Low-Mach-number Euler equations with solid-wall boundary condition and general initial data

TL;DR

This paper proves that the divergence-free part of the compressible Euler equations with solid-wall boundary conditions converges to the incompressible Euler equations at the same order as the Mach number, even with general initial data.

Contribution

It demonstrates convergence without requiring initial data close to divergence-free states and analyzes the interaction of fast oscillations with slow dynamics.

Findings

Strong convergence of divergence-free component to incompressible Euler equations.

Persistence of large amplitude fast oscillations without dissipation.

Nonlinear coupling of oscillations contributes at the order of Mach number when averaged.

Abstract

We prove that the divergence-free component of the compressible Euler equations with solid-wall boundary condition converges strongly towards the incompressible Euler equations at the same order as the Mach number. General initial data are considered and are not necessarily close to the divergence-free state. Thus, large amplitude of fast oscillations persist and interact through nonlinear coupling without any dissipative or dispersive mechanism. It is then shown, however, that the contribution from fast oscillations to the slow dynamics through nonlinear coupling is of the same order as the Mach number when averaged in time. The structural condition of a vorticity equation plays a key role in our argument.