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 \[ \partial_t v = -\tfrac12 (-Δ)^θv - λv \cdot \nabla v + \nabla p - \nabla^{\perp} (-Δ)^{\frac{θ-1}{2}} ξ, \qquad \nabla \cdot v = 0 \, , \] 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 $\lambda =0$.