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

Abstract

In this paper, we consider the 1D compressible Euler equation with the damping coefficient $\lambda /(1+t{)}^{\mu }$. Under the assumption that $0\le \mu <1$ 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 ${\lim }_{|x|\to \infty }(u(0,x),v(0,x))$, assumed in previous results.

9 pages. To appear in to appear in Advanced Studies in Pure Mathematics related to MSJ-SI 2018. arXiv admin note: text overlap with arXiv:1703.09821