Global smooth axisymmetric solutions to 2D compressible Euler equations of Chaplygin gases with non-zero vorticity

TL;DR

This paper proves the global existence of smooth solutions for 2D compressible Chaplygin gas Euler equations with small perturbations, contrasting with finite-time blow-up in polytropic gases, by analyzing wave equations satisfying null conditions.

Contribution

It demonstrates the global existence of smooth solutions for 2D Chaplygin gases with small perturbations, utilizing null conditions and detailed energy estimates, which is a novel extension beyond previous blow-up results.

Findings

Global smooth solutions exist for small perturbations.

Wave equations satisfy null conditions enabling global existence.

Distinct decay behaviors are identified and handled in energy estimates.

Abstract

For 2D compressible isentropic Euler equations of polytropic gases, when the rotationally invariant data are a perturbation of size $\ve>0$ of a rest state, S.~Alinhac in \cite{Alinhac92} and \cite{Alinhac93} establishes that the smooth solution blows up in finite time and the lifespan $T_{\ve}$ satisfies $\ds\lim_{\ve\to 0}\ve^2 T_{\ve}=\tau_{0}^2>0$. In the present paper, for 2D compressible isentropic Euler equations of Chaplygin gases, we shall show that the small perturbed smooth solution exists globally when the rotationally invariant data are a perturbation of size $\ve>0$ of a rest state. Near the light cone, 2D Euler equations of Chaplygin gases can be transformed into a second order quasilinear wave equation of potential, which satisfies both the first and the second null conditions. This will lead to that the corresponding second order quasilinear wave equation admits a global smooth solution near the light cone (see \cite{Alinhac01}). However, away from the light cone, the hydrodynamical waves of 2D Chaplygin gases have no decay in time and strongly affect the related acoustical waves. Thanks to introducing a nonlinear ODE and taking some delicate observations, we can distinguish the fast decay part and non-decay part explicitly so that the global energy estimates with different weights can be derived by involved analysis.