Stochastic Navier-Stokes-Fourier equations

TL;DR

This paper investigates the stochastic Navier-Stokes-Fourier system for viscous, heat-conducting, compressible fluids, establishing the existence of weak solutions under random influences and analyzing stationary solutions.

Contribution

It introduces a framework for stochastic perturbations in the full Navier-Stokes-Fourier equations and proves the existence of weak martingale solutions with physical assumptions.

Findings

Existence of weak martingale solutions under stochastic effects.

Stationary solutions only exist in trivial cases.

Provides a mathematical foundation for stochastic fluid dynamics.

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. Stochastic effects are implemented through (i) random initial data, (ii) a forcing term in the momentum equation represented by a multiplicative white noise, (iii) random heat source in the internal energy balance. We establish existence of a weak martingale solution under physically grounded structural assumptions. As a byproduct of our theory we can show that stationary martingale solutions only exist in certain trivial cases.