Dissipative solutions to randomly forced 3D Euler equations

TL;DR

This paper constructs probabilistically strong, dissipative solutions to the 3D Euler equations with additive noise, and demonstrates non-uniqueness and ergodic properties of these solutions under external forcing.

Contribution

It introduces a method to construct dissipative solutions to stochastic 3D Euler equations and proves non-uniqueness and ergodicity results for these solutions.

Findings

Constructed solutions are almost surely continuous in time and Hölder continuous in space.

Solutions satisfy the local energy inequality up to large stopping times.

Proved non-uniqueness and ergodic properties for the forced Euler equations.

Abstract

The purpose of this work is twofold. First, we construct probabilistically strong solutions to the three-dimensional Euler equations perturbed by additive noise that are $\mathbb{P}$-almost surely continuous in time, H\"older in space, and satisfy the local energy inequality up to an arbitrarily large stopping time. Second, we prove several non-unique ergodicity results for the forced Euler equations with continuous-in-time external forcing. The solutions we construct are genuinely random and, almost surely, strictly dissipative and not steady states.