Mach limits in analytic spaces on exterior domains

TL;DR

This paper studies the behavior of solutions to the Euler equations in exterior domains with analytic boundaries as the Mach number approaches zero, establishing uniform bounds and convergence in analytic and Gevrey spaces.

Contribution

It introduces a method to handle Mach limits in exterior domains using analytic vector fields and norms, extending results to Gevrey initial data.

Findings

Uniform boundedness of solutions in analytic spaces independent of Mach number

Mach limit convergence in analytic and Gevrey norms

Extension of results to exterior domains with analytic boundaries

Abstract

We address the Mach limit problem for the Euler equations in an exterior domain with analytic boundary. We first prove the existence of tangential analytic vector fields for the exterior domain with constant analyticity radii, and introduce an analytic norm in which we distinguish derivatives taken from different directions. Then we prove the uniform boundedness of the solutions in the analytic space on a time interval independent of the Mach number, and Mach limit holds in the analytic norm. The results extends more generally to Gevrey initial data with convergence in a Gevrey norm.