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Compressible fluids excited by space-dependent transport noise

D. Breit, E. Feireisl, M. Hofmanova, P. B. Mucha

TL;DR

The paper addresses the existence of weak solutions to the compressible Navier–Stokes equations with space dependent transport noise on a periodic domain. It introduces a flow transformation that converts the stochastic system into a random PDE with coefficients driven by the noise, enabling a multi-layer approximation and stochastic compactness via $u$-stability and Jakubowski’s representation. A key technical advance is an extended transformed div–curl lemma and pathwise pressure estimates that yield identification of the limit pressure as $p( ilde{ ho})$. By passing to vanishing viscosity and artificial pressure limits, the authors establish the existence of dissipative martingale solutions for $oldsymbol{oldsymbol{oldsymbol{ ilde{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{}}}}}}}}}}}}}$, broadening the class of transport noise models in compressible fluids and providing a robust framework for stochastic compactness with non differentiable random coefficients.

Abstract

We study the compressible Navier-Stokes system driven by physically relevant transport noise, where the noise influences both the continuity and momentum equations. Our approach is based on transforming the system into a partial differential equation with random, time- and space-dependent coefficients. A key challenge arises from the fact that these coefficients are non-differentiable in time, rendering standard compactness arguments for the identification of the pressure inapplicable. To overcome this difficulty, we develop a novel multi-layer approximation scheme and introduce a precise localization strategy with respect to both the sample space and time variable. The limit pressure is then identified via the corresponding effective viscous flux identity. By means of stochastic compactness methods, particularly Skorokhod's representation theorem and its generalization by Jakubowski, we ensure the progressive measurability required to return to the original system. Our results broaden the applicability of transport noise models in fluid dynamics and offer new insights into the interaction between stochastic effects and compressibility.

Compressible fluids excited by space-dependent transport noise

TL;DR

The paper addresses the existence of weak solutions to the compressible Navier–Stokes equations with space dependent transport noise on a periodic domain. It introduces a flow transformation that converts the stochastic system into a random PDE with coefficients driven by the noise, enabling a multi-layer approximation and stochastic compactness via -stability and Jakubowski’s representation. A key technical advance is an extended transformed div–curl lemma and pathwise pressure estimates that yield identification of the limit pressure as . By passing to vanishing viscosity and artificial pressure limits, the authors establish the existence of dissipative martingale solutions for , broadening the class of transport noise models in compressible fluids and providing a robust framework for stochastic compactness with non differentiable random coefficients.

Abstract

We study the compressible Navier-Stokes system driven by physically relevant transport noise, where the noise influences both the continuity and momentum equations. Our approach is based on transforming the system into a partial differential equation with random, time- and space-dependent coefficients. A key challenge arises from the fact that these coefficients are non-differentiable in time, rendering standard compactness arguments for the identification of the pressure inapplicable. To overcome this difficulty, we develop a novel multi-layer approximation scheme and introduce a precise localization strategy with respect to both the sample space and time variable. The limit pressure is then identified via the corresponding effective viscous flux identity. By means of stochastic compactness methods, particularly Skorokhod's representation theorem and its generalization by Jakubowski, we ensure the progressive measurability required to return to the original system. Our results broaden the applicability of transport noise models in fluid dynamics and offer new insights into the interaction between stochastic effects and compressibility.

Paper Structure

This paper contains 15 sections, 9 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Problem formulation and main results
  3. Stratonovich integration
  4. The concept of solution
  5. Flow transformation
  6. Main result
  7. Transformed div-curl lemma
  8. The approximate system
  9. Vanishing viscosity limit
  10. Stochastic compactness
  11. Pressure estimates
  12. Strong convergence of the density
  13. Proof of Theorem \ref{['thm:main']}
  14. The limit $l\rightarrow0$
  15. The vanishing artificial pressure limit

Key Result

Lemma 2.4

The quantity $((\Omega,\mathfrak{F},(\mathfrak{F}_t)_{t\geq0},\mathbb P),\varrho,\mathbf{u},W)$ is a dissipative martingale solution to 1.1--eq:transport with initial data $(\varrho_0,\mathbf{q}_0)$ if and only if $((\Omega,\mathfrak{F},(\mathfrak{F}_t)_{t\geq0},\mathbb P),\eta,\mathbf{v},W))$, wher

Theorems & Definitions (16)

  • Definition 2.1: Dissipative martingale solution
  • Remark 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • Lemma 2.6
  • proof
  • Theorem 3.1
  • Definition 4.1
  • ...and 6 more