Global Mild Solutions to a BGK Model for Barotropic Gas Dynamics

TL;DR

This paper proves the global existence of mild solutions for a BGK model of barotropic gas dynamics with minimal initial data assumptions and derives a kinetic entropy inequality leading to the hydrodynamic limit.

Contribution

It establishes the first global existence results for this BGK model under minimal assumptions and connects kinetic solutions to the barotropic Euler equations.

Findings

Proved global existence of mild solutions.

Derived a kinetic entropy inequality.

Established the hydrodynamic limit to Euler equations.

Abstract

We establish global existence of mild solutions to the BGK model proposed by Bouchut [J. Stat. Phys., 95, (1999), 113--170] under the minimal assumption of finite kinetic entropy initial data. Moreover we rigorously derive a kinetic entropy inequality, which combined with the theory developed by Berthelin and Vasseur [SIAM J. Math. Anal., 36, (2005), 1807--1835] leads to the hydrodynamic limit to the barotropic Euler equations. The main tools employed in the analysis are stability estimates for the Maxwellian and a velocity averaging lemma.