The isothermal limit for the compressible Euler equations with damping

TL;DR

This paper analyzes the isothermal limit of the compressible Euler equations with damping, demonstrating convergence of solutions to a Gaussian profile and detailing the asymptotic behavior of solutions with Gaussian initial data.

Contribution

It rigorously proves the convergence of Barenblatt solutions to a Gaussian profile as the adiabatic index approaches 1 and characterizes the asymptotic behavior of solutions with Gaussian initial data.

Findings

Barenblatt solutions converge to a Gaussian profile as γ approaches 1

Weak L1 convergence of the density is established

First and second moments of the density exhibit specific asymptotic behavior

Abstract

We consider the isothermal Euler system with damping. We rigorously show the convergence of Barenblatt solutions towards a limit Gaussian profile in the isothermal limit $\gamma$ $\rightarrow$ 1, and we explicitly compute the propagation and the behavior of Gaussian initial data. We then show the weak L 1 convergence of the density as well as the asymptotic behavior of its first and second moments. Contents 1. Introduction 1 2. Assumptions and main results 3 3. The limit $\gamma$ $\rightarrow$ 1 of Barenblatt's solutions 6 4. Gaussian solutions 9 5. Evolution of certain quantities 10 6. Convergence 15 7. Conclusion 17 References 17