# Stationary solutions of the Navier-Stokes-Fourier system in planar domains with impermeable boundary

## TL;DR

This paper proves the existence of stationary weak solutions for the Navier-Stokes-Fourier system in 2D bounded domains, considering realistic fluid models with temperature-dependent properties and new a priori bounds.

## Contribution

It establishes existence results for weak solutions with general constitutive relations and a bounded density range, using novel a priori bounds from Trudinger-Moser inequality.

## Key findings

- Existence of weak solutions in 2D bounded domains.
- Inclusion of realistic fluid equations of state.
- Development of new a priori bounds using Trudinger-Moser inequality.

## Abstract

The existence of weak solutions to the Navier-Stokes-Fourier system describing the stationary states of a compressible, viscous, and heat conducting fluid in bounded 2D-domains is shown under fairly general and physically relevant constitutive relations. The equation of state of a real fluid is considered, where the admissible range of density is confined to a bounded interval (hard sphere model). The transport coefficients depend on the temperature in a general way including both gases and liquids behavior. The heart of the paper are new a priori bounds resulting from Trudinger-Moser inequality.