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

TL;DR

This paper proves the global nonlinear stability of vortex sheets in the Navier-Stokes equations with large data and small Mach number, revealing key cancellation properties and decay rates that justify the incompressible limit.

Contribution

It introduces an auxiliary flow and a novel decomposition approach to handle large amplitude vortex sheets, establishing uniform global stability results.

Findings

Established global stability of vortex sheets with large data

Discovered cancellation properties among zero modes

Justified the incompressible limit for the Navier-Stokes equations

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.