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Probabilistically Strong Solutions to Stochastic Euler Equations

Benjamin Gess, Robert Lasarzik

TL;DR

The authors address the problem of constructing probabilistically strong solutions to the stochastic Euler equations with energy inequality for general $L^2$ initial data by first establishing probabilistically strong, measure-valued solutions for the 3D stochastic Navier–Stokes equations and then proving convergence to probabilistically strong Euler solutions in the vanishing viscosity limit. The methodological core combines Galerkin approximations, a priori estimates, and compactness via Young measures and a Skorokhod–Jakubowski type representation to handle nonlinear limits without changing the underlying probability space, complemented by the energy-variational framework in the stochastic setting. A key innovation is the introduction of energy-variational solutions, which naturally yield dissipative weak solutions with Reynolds stresses and permit a robust vanishing viscosity analysis, extended to transport noise. Collectively, these results resolve open questions about probabilistically strong solutions for stochastic Euler under energy inequalities and provide a versatile framework for stochastic fluid models with additive and transport noise, with potential implications for understanding dissipation and uniqueness in stochastic turbulence.

Abstract

In this paper, we establish the existence of probabilistically strong, measure-valued solutions for the stochastic incompressible Navier--Stokes equations and prove their convergence, in the vanishing viscosity limit, to probabilistically strong solutions for the stochastic incompressible Euler equations. In particular, this solves the open problem of constructing probabilistically strong solutions for the stochastic Euler equations that satisfy the energy inequality for general $L^2$ initial data. We introduce the concept of energy-variational solutions in the stochastic context in order to treat the nonlinearities without changing the probability space. Furthermore, we extend these results to fluids driven by transport noise.

Probabilistically Strong Solutions to Stochastic Euler Equations

TL;DR

The authors address the problem of constructing probabilistically strong solutions to the stochastic Euler equations with energy inequality for general initial data by first establishing probabilistically strong, measure-valued solutions for the 3D stochastic Navier–Stokes equations and then proving convergence to probabilistically strong Euler solutions in the vanishing viscosity limit. The methodological core combines Galerkin approximations, a priori estimates, and compactness via Young measures and a Skorokhod–Jakubowski type representation to handle nonlinear limits without changing the underlying probability space, complemented by the energy-variational framework in the stochastic setting. A key innovation is the introduction of energy-variational solutions, which naturally yield dissipative weak solutions with Reynolds stresses and permit a robust vanishing viscosity analysis, extended to transport noise. Collectively, these results resolve open questions about probabilistically strong solutions for stochastic Euler under energy inequalities and provide a versatile framework for stochastic fluid models with additive and transport noise, with potential implications for understanding dissipation and uniqueness in stochastic turbulence.

Abstract

In this paper, we establish the existence of probabilistically strong, measure-valued solutions for the stochastic incompressible Navier--Stokes equations and prove their convergence, in the vanishing viscosity limit, to probabilistically strong solutions for the stochastic incompressible Euler equations. In particular, this solves the open problem of constructing probabilistically strong solutions for the stochastic Euler equations that satisfy the energy inequality for general initial data. We introduce the concept of energy-variational solutions in the stochastic context in order to treat the nonlinearities without changing the probability space. Furthermore, we extend these results to fluids driven by transport noise.
Paper Structure (8 sections, 11 theorems, 130 equations)

This paper contains 8 sections, 11 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The incompressible Navier--Stokes and Euler equations with stochastic forcing
  4. Notation
  5. Preliminary lemmata
  6. Main results for the stochastic incompressible Navier--Stokes equations
  7. Energy-variational solutions for the stochastic incompressible Euler equations
  8. Acknowledgments

Key Result

Lemma 2.1

Let $f\in L^p(\Omega;L^1(0,T))$, $g\in L^p_{w^*}(\Omega;L^\infty(0,T ))$, $g_0\in L^p(\Omega)$ for $p\geq 1$, and let $h\in L^2(\Omega, L^2(0,T;L_2(\mathfrak{U};\mathbb{R}))$ be $(\mathcal{F}_t)$-progressively measurable. Then the following two statements are equivalent: If one of these conditions is satisfied, then one representative of $g$ can be identified with a function in $\mathfrak{D}([0,T

Theorems & Definitions (30)

  • Lemma 2.1
  • proof
  • Lemma 2.2: Fundamental lemma of variational calculus
  • Theorem 2.3: Fundamental theorem for Young measures on $\Omega\times (0,T)\times D$
  • proof
  • Lemma 2.4
  • proof
  • Definition 3.2: Weak-weak solutions to stochastic Navier--Stokes equations
  • Definition 3.3: Test process
  • Remark 3.4
  • ...and 20 more