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 $$\mathbb {T}\times [-1,1]$$ T × [ - 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 $$\nu ^{5/6} \Phi \textrm{d}V_t + \nu ^{2/3+\alpha } \Psi \textrm{d}W_t$$ ν 5 / 6 Φ d V t + ν 2 / 3 + α Ψ d W t , where $$\Phi \textrm{d}V_t$$ Φ d V t has spatial correlation in $$H_0^3$$ H 0 3 and acts only on x -independent modes of the vorticity, while $$\Psi \textrm{d}W_t$$ Ψ d W t has spatial correlation in a lower order, anisotropic, Sobolev space $$\mathcal {H}$$ H and acts on x -dependent-modes. We take the initial x -independent modes in the perturbation to be small in $$H_0^3$$ H 0 3 in a $$\nu $$ ν -independent sense, while the non-zero x -modes are taken to be $$O(\nu ^{1/2 + \alpha })$$ O ( ν 1 / 2 + α ) in $$\mathcal {H}$$ H . The parameter $$\alpha $$ α is taken to be $$\alpha > 1/12$$ α > 1 / 12 . Letting