A proof of Onsager's conjecture for the stochastic 3D Euler equations
A talk in the Bielefeld Stochastic Afternoon series by
Lin Lyu
| Abstract: | We investigate the three-dimensional incompressible Euler equations
on a periodic domain, driven by an additive stochastic forcing of
trace class. First, for any $\vartheta<1/3$, we construct infinitely
global-in-time probabilistically strong and analytically weak
solutions in $C_tC^{\vartheta}_x$. These solutions dissipate the
energy pathwisely up to a stopping time, which can be chosen
arbitrarily large with high probability. In addition, we provide a
brief proof of energy conservation for $\vartheta>1/3$ based on
Constantin, E and Titi (Comm. Math. Phys 165: 207–209, 1994),
thereby confirming the Onsager theorem for the stochastic 3D Euler
equations. Second, let $0<\bar{\vartheta}<\bar{\beta}<1/3$, we
construct infinitely many global-in-time probabilistically strong
and analytically weak solutions in $C_tC_x^{\bar{\vartheta}}$ for
any given divergence-free initial condition in $C^{\bar{\beta}}_x$.
Our construction relies on the convex integration method developed
in the deterministic setting by Isett (Ann. Math. 188: 871–963,
2018), adapting it to the stochastic context by introducing a novel
energy inequality into the convex integration scheme and combining
stochastic analysis arguments with a Wong--Zakai type estimate. Within the CRC this talk is associated to the project(s): B1 |