Long time existence of smooth solutions to 2D compressible Euler equations of Chaplygin gases with non-zero vorticity
Fei Hou, Huicheng Yin
TL;DR
This paper proves long-time existence for smooth solutions to the two-dimensional compressible Euler equations of Chaplygin gases with nonzero vorticity, showing the lifespan scales as $T_\delta=\frac{\kappa}{\delta}$ with initial vorticity size $\delta$. The authors introduce a good unknown $g=u-\omega\sigma$, exploit the Chaplygin null structure, and develop weighted $L^{\infty}$-$L^{\infty}$ estimates for the linear wave components and the potential formulation, complemented by Helmholtz decompositions and $L^p$ vorticity control to close a bootstrap argument. Key contributions include identifying null-condition structures in Chaplygin-gas Euler flow, a novel weighted energy framework with ghost weights, and a robust treatment of vorticity via $L^p$ estimates that yields lifespan estimates $T_\delta=O(1/\delta)$. This work advances understanding of how vorticity governs long-time behavior in multidimensional compressible flows and provides a methodological template for similar nonlinear hyperbolic systems.
Abstract
For the 2D compressible isentropic Euler equations of polytropic gases with an initial perturbation of size $\ve$ of a rest state, it has been known that if the initial data are rotationnally invariant or irrotational, then the lifespan $T_{\ve}$ of the classical solutions is of order $O(\f{1}{\ve^2})$; if the initial vorticity is of size $\ve^{1+\al}$ ($0\le\al\le 1$), then $T_{\ve}$ is of $O(\f{1}{\ve^{1+\al}})$. In the present paper, for the 2D compressible isentropic Euler equations of Chaplygin gases, if the initial data are a perturbation of size $\ve$, and the initial vorticity is of any size $\dl$ with $0<\dl\le \ve$, we will establish the lifespan $T_{\dl}=O(\f{1}{\dl})$. For examples, if $\dl=e^{-\f{1}{\ve^2}}$ or $\dl=e^{-e^{\f{1}{\ve^2}}}$ are chosen, then $T_{\dl}=O(e^{\f{1}{\ve^2}})$ or $T_{\dl}=O(e^{e^{\f{1}{\ve^2}}})$ although the perturbations of the initial density and the divergence of the initial velocity are only of order $O(\ve)$. Our main ingredients are: finding the null condition structures in 2D compressible Euler equations of Chaplygin gases and looking for the good unknown; establishing a new class of weighted space-time $L^\infty$-$L^\infty$ estimates for the solution itself and its gradients of 2D linear wave equations; introducing some suitably weighted energies and taking the $L^p$ $(1<p<\infty)$ estimates on the vorticity.