Global in time justification of a two-phase averaged system for heat-conducting ideal gases

TL;DR

This paper provides a rigorous mathematical justification for a two-phase averaged system modeling heat-conducting ideal gases, establishing global solutions and deriving the system via homogenization techniques.

Contribution

It introduces a novel framework for global existence of solutions between weak and strong, and derives the system from Navier-Stokes equations using homogenization and Young measures.

Findings

Established global existence of solutions in an intermediate framework.

Derived the two-phase system through homogenization techniques.

Proved uniform bounds despite oscillating coefficients.

Abstract

In this article, we mathematically justify (globally in time) a Baer-Nunziato type system from the non-isentropic compressible Navier-Sokes equations for heat conducting ideal gases posed over the torus and in one space dimension. The breakthrough in this paper is to define and prove the global existence of solutions in a framework intermediate between weak and strong solutions and then to derive the system through homogenization and Young measures characterization. Note that the main difficulty is to derive a priori uniform bounds on appropriate unknowns in the presence of piecewise constant coefficients (viscosity and adiabatic constants) exhibiting rapid oscillations between two positive values.