The Incompressible Navier--Stokes--Fourier System with Thermal Noise

TL;DR

This paper develops a solution theory for the stochastic incompressible Navier--Stokes--Fourier system with thermal noise, overcoming new analytical challenges through innovative transformations and entropy estimates.

Contribution

It introduces a novel approach to handle nonlinear gradient noise and establishes existence and uniqueness results for solutions to the stochastic system.

Findings

Proved existence of global weak solutions for L^2 initial data.

Established local strong solutions for regular initial data.

Demonstrated weak-strong uniqueness for the stochastic system.

Abstract

We establish a solution theory for the incompressible Navier--Stokes--Fourier system with thermal noise, posed on the three-dimensional torus. While in the incompressible deterministic setting the equation for the velocity can be solved independently of the temperature, the inclusion of the effects of thermal fluctuations by means of the GENERIC framework leads to a nonlinear gradient noise term, which couples the dynamics of both variables. Therefore, the analysis poses new challenges, which are absent in the deterministic incompressible Navier--Stokes--Fourier equations. In particular, the a priori estimates used in the deterministic setting are not readily generalizable, the noise introduces strongly nonlinear gradient terms and the total energy lacks convexity. These challenges are overcome in the present work by a novel variable transformation, and novel entropy dissipation estimates. Thereby, the existence of global-in-time weak solutions for $L_x^2$ initial data, the existence of local-in-time strong solutions for regular initial data, and weak-strong uniqueness are obtained.