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
Summary
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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