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

TL;DR

This paper demonstrates shock formation in 3D compressible Euler equations with damping, showing how damping influences the timing and nature of shock development for large initial data.

Contribution

It extends the understanding of shock formation to damped 3D Euler equations, highlighting the impact of damping on shock timing and the behavior of characteristic hypersurfaces.

Findings

Shock forms via collapse of characteristic hypersurfaces

Damping shifts the shock formation time

Inverse foliation density function behavior is altered by damping

Abstract

In this paper, we show the shock formation of the solutions to the 3-dimensional (3D) compressible isentropic and irrotational Euler equations with damping for the initial short pulse data which was first introduced by D.Christodoulou\cite{christodoulou2007}. Due to the damping effect, the largeness of the initial data is necessary for the shock formation and we will work on the class of large data (in energy sense). Similar to the undamped case, 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. However, the damping effect changes the asymptotic behavior of the inverse foliation density function $\mu$ and then shifts the time of shock formation compared with the undamped case. The methods in the paper can also be extended to a class of $3D$ quasilinear wave equations for the short pulse initial data.