Formation of shifted shock for the 3D compressible Euler equations with time-dependent damping

TL;DR

This paper demonstrates the formation of shocks in 3D compressible Euler equations with time-dependent damping, showing how damping influences shock timing and characterizing the blow-up of derivatives without symmetry assumptions.

Contribution

It extends shock formation analysis to 3D Euler equations with time-decaying damping, revealing how damping shifts shock formation time and characterizes the collapse of characteristic hypersurfaces.

Findings

Shock formation occurs with exponential lifespan depending on damping parameters.

Damping shifts the shock formation time $T_{*}$.

Method extends to Euler equations with general time-decay damping.

Abstract

In this paper, we show the shock formation to the compressible Euler equations with time-dependent damping $\frac{a\p u}{(1+t)^{\lam}}$ in three spatial dimensions without any symmetry conditions. It's well-known that for $\lam>1$, the damping is too weak to prevent the shock formation for suitably large data. However, the classical results only showed the finite existence of the solution. Follow the work by D.Christodoulou in\cite{christodoulou2007}, starting from the initial isentropic and irrotational short pulse data, we show the formation of shock is characterized by the collapse of the characteristic hypersurfaces and the vanishing of the inverse foliation density function $\mu$, at which the first derivatives of the velocity and the density blow up, and the lifespan $T_{\ast}(a,\lam)$ is exponentially large. Moreover, the damping effect will shift the time of shock formation $T_{\ast}$. The methods in the paper can also be extended to the Euler equations with general time-decay damping.