Delayed singularity formation for the three dimensional compressible Euler equations with non-zero vorticity
Fei Hou, Huicheng Yin
TL;DR
This work proves that small perturbations of a constant state for the 3D compressible Euler equations with nonzero vorticity have lifespans that depend crucially on initial vorticity: $T_{\delta}=O\left(\min\{e^{1/\varepsilon},1/\delta\}\right)$ for polytropic gases and $T_{\delta}=O(1/\delta)$ for Chaplygin gases. The authors develop a weighted space–time energy framework and introduce a good unknown $g=u-\sigma\omega$ to exploit decay and null structures, complemented by careful vorticity estimates via Helmholtz decomposition. The analysis yields explicit relationships between the vorticity scale $\delta$ and the lifespan, including exponential-in-$\varepsilon$ lifespans when $\delta$ is as small as $e^{-1/\varepsilon^2}$, thus clarifying how vorticity controls the delay of singularity formation. The results extend the understanding of multi-dimensional shock formation in near-constant states and provide robust techniques (weighted energies, ghost weights, and vorticity control) applicable to related hyperbolic systems. The findings have implications for the stability of near-rest states under rotational perturbations and for the interplay between vorticity and nonlinear wave propagation in fluids.
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 $T_{\dl}=O(\min\{e^\frac{1}{\ve},\frac{1}δ\})$ for the polytropic gases, and $T_{\dl}=O(\frac{1}δ)$ 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.