Stability of Stochastically Driven Couette Flow in 2D with Navier Boundary Conditions at high Reynolds number via Averaging Principle

Abstract

We characterize the behavior of stochastic Navier-Stokes on $\mathbb{T} \times [-1,1]$ with Navier boundary conditions at high Reynolds number when initialized near Couette flow subject to small additive stochastic forcing. We take additive noise of strength $\nu^{5/6} \Phi dV_t + \nu^{2/3+\alpha} \Psi dW_t$, where $\Phi dV_t$ has spatial correlation in $H_0^3$ and acts only on $x$-independent modes of the vorticity, while $\Psi dW_t$ has spatial correlation in a lower order, anisotropic, Sobolev space $\mathcal{H}$ and acts on $x$-dependent-modes. We take the initial $x$-independent modes in the perturbation to be small in $H_0^3$ in a $\nu$-independent sense, while the non-zero $x$-modes are taken to be $O(\nu^{1/2 + \alpha})$ in $\mathcal{H}$. The parameter $\alpha$ is taken to be $\alpha > 1/12$. Letting $\omega$ solve the resulting perturbation equation, we split $\omega$ into the zero $x$-modes $\omega_0$ and the non-zero $x$-modes $\omega_{\neq}$. We demonstrate an averaging principle holds wherein $\omega_{\neq}$ is the fast variable and $\omega_0$ is the slow variable, deriving a closed nonlinear evolution equation on $\omega_0$ that holds over long time-scales (while the fast $\omega_{\neq}$ modes solve a `pseudo-linearized' equation to leading order with dynamics dominated by inviscid damping and enhanced dissipation). This work can also be considered the stochastic analogue of the stability threshold problem for shear flows. Furthermore, we explain the connections to the Stochastic Structural Stability Theory (S3T) in the physics literature.