Initial Layer and Relaxation Limit of Non-Isentropic Compressible Euler Equations with Damping

TL;DR

This paper analyzes the relaxation limit of non-isentropic compressible Euler equations with damping, demonstrating convergence behaviors of variables and initial layer phenomena for different data preparations.

Contribution

It establishes the weak convergence of velocity and strong convergence of other variables in the relaxation limit, including initial layer analysis for ill-prepared data.

Findings

Velocity converges weakly to the relaxed equations' velocity.

Other variables converge strongly to their relaxed counterparts.

Initial layer phenomena depend on data preparation.

Abstract

In this paper, we study the relaxation limit of the relaxing Cauchy problem for non-isentropic compressible Euler equations with damping in multi-dimensions. We prove that the velocity of the relaxing equations converges weakly to that of the relaxed equations, while other variables of the relaxing equations converges strongly to the corresponding variables of the relaxed equations. We show that as relaxation time approaches 0, there exists an initial layer for the ill-prepared data, the convergence of the velocity is strong outside the layer; while there is no initial layer for the well-prepared data, the convergence of the velocity is strong near t=0.