Two-phase averaged system justification for ideal gases without conductivity

TL;DR

This paper mathematically justifies a homogenized two-phase model for ideal gases without heat conduction, accounting for oscillations in density and temperature, using homogenization and strong convergence techniques.

Contribution

It introduces a novel homogenization approach for a complex two-phase system with oscillating density and temperature, deriving a six-equation model.

Findings

Proved strong convergence of the stress tensor in L^2 space.

Established uniform estimates despite oscillating coefficients.

Derived a six-equation model capturing oscillations in density and temperature.

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 and the difficulty 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-equations, two-pressures, two-temperatures 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 to obtain uniform estimates in spite of the presence of oscillating coefficients in the energy equation. It requires to look at solutions with low regularity for the density and the temperature.