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

Benjamin Gess, Max Sauerbrey, Zhengyan Wu

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.

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

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 initial data, the existence of local-in-time strong solutions for regular initial data, and weak-strong uniqueness are obtained.

Paper Structure

This paper contains 26 sections, 21 theorems, 341 equations.

Table of Contents

  1. Introduction
  2. Derivation from the GENERIC framework
  3. Noise regularity and Itô-to-Stratonovich corrections
  4. The necessity of the transformation $(u,\vartheta) \rightarrow(u,\psi)$
  5. Definition of weak and strong solutions
  6. Proof strategies
  7. Comparison to the literature
  8. Notation
  9. Properties of weak and strong solutions
  10. Weak-strong uniqueness
  11. Proof of the relative energy estimate
  12. Global existence of weak martingale solutions
  13. Construction of weak solutions to the ($\delta,\epsilon)$-regularization.
  14. Construction of weak solutions to the $\epsilon$-regularization.
  15. Uniform estimates and tightness
  16. ...and 11 more sections

Key Result

Lemma 2.1

Let Assumption Ass:noise hold and $(\Omega,\mathfrak{A},\mathbb{P})$, $\mathscr{F}$, $(W_n)_{n\in \mathbb{Z}^3}$, $(B_n)_{n\in \mathbb{Z}^3}$, $(u,\psi ,q)$ be a weak martingale solution to Eq_new_var with divergence-free initial value $u_0 \in L^2(\mathbb{T}^3; \mathbb{R}^3)$ and $\psi_0\in L^2(\ma this version satisfies moreover, $\mathbb{P}$-a.s., and for all $\eta \in C^{{\infty}}(\mathbb{T}

Theorems & Definitions (48)

  • Definition 1.3: Globally admissible weak martingale solution to \ref{['Eq_new_var']}
  • Remark 1.4
  • Definition 1.5: Local strong solution
  • Remark 1.6
  • Lemma 2.1: Càdlàg version of $\psi$ for weak solutions
  • proof
  • Proposition 2.2: Characterization of strong solutions by the temperature formulation
  • proof
  • Lemma 2.3: Total energy conservation of strong solutions
  • Theorem 3.1: Weak-strong uniqueness
  • ...and 38 more