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Compressible Euler equations with transport noise

Richard Boadi, Dominic Breit, Thamsanqa Castern Moyo

TL;DR

The paper addresses the stochastic isentropic compressible Euler equations perturbed by transport-type noise in $N=2,3$ dimensions. It develops a rigorous framework for dissipative measure-valued martingale solutions, proves a weak-strong uniqueness principle, and constructs Markov selections for the stochastic Euler dynamics by leveraging a vanishing-viscosity approximation from Navier–Stokes with transport noise. Central to the analysis are Stratonovich formulations, Jakubowski–Skorokhod compactness, and defect measures that capture nonlinear oscillations and concentrations, with the transport noise yielding a deterministic energy balance that simplifies limiting procedures. The results provide a solid analytic basis for turbulent transport models and establish Markovian well-posedness in a stochastic PDE setting where uniqueness may fail.

Abstract

We study the isentropic compressible Euler equations in multi-dimensions with stochastic perturbation of transport type. On the one hand, this is motivated by the physical modelling in turbulence theory. On the other hand, it has been shown recently that this type of noise can have regularising effects. In this paper, we prove the existence of dissipative measure-valued martingale solutions, the weak-strong uniqueness property and the existence of Markov selections.

Compressible Euler equations with transport noise

TL;DR

The paper addresses the stochastic isentropic compressible Euler equations perturbed by transport-type noise in dimensions. It develops a rigorous framework for dissipative measure-valued martingale solutions, proves a weak-strong uniqueness principle, and constructs Markov selections for the stochastic Euler dynamics by leveraging a vanishing-viscosity approximation from Navier–Stokes with transport noise. Central to the analysis are Stratonovich formulations, Jakubowski–Skorokhod compactness, and defect measures that capture nonlinear oscillations and concentrations, with the transport noise yielding a deterministic energy balance that simplifies limiting procedures. The results provide a solid analytic basis for turbulent transport models and establish Markovian well-posedness in a stochastic PDE setting where uniqueness may fail.

Abstract

We study the isentropic compressible Euler equations in multi-dimensions with stochastic perturbation of transport type. On the one hand, this is motivated by the physical modelling in turbulence theory. On the other hand, it has been shown recently that this type of noise can have regularising effects. In this paper, we prove the existence of dissipative measure-valued martingale solutions, the weak-strong uniqueness property and the existence of Markov selections.

Paper Structure

This paper contains 14 sections, 14 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Mathematical framework
  3. Stratonovich integration
  4. Navier--Stokes equations
  5. Euler equations
  6. Existence of solutions
  7. Compactness Argument
  8. The new probability space
  9. Passage to the limit
  10. Markov selection
  11. Basics
  12. Martingale solutions as measures on the space of trajectories
  13. Weak-strong uniqueness
  14. The main theorem

Key Result

Theorem 2.2

Let $\boldsymbol{\sigma}_k \in C^\infty (\mathcal{T}^N, R^N)$, ${\rm div}\boldsymbol{\sigma}_k = 0$, $k=1,\dots,K$, and let $\gamma > \frac{N}{2}$, $N= 2,3$ be given. Suppose that $\varrho_0\in L^\gamma(\mathcal{T}^N),\mathbf m_0\in L^{\frac{2\gamma}{\gamma+1}}(\mathcal{T}^N)$ with $\varrho_0\geq0$

Theorems & Definitions (28)

  • Definition 2.1
  • Theorem 2.2
  • Definition 2.3: Dissipative measure-valued martingale solution
  • Theorem 2.4
  • Corollary 2.5
  • Definition 2.6: Strong solution
  • Proposition 3.1
  • Theorem 4.1: Disintegration
  • Theorem 4.2: Reconstruction
  • Definition 4.3
  • ...and 18 more