Low Mach number limit for the isentropic Euler system with axisymmetric initial data

TL;DR

This paper investigates the low Mach number limit of the isentropic Euler system with axisymmetric initial data without swirl, establishing lifespan bounds and convergence to incompressible Euler solutions under different regularity conditions.

Contribution

It provides new lifespan estimates and convergence results for the low Mach number limit in both subcritical and critical regularity settings, utilizing Strichartz estimates and vorticity structure.

Findings

Lifespan of solutions is bounded below by a triple logarithm of inverse Mach number.

Solutions converge to incompressible Euler solutions as Mach number approaches zero.

Critical regularity case presents additional challenges due to the Beale-Kato-Majda criterion and Strichartz estimate scaling.

Abstract

This paper is devoted to the study of the low Mach number limit for the isentropic Euler system with axisymmetric initial data without swirl. In the first part of the paper we analyze the problem corresponding to the subcritical regularities, that is $H^s$ {with $s>\frac52$.} Taking advantage of the Strichartz estimates and using the special structure of the vorticity we show that the {lifespan $T_\epsilon$} of the solutions is bounded below by $\log\log\log\frac1\epsilon$, where $\epsilon$ denotes the Mach number. Moreover, we prove that the incompressible parts converge to the solution of the incompressible Euler system, when the parameter $\epsilon$ goes to zero. In the second part of the paper we address the same problem but for the Besov critical regularity $B_{2,1}^{\frac52}$. This case turns out to be more subtle at least due to two facts. The first one is related to the Beale-Kato-Majda criterion which is not known to be valid for rough regularities. The second one concerns the critical aspect of the Strichartz estimate $L^1_TL^\infty$ for the acoustic parts $(\nabla\Delta^{-1}\diver\vepsilon,\cepsilon)$: it scales in the space variables like the space of the initial data.