Dissipative solutions to Stochastic 3D Euler equations

TL;DR

The paper extends convex integration to the stochastic 3D Euler equations with additive noise, proving the existence of probabilistically strong solutions that are time-continuous and Hölder continuous in space while obeying a local energy inequality up to a stopping time. The authors implement a Da Prato–Debussche decomposition to handle the stochastic term, and build a dissipative Euler–Reynolds flow that is iteratively corrected by a velocity perturbation constructed from Mikado flows, partition of unity, and carefully designed amplitudes. A central achievement is showing that the Reynolds stress and current can be driven to zero in the limit, yielding a solution that satisfies the local energy balance almost surely up to the stopping time, with non-uniqueness in law within this class. The work thereby extends deterministic convex integration results to the stochastic setting, providing a framework for dissipative, non-unique weak solutions with controlled energy behavior in the presence of noise.

Abstract

We construct probabilistically strong solutions to the three-dimensional Euler equations perturbed by additive noise that are $\mathbb{P}$-almost surely continuous in time, Hölder in space, and satisfy the local energy inequality up to an arbitrarily large stopping time.

Theorems & Definitions (47)

• Theorem 1.1
• Definition 2.1
• Lemma 2.2: Iteration Lemma
• proof : Proof of \ref{['thm:main1']}
• Remark 2.3
• Lemma 3.1
• Lemma 3.2
• Proposition 3.3
• Lemma 3.4
• proof