Existence of dissipative solutions to the compressible Navier-Stokes system with potential temperature transport

TL;DR

This paper introduces dissipative solutions for the compressible Navier-Stokes system with potential temperature, proving their global existence and convergence of numerical methods, even in cases where classical solutions are unknown.

Contribution

It establishes the existence of dissipative solutions using Young measures and demonstrates convergence of a mixed finite element-finite volume method.

Findings

Global-in-time existence of dissipative solutions.

Strong convergence of numerical solutions when classical solutions exist.

Applicability to physically relevant cases with open existence of weak solutions.

Abstract

We introduce dissipative solutions to the compressible Navier-Stokes system with potential temperature transport motivated by the concept of Young measures. We prove their global-in-time existence by means of convergence analysis of a mixed finite element-finite volume method. If a classical solution to the compressible Navier-Stokes system with potential temperature transport exists, we prove the strong convergence of numerical solutions. Our results hold for the full range of adiabatic indices including the physically relevant cases in which the existence of global-in-time weak solutions is open.