Singularity formation for compressible Euler equations with time-dependent damping

TL;DR

This paper investigates finite-time singularity formation in compressible Euler equations with time-dependent damping, establishing conditions for shock-like blow-up and providing density bounds without initial data restrictions.

Contribution

It introduces Riccati-type equations to analyze solution breakdown, covering cases with various damping rates and initial data conditions, and offers density bounds for specific adiabatic indices.

Findings

Solutions blow up in finite time under certain initial conditions.

Derivatives become unbounded, indicating shock formation.

Density bounds are established for 1<γ<3 without extra assumptions.

Abstract

In this paper, we consider the compressible Euler equations with time-dependent damping \frac{\a}{(1+t)^\lambda}u in one space dimension. By constructing 'decoupled' Riccati type equations for smooth solutions, we provide some sufficient conditions under which the classical solutions must break down in finite time. As a byproduct, we show that the derivatives blow up, somewhat like the formation of shock wave, if the derivatives of initial data are appropriately large at a point even when the damping coefficient goes to infinity with a algebraic growth rate. We study the case \lambda\neq1 and \lambda=1 respectively, moreover, our results have no restrictions on the size of solutions and the positivity/monotonicity of the initial Riemann invariants. In addition, for 1<\gamma<3 we provide time-dependent lower bounds on density for arbitrary classical solutions, without any additional assumptions on the initial data.