Stationary stochastic Navier-Stokes on the plane at and above criticality

Abstract

In the present paper, we study the fractional incompressible Stochastic Navier-Stokes equation on $\mathbb{R}^2$, formally defined as \[ \partial_t v = -\tfrac12 (-\Delta)^\theta v - \lambda v \cdot \nabla v + \nabla p - \nabla^{\perp} (-\Delta)^{\frac{\theta-1}{2}} \xi, \qquad \nabla \cdot v = 0 \, , \] where $\theta\in(0,1]$, $\xi$ is the space-time white noise on $\mathbb{R}_+\times\mathbb{R}^2$ and $\lambda$ is the coupling constant. For any value of $\theta$ the previous equation is ill-posed due to the singularity of the noise, and is critical for $\theta=1$ and supercritical for $\theta\in(0,1)$. For $\theta=1$, we prove that the weak coupling regime for the equation, i.e. regularisation at scale $N$ and coupling constant $\lambda=\hat\lambda/\sqrt{\log N}$, is meaningful in that the sequence $\{v^N\}_N$ of regularised solutions is tight and the nonlinearity does not vanish as $N\to\infty$. Instead, for $\theta\in(0,1)$ we show that the large scale behaviour of $v$ is trivial, as the nonlinearity vanishes and $v$ is simply converges to the solution of the original equation but with $\lambda=0$.