Blow-up for compressible Euler system with space-dependent damping in 1-D

Abstract

This article considers the Cauchy problem for compressible Euler system in R{\bf{R}} with damping, in which the coefficient depends on the space variable. Assuming the initial density has a small perturbation around a constant state and both the small perturbation and the small initial velocity field are compact supported, finite-time blow-up result will be established. This result reveals the fact that if the space-dependent damping coefficient decays fast enough in the far field (belongs to L1(R){L}^{1}\left({\bf{R}})), then the damping is non-effective to the long-time behavior of the solution.

Keywords

• compressible euler equations
• damping
• blow-up
• space-dependent
• 35q31
• 35l65
• 35l67
• 76n15