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Global nonlinear stability of vortex sheets for the Navier-Stokes equations with large data

Qian Yuan, Wenbin Zhao

Abstract

This paper concerns the global nonlinear stability of vortex sheets for the Navier-Stokes equations. When the Mach number is small, we allow both the amplitude and vorticity of the vortex sheets to be large. We introduce an auxiliary flow and reformulate the problem as a vortex sheet with small vorticity but subjected to a large perturbation. Based on the decomposition of frequency, the largeness of the perturbation is encoded in the zero modes of the tangential velocity. We discover an essential cancellation property that there are no nonlinear interactions among these large zero modes in the zero-mode perturbed system. This cancellation is owing to the shear structure inherent in the vortex sheets. Furthermore, with the aid of the anti-derivative technique, we establish a faster decay rate for the large zero modes. These observations enable us to derive the global estimates for strong solutions that are uniform with respect to the Mach number. As a byproduct, we can justify the incompressible limit.

Global nonlinear stability of vortex sheets for the Navier-Stokes equations with large data

Abstract

This paper concerns the global nonlinear stability of vortex sheets for the Navier-Stokes equations. When the Mach number is small, we allow both the amplitude and vorticity of the vortex sheets to be large. We introduce an auxiliary flow and reformulate the problem as a vortex sheet with small vorticity but subjected to a large perturbation. Based on the decomposition of frequency, the largeness of the perturbation is encoded in the zero modes of the tangential velocity. We discover an essential cancellation property that there are no nonlinear interactions among these large zero modes in the zero-mode perturbed system. This cancellation is owing to the shear structure inherent in the vortex sheets. Furthermore, with the aid of the anti-derivative technique, we establish a faster decay rate for the large zero modes. These observations enable us to derive the global estimates for strong solutions that are uniform with respect to the Mach number. As a byproduct, we can justify the incompressible limit.

Paper Structure

This paper contains 35 sections, 36 theorems, 351 equations.

Table of Contents

  1. Introduction
  2. Prior works on instability/stability of vortex sheets
  3. Large time behavior of vortex sheets for Navier-Stokes equations
  4. Main Results
  5. Difficulties and novelties in the proof
  6. Outline of the paper
  7. Ansatz and reformulation of problem
  8. An auxiliary flow
  9. An ansatz and the anti-derivative variables
  10. Reformulation of the Problem
  11. Proof of the main result
  12. Useful lemmas
  13. Local existence and a priori estimates
  14. Local existence
  15. A priori estimates
  16. ...and 20 more sections

Key Result

Theorem 1.1

Given a vortex layer profile-t0, where $\bar{\rho}>0, t_0>0, \bar{u}_1$ and $\bar{u}_2$ in profile-t0 are arbitrary constants. Assume that the initial perturbations $(b_0, \mathbf{v}_0) \in H^3_{3/2}$, and denote Then there exist such that given any $M_0 >0,$ if then the Cauchy problem NS, ic admits a unique strong solution, $(\rho^{\textcolor{red}{\varepsilon}},\mathbf{u}^{\textcolor{red}{\var

Theorems & Definitions (63)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 53 more