Abstract

We study the three-dimensional incompressible Euler equations subject to stochastic forcing. We develop a concept of dissipative martingale solutions, where the nonlinear terms are described by generalised Young measures. We construct these solutions as the vanishing viscosity limit of solutions to the corresponding stochastic Navier–Stokes equations. This requires a refined stochastic compactness method incorporating the generalised Young measures. As a main novelty, our solutions satisfy a form of the energy inequality which gives rise to a weak–strong uniqueness result (pathwise and in law). A dissipative martingale solution coincides (pathwise or in law) with the strong solution as soon as the latter exists.

Highlights

  • We are interested in the stochastic Euler equations describing the motion of an incompressible inviscid fluid in the three-dimensional torus T3

  • The energy inequality implies a weak–strong uniqueness principle for measure-valued solutions as shown in [7]: A measure-valued solution coincides with the strong solution as soon as the strong solution exists

  • In a pathwise approach we prove that a dissipative martingale solution agrees with the strong solution if both exist on the same probability space

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Summary

Introduction

We are interested in the stochastic Euler equations describing the motion of an incompressible inviscid fluid in the three-dimensional torus T3. In the deterministic case a series of counter examples concerning uniqueness of solutions to the Euler equations have been accomplished recently. These solutions are called wild solutions and are constructed by the method of convex integration pioneered by the work of De Lellis and Szekelyhidi [16,17]. A natural approach to deal with such situations is the concept of measure-valued solutions as introduced by Di Perna and Majda [19] (see [18]) These solutions are constructed by compactness methods and the nonlinearities are described by generalised Young measures.

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Generalised Young Measures
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Random Distributions
Stochastic Analysis
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Stochastic Navier–Stokes Equations
Dissipative Solutions
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A Priori Estimates
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Compactness
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Concerning the New Probability Space
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Weak–Strong Uniqueness
Pathwise Weak–Strong Uniqueness
Weak–Strong Uniqueness in Law
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