Vacuum and singularity formation for compressible Euler equations with time-dependent damping

TL;DR

This paper investigates vacuum and singularity formation in compressible Euler equations with time-dependent damping, establishing new lower bounds on density and proving singularity formation for a range of parameters.

Contribution

It introduces novel control functions to derive density lower bounds and proves singularity formation for all damping parameters, extending previous results.

Findings

Lower bounds on density for classical solutions

Proof of singularity formation for all damping parameters

Analysis of singularity formation when γ=3

Abstract

In this paper, vacuum and singularity formation are considered for compressible Euler equations with time-dependent damping. For $1<\gamma\leq 3$, by constructing some new control functions ingeniously, we obtain the lower bounds estimates on density for arbitrary classical solutions. Basing on these lower estimates, we succeed in proving the singular formation theorem for all $\lambda$, which was open in [1] for some cases.Moreover, the singularity formation of the compressible Euler equations when $\gamma=3$ is investigated, too.