Remark on global existence of solutions to the 1D compressible Euler equation with time-dependent damping

TL;DR

This paper proves the global existence of solutions to the 1D compressible Euler equation with time-dependent damping under certain conditions, removing previous restrictions on initial data behavior at infinity.

Contribution

It establishes global solutions for the 1D compressible Euler equation with time-dependent damping without requiring conditions on initial data at infinity.

Findings

Solutions exist globally for small initial data when $0\\leq \\mu <1$ and $\\lambda >0$ or $\\mu=1$ and $\\lambda > 2$.

Previous assumptions on the limit behavior of initial data at infinity are removed.

The results extend the understanding of damping effects on the Euler equations.

Abstract

In this paper, we consider the 1D compressible Euler equation with the damping coefficient $\lambda/(1+t)^{\mu}$. Under the assumption that $0\leq \mu <1$ and $\lambda >0$ or $\mu=1$ and $\lambda > 2$, we prove that solutions exist globally in time, if initial data are small $C^1$ perturbation near constant states. In particular, we remove the conditions on the limit $\lim_{|x| \rightarrow \infty} (u (0,x), v (0,x))$, assumed in previous results.