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

TL;DR

This paper investigates the stochastic Navier-Stokes equations on the plane for different fractional dissipation levels, establishing well-posedness and behavior at critical and supercritical regimes with implications for turbulence modeling.

Contribution

It provides the first rigorous analysis of stationary solutions for fractional stochastic Navier-Stokes equations at and above criticality, including regularization effects and triviality results.

Findings

Weak coupling regime yields tight solutions at criticality.

Nonlinearity persists in the critical case with regularization.

Solutions become trivial in the supercritical regime.

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 $\lambda=0$.