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

We prove existence of infinitely many stationary solutions as well as ergodic stationary solutions for the stochastic Navier-Stokes equations on T² {align*} u+÷(u⊗ u) t+∇ p t&=Δ u t + (-Δ)/2 B_t,\ \ \ \ ÷ u=0, {align*} driven by derivative of space-time white noise, where ∈[0,13). 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(0,12(-Δ)⁻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.