Isothermal Limit of Entropy Solutions of the Euler Equations for Isentropic Gas Dynamics

TL;DR

This paper proves that entropy solutions of the isentropic Euler equations converge strongly to those of the isothermal Euler equations as the adiabatic exponent approaches 1, using entropy analysis and compensated compactness.

Contribution

It provides a rigorous proof of the isothermal limit of entropy solutions for the Euler equations, including vacuum states, with refined kinetic formulation and explicit Riemann solution analysis.

Findings

Strong convergence of entropy solutions as γ→1

Uniform estimates for entropy dissipation measures

Explicit asymptotic analysis of Riemann solutions with vacuum

Abstract

We are concerned with the isothermal limit of entropy solutions in $L^\infty$, containing the vacuum states, of the Euler equations for isentropic gas dynamics. We prove that the entropy solutions in $L^\infty$ of the isentropic Euler equations converge strongly to the corresponding entropy solutions of the isothermal Euler equations, when the adiabatic exponent $\gamma \rightarrow 1$. This is achieved by combining careful entropy analysis and refined kinetic formulation with compensated compactness argument to obtain the required uniform estimates for the limit. The entropy analysis involves careful estimates for the relation between the corresponding entropy pairs for the isentropic and isothermal Euler equations when the adiabatic exponent $\gamma\to 1$. The kinetic formulation for the entropy solutions of the isentropic Euler equations with the uniformly bounded initial data is refined, so that the total variation of the dissipation measures in the formulation is locally uniformly bounded with respect to $\gamma>1$. The explicit asymptotic analysis of the Riemann solutions containing the vacuum states is also presented.