Delayed singularity formation for the three dimensional compressible Euler equations with non-zero vorticity

TL;DR

This paper investigates how initial vorticity influences the lifespan of smooth solutions to 3D compressible Euler equations, revealing that smaller vorticity extends the existence time significantly, especially for specific gas types.

Contribution

It establishes precise lifespan estimates for solutions based on initial vorticity size, highlighting the crucial role of vorticity in the formation of singularities.

Findings

Lifespan depends on initial vorticity size and type of gas.

For small vorticity, solutions exist longer, with exponential dependence.

Vorticity controls the delay in singularity formation.

Abstract

For the 3D compressible isentropic Euler equations with an initial perturbation of size $\ve$ of a rest state, if the initial vorticity is of size $\dl$ with $0<\dl\le \ve$ and $\ve$ is small, we establish that the lifespan of the smooth solutions is $T_{\dl}=O(\min\{e^\frac{1}{\ve},\frac{1}{\delta}\})$ for the polytropic gases, and $T_{\dl}=O(\frac{1}{\delta})$ for the Chaplygin gases. For example, when $\dl=e^{-\f{1}{\ve^2}}$ is chosen, then $T_{\dl}=O(e^{\f{1}{\ve}})$ for the polytropic gases and $T_{\dl}=O(e^{\f{1}{\ve^2}})$ for the Chaplygin gases although the perturbations of the initial density and the divergence of the initial velocity are only of order $O(\ve)$. Our result illustrates that the time of existence of smooth solutions depends crucially on the size of the vorticity of the initial data, as long as the initial data is sufficiently close to a constant. The main ingredients in the paper are: introducing some suitably weighted energies, deriving the pointwise space-time decay estimates of solutions, looking for the good unknown instead of the velocity, and establishing the required weighted estimates on the vorticty.