Stationary solutions in thermodynamics of stochastically forced fluids

TL;DR

This paper demonstrates the existence of stationary solutions in the thermodynamics of stochastically forced fluids by analyzing the Navier--Stokes--Fourier system with open boundary conditions and stochastic perturbations.

Contribution

It introduces a novel approach using global-in-time estimates to establish stationary solutions for open, stochastic fluid systems, contrasting with closed systems.

Findings

Existence of stationary solutions in open, stochastic fluid systems.

Development of new global-in-time estimates for such systems.

Analysis of the impact of boundary conditions on system behavior.

Abstract

We study the full Navier--Stokes--Fourier system governing the motion of a general viscous, heat-conducting, and compressible fluid subject to stochastic perturbation. The system is supplemented with non-homogeneous Neumann boundary conditions for the temperature and hence energetically open. We show that, in contrast with the energetically closed system, there exists a stationary solution. Our approach is based on new global-in-time estimates which rely on the non-homogeneous boundary conditions combined with estimates for the pressure.