type techniques to high-dimensional PDEs with large forcing and contribute new insights into non-resistive MHD dynamics under quasi-periodic forcing.

Abstract

We prove the existence of large amplitude bi-periodic traveling waves (stationary in a moving frame) of the two-dimensional non-resistive Magnetohydrodynamics (MHD) system with a traveling wave external force with large velocity speed and of amplitude of order where is a large parameter. For most values of and for large enough, we construct bi-periodic traveling wave solutions of arbitrarily large amplitude as . More precisely, we show that the velocity field is of order , whereas the magnetic field is close to a constant vector as . Due to the presence of small divisors, the proof is based on a nonlinear Nash-Moser scheme adapted to construct nonlinear waves of large amplitude. The main difficulty is that the linearized equation at any approximate solution is an unbounded perturbation of large size of a diagonal operator and hence the problem is not perturbative. The invertibility of the linearized operator is then performed by using tools from micro-local analysis and normal forms together with a sharp analysis of high and low frequency regimes w.r. to the large parameter . To the best of our knowledge, this is the first result in which global in time, large amplitude solutions are constructed for the 2D non-resistive MHD system with periodic boundary conditions and also the first existence results of large amplitude quasi-periodic solutions for a nonlinear PDE in higher space dimension.
Paper Structure (25 sections, 45 theorems, 443 equations)

This paper contains 25 sections, 45 theorems, 443 equations.

Table of Contents

  1. Introduction
  2. Main result
  3. Main ideas, novelties and strategy of the proof
  4. Functions spaces, norms and linear operators
  5. Diophantine equations
  6. Linear operators
  7. Pseudo-differential operators and norms
  8. A quantitative Egorov Theorem
  9. The linearized operator
  10. Decoupling of linearized operator up to a smoothing remainder
  11. Decoupling of the first order part
  12. Construction of $\Psi$
  13. The iterative step
  14. Inversion of the first equation
  15. Inversion of the operator ${\mathcal{L}}_1^{(1)}$
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $f \in {\mathcal{C}}^\infty(\mathbb{T}^2, \mathbb{R}^2)$ with zero average and $\mathbf b \in \mathbb{R}^2 \setminus \{ 0 \}$ satisfies assumption b f. There exist $\bar{S}>0$, $\eta \in (0, 1)$ such that for any $S\geq\bar{S}$, there exist $\lambda_0\equiv\lambda_0(f, \eta, S, \mathbf b)\gg 1$

Theorems & Definitions (81)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1: Weighted norms
  • Lemma 2.2: Smoothing estimates
  • Lemma 2.3
  • Lemma 2.4
  • Definition 2.5
  • Lemma 2.6
  • Definition 2.7
  • Definition 2.8: Pseudo-differential operators and symbols
  • ...and 71 more