Non-unique Ergodicity for the 2D Stochastic Navier-Stokes Equations with Derivative of Space-Time White Noise

Abstract

We prove existence of infinitely many stationary solutions as well as ergodic stationary solutions for the stochastic Navier-Stokes equations on $\mathbb{T}^2$ \begin{align*} \dif u+\div(u\otimes u)\dif t+\nabla p\dif t&=\Delta u\dif t + (-\Delta)^{\fa/2}\dif B_t,\ \ \ \ \div u=0,\notag \end{align*} driven by derivative of space-time white noise, where $\fa\in[0,\frac13)$. In this setting, the solutions are not function valued and probabilistic renormalization is required to give a meaning to the equations. Finally, we show that the stationary distributions are not Gaussian distribution $N(0,\frac12(-\Delta)^{\fa-1})$. The proof relies on a time-dependent decomposition and a stochastic version of the convex integration method which provides uniform moment bounds in some function spaces.