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

Abstract

AbstractIn the present paper, we study the fractional incompressible Stochastic Navier–Stokes equation on $${\mathbb {R}}^2$$ R 2 , formally defined as $$\begin{aligned} \partial _t v = -\tfrac{1}{2} (-\Delta )^\theta v - \lambda v \cdot \nabla v + \nabla p + \nabla ^{\perp }(-\Delta )^{\frac{\theta -1}{2}} \xi , \qquad \nabla \cdot v = 0 \, , \end{aligned}$$ ∂ t v = - 1 2 ( - Δ ) θ v - λ v · ∇ v + ∇ p + ∇ ⊥ ( - Δ ) θ - 1 2 ξ , ∇ · v = 0 , where $$\theta \in (0,1]$$ θ ∈ ( 0 , 1 ] , $$\xi $$ ξ is the space-time white noise on $${\mathbb {R}}_+\times {\mathbb {R}}^2$$ R + × R 2 and $$\lambda $$ λ is the coupling constant. For any value of $$\theta $$ θ the previous equation is ill-posed due to the singularity of the noise, and is critical for $$\theta =1$$ θ = 1 and supercritical for $$\theta \in (0,1)$$ θ ∈ ( 0 , 1 ) . For $$\theta =1$$ θ = 1 , we prove that the weak coupling regime for the equation, i.e. regularisation at scale N and coupling constant $$\lambda ={{\hat{\lambda }}}/\sqrt{\log N}$$ λ = λ ^ / log N , is meaningful in that the sequence $$\{v^N\}_N$$ { v N } N of regularised solutions is tight and the nonlinearity does not vanish as $$N\rightarrow \infty $$ N → ∞ . Instead, for $$\theta \in (0,1)$$ θ ∈ ( 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 (0.1) with $$\lambda =0$$ λ = 0 .