Justification of a two-phase averaged system for ideal gases without heat conduction

Abstract

This article concerns the mathematical justification of an averaged system of partial differential equations governing the evolution of a two-phase mixture of compressible ideal fluids, with viscosity and without conductivity, in space dimension $1$ with periodic boundary conditions. The derivation is done by some homogenization procedure. The originality of the paper consists in the fact that both the density and temperature are allowed to oscillate (because of the absence of heat conduction), so that the limiting model is a six-equation, two-pressure, two-temperature model. The key point is to show the strong convergence of the stress tensor in $L^2((0,T)\times (0,1))$. The main difficulties are obtaining uniform estimates in spite of the presence of oscillating coefficients in the energy equation. It requires looking at solutions with low regularity for the density and the temperature.