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

TL;DR

This work proves global existence of smooth axisymmetric solutions to the 2D compressible Euler equations for Chaplygin gases with nonzero vorticity under small perturbations of a rest state. The authors recast the problem in axisymmetric variables $(v,f,g)$, introduce a nonlinear decomposition $v= ilde{v}+G$ via $(1+v)rac{ abla_r G}{dr}+rac{g^2}{r}=0$, and develop a mixed energy framework with region-specific weights to separately handle decaying and non-decaying parts. Near the light cone, the system reduces to a quasilinear wave equation with first and second null conditions, enabling ghost-weight energy estimates; away from the light cone, strong hydrodynamic effects are managed by the nonlinear decomposition and weighted energies, together with detailed $L^p$-type controls of vorticity. The combination of null-condition analysis, ghost weights, and a nonlinear ODE-based decoupling yields global smooth axisymmetric solutions for small data, advancing understanding of Chaplygin-gas dynamics with nonzero vorticity in 2D.

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 $\ds\lim_{\ve\to 0}\ve^2 T_{\ve}=τ_{0}^2>0$. 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.