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Stochastically forced compressible Navier-Stokes equations with slip boundary conditions of friction type

Reo Tsuboya

TL;DR

The paper proves global-in-time existence of dissipative martingale solutions to the stochastically forced, compressible Navier–Stokes equations with slip boundary conditions of friction type on bounded $C^{2+ u}$ domains, for isentropic pressure $p( ho)=a ho^eta$ with $eta> rac{3}{2}$. It achieves this via a novel five-layer approximation scheme that combines the BFH four-layer approach with a convex-approximation strategy for the boundary term, while regularizing the diffusion to manage stochastic effects without heavy projections. A careful chain of uniform energy estimates and compactness arguments (including Jakubowski’s extension of the Skorokhod representation) enables limit passages in $R o ty$, $ u o 0$, and $m o ty$, yielding a global weak solution that is consistent with both the analytic structure and the probabilistic framework. The work also clarifies equivalence with stronger martingale notions when density is bounded away from zero and extends the method to alternative slip conditions, highlighting the role of friction modulus and boundary terms in the energy balance and momentum equations.

Abstract

We study a mathematical model of a compressible viscous fluid driven by stochastic forces under slip boundary conditions of friction type. We introduce a notion of a weak solution that is analytically and probabilistically consistent with this model. Our main result establishes the existence of such weak solutions under slip boundary conditions on bounded domains with $C^{2+ν}$-boundary ($ν>0$). The proof of this result combines an extended version of the four-layer approximation scheme on the torus by Breit/Feireisl/Hofmanová (2018) with the convex approximation method for absolute value functions studied by Nečasová/Ogorzaly/Scherz (2023).

Stochastically forced compressible Navier-Stokes equations with slip boundary conditions of friction type

TL;DR

The paper proves global-in-time existence of dissipative martingale solutions to the stochastically forced, compressible Navier–Stokes equations with slip boundary conditions of friction type on bounded domains, for isentropic pressure with . It achieves this via a novel five-layer approximation scheme that combines the BFH four-layer approach with a convex-approximation strategy for the boundary term, while regularizing the diffusion to manage stochastic effects without heavy projections. A careful chain of uniform energy estimates and compactness arguments (including Jakubowski’s extension of the Skorokhod representation) enables limit passages in , , and , yielding a global weak solution that is consistent with both the analytic structure and the probabilistic framework. The work also clarifies equivalence with stronger martingale notions when density is bounded away from zero and extends the method to alternative slip conditions, highlighting the role of friction modulus and boundary terms in the energy balance and momentum equations.

Abstract

We study a mathematical model of a compressible viscous fluid driven by stochastic forces under slip boundary conditions of friction type. We introduce a notion of a weak solution that is analytically and probabilistically consistent with this model. Our main result establishes the existence of such weak solutions under slip boundary conditions on bounded domains with -boundary (). The proof of this result combines an extended version of the four-layer approximation scheme on the torus by Breit/Feireisl/Hofmanová (2018) with the convex approximation method for absolute value functions studied by Nečasová/Ogorzaly/Scherz (2023).
Paper Structure (34 sections, 65 theorems, 574 equations)

This paper contains 34 sections, 65 theorems, 574 equations.

Table of Contents

  1. Introduction
  2. Motivation and previous results
  3. Notation
  4. Weak formulation and main result
  5. Equivalence of definitions
  6. Application to other slip boundary conditions
  7. Preliminaries
  8. Elements of functional analysis
  9. Elements of stochastic analysis
  10. Approximation system
  11. Outline of the proof of the main result
  12. Solvability of the basic approximate problem
  13. Iteration scheme
  14. The limit for vanishing time step
  15. Well-posedness of the basic approximate problem
  16. ...and 19 more sections

Key Result

Theorem 1.3

Let $T>0$ and let $D\subset \mathbb{R}^3$ be a bounded domain of class $C^{2+\nu}$ for some $\nu>0$. Let $\gamma>\frac{3}{2}$ and let $\Lambda$ be a Borel probability measure defined on $L^1(D)\times L^1(D)$ such that for some deterministic constants $\underline{\rho},\overline{\rho}>0$ and for some $r\geq 4$. Assume that the diffusion coefficients $\mathbb{G}=\mathbb{G}(x,\rho,q)$ are continuou

Theorems & Definitions (136)

  • Definition 1.1: Dissipative martingale solution
  • Remark 1.2
  • Theorem 1.3
  • Definition 1.4
  • Proposition 1.5
  • Remark 1.6
  • proof : Proof of Proposition \ref{['prop:1.5']}
  • Definition 1.7
  • Theorem 1.8
  • Theorem 2.1
  • ...and 126 more