Dissipative weak solutions to the three-dimensional stochastic Euler equations

This paper is concerned with the dissipative weak solutions to the three-dimensional stochastic Euler equations. For any $γ\in \left[ {0,\frac{1}{5}} \right)$, we establish the existence of $γ$-Hölder continuous weak solutions that exhibit a strict energy dissipation in expectation. For any $γ> \frac{1}{3}$, we prove that the three-dimensional stochastic Euler equations do not possess dissipative solutions satisfying the corresponding spatial $γ$-Hölder regularity. It is well known that the stochastic forcing inevitably gives rise to additional stochastic effects to the Reynolds stresses, which cannot be treated by the differential estimates in the classical convex integration scheme introduced by Buckmaster, De Lellis, Isett and Székelyhidi (Ann. Math., 182(1):127--172, 2015). To address this difficulty, we develop a novel material integration technique into the stochastic convex integration scheme, so that those stochastic Reynolds stresses are quantified.