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Global current-vortex sheets in the two-dimensional ideal incompressible MHD

Yuan Cai, Zhen Lei

Abstract

The magnetohydrodynamic current-vortex sheet is a free boundary problem involving a moving free surface separating two plasma regions. We prove the global nonlinear stability of current-vortex sheet in the two dimensional ideal incompressible magnetohydrodynamics under the strong horizontal background magnetic field. This appears to be the first result on the global solutions of the free boundary problems for the ideal (inviscid and non-resistive) incompressible rotational fluids. The strong magnetic field plays a crucial role in the global in time stabilization effect. The proof relies on the understanding of the interplay between the dynamics of the fluids inside the domain and on the free interface, a design of multiple-level energy estimates with different weights, and the inherent structures of the problem.

Global current-vortex sheets in the two-dimensional ideal incompressible MHD

Abstract

The magnetohydrodynamic current-vortex sheet is a free boundary problem involving a moving free surface separating two plasma regions. We prove the global nonlinear stability of current-vortex sheet in the two dimensional ideal incompressible magnetohydrodynamics under the strong horizontal background magnetic field. This appears to be the first result on the global solutions of the free boundary problems for the ideal (inviscid and non-resistive) incompressible rotational fluids. The strong magnetic field plays a crucial role in the global in time stabilization effect. The proof relies on the understanding of the interplay between the dynamics of the fluids inside the domain and on the free interface, a design of multiple-level energy estimates with different weights, and the inherent structures of the problem.

Paper Structure

This paper contains 41 sections, 47 theorems, 551 equations.

Table of Contents

  1. Introduction
  2. Review of related works
  3. Elsässer variables
  4. Main result
  5. Main ideas and sketch of the proof
  6. Alfvén waves and the null conditions
  7. Weighted energy estimates
  8. Further technical issues
  9. Further discussions
  10. Equations for the free surface, null condition and various energies
  11. Second order equations of the free surface
  12. Equations restricted to the free surface
  13. Perturbation variables, Alfvén waves, equations for the pressure, null conditions
  14. Energy notations and five parts energy estimate
  15. Preliminary weighted estimates
  16. ...and 26 more sections

Key Result

Theorem 1.1

Let $s\geq 4$ be an integer, $1/2< \mu \leq 3/5$ and $0 < c_0 < \frac{1}{2}$. Assume $\Lambda_{\pm}^0\in H^s(\Omega_{f_0}),\, \hat{\Lambda}_{\pm}^0\in H^s(\hat{\Omega}_{f_0})$, $f^0 \in \dot{H}^1({\mathbb{R}}) \cap \dot{H}^{s+\frac{1}{2}}({\mathbb{R}})\cap L^{\infty}({\mathbb{R}})$, $v^0\in H^{s-\f There exists a positive constant $\epsilon_0$ which depends only on $s,\ \mu$ and $c_0$ such that,

Theorems & Definitions (88)

  • Remark 1.1
  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 78 more