Navier-Stokes-Fourier system with Dirichlet boundary conditions

TL;DR

This paper establishes the existence and uniqueness of global weak solutions for the Navier-Stokes-Fourier system with non-homogeneous Dirichlet boundary conditions, advancing the mathematical understanding of compressible heat-conducting fluids.

Contribution

It introduces a new weak solution concept based on entropy and energy inequalities, proving global existence and weak-strong uniqueness for the system.

Findings

Existence of global-in-time weak solutions

Weak-strong uniqueness principle established

New weak solution framework based on entropy and energy balance

Abstract

We consider the Navier--Stokes--Fourier system describing the motion of a compressible, viscous, and heat conducting fluid in a bounded domain with general non-homogeneous Dirichlet boundary conditions for the velocity and the absolute temperature, with the associated boundary conditions for the density on the inflow part. We introduce a new concept of weak solution based on the satisfaction of the entropy inequality together with a balance law for the ballistic energy. We show the weak-strong uniqueness principle as well as the existence of global-in-time solutions.