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

Arch. Rational Mech. Anal. (2026) 250:31 Digital Object Identifier (DOI) https://doi.org/10.1007/s00205-026-02187-3 Stability of Stochastically Driven Couette Flow in 2D with Navier Boundary Conditions at High Reynolds Number via Averaging Principle Ryan Arbon & Jacob Bedrossian Communicated by V. Vicol Abstract We characterize the behavior of stochastic Navier–Stokes on 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 ν 5/6 dVt + ν 2/3+α dWt , where dVt has spatial correlation in H03 and acts only on x-independent modes of the vorticity, while dWt has spatial correlation in a lower order, anisotropic, Sobolev space H and acts on x-dependent-modes. We take the initial x-independent modes in the perturbation to be small in H03 in a ν-independent sense, while the non-zero x-modes are taken to be O(ν 1/2+α ) in H. The parameter α is taken to be α > 1/12. Letting ω solve the resulting perturbation equation, we split ω into the zero x-modes ω0 and the non-zero xmodes ω= . We demonstrate that an averaging principle holds wherein ω= is the fast variable and ω0 is the slow variable, deriving a closed nonlinear evolution equation on ω0 that holds over long time-scales (while the fast ω= 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. Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1. Assumptions and Problem Set-Up . . . . . . . . . . . . . . . . . . 1.2. The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3. Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Quantitative Stability . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Fast–Slow Systems . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Relationship with Stochastic Structural Stability Theory (S3T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 5 7 11 11 12 14 31 Page 2 of 69 Arch. Rational Mech. Anal. (2026) 250:31 1.3.4 Physical Interpretation . . . . . . . . . . . . . 1.4. Outline . . . . . . . . . . . . . . . . . . . . . . . . . 2. Control of the Energy Structure and a Priori Estimates . . 2.1. The Random Shear Flow . . . . . . . . . . . . . . . 2.2. The Fast–Slow Energy . . . . . . . . . . . . . . . . 2.3. New Dissipation Estimates . . . . . . . . . . . . . . 3. Analysis of the Fast Process with Frozen Slow Component 4. Analysis of the Averaged Equation . . . . . . . . . . . . 5. Proof of the Main Theorem . . . . . . . . . . . . . . . . 5.1. Proof of Key Propositions . . . . . . . . . . . . . . . 5.2. Explicit Rate of Convergence in Theorem 1.3 . . . . A. Appendix: Well-Posedness . . . . . . . . . . . . . . . . . B. Appendix: Technical Estimate on Nonlinearity . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 15 18 19 21 31 32 40 43 43 58 61 66 66 1. Introduction Consider the incompressible stochastic Navier–Stokes system on T × [−1, 1] with Navier boundary conditions, expressed in vorticity formulation as ⎧ 5/6 2/3+α dW , ⎪ t ⎨dw = νwdt − v · ∇wdt + ν dVt + ν (1.1) v = ∇ ⊥ −1 w = (∂ y −1 w, −∂x −1 w), w(t, x, y = ±1) = 1, ⎪ ⎩ w(t = 0, x, y) = win (x, y). Here, v = (v 1 , v 2 ) is the velocity field, w is the scalar vorticity, ν > 0 is the viscosity. We take the torus to be length [0, 2π ] with periodic boundary conditions. The noise terms are white-in-time and colored-in-space, and Vt is taken independent of Wt . The noise ν 5/6 dVt acts only on the x-independent component of w, while ν 2/3+α dWt acts only on the x-dependent modes of w. The parameter α > 1/12 is related to a rescaling defined in (1.5). We consider dVt to have spatial correlation in H03 ([−1, 1]), while dWt has spatial correlation in a lower order anisotropic Sobolev space. We will make this precise in Subsection 1.1. The well-posedness of (1.1) under suitable assumptions was considered in [17]. In the deterministic setting of  ≡ 0,  ≡ 0, the unique steady state of (1.1) is the Couette flow w E :=1, v E :=(y, 0). We consider the stability of w E under a small initial perturbation, in the manner of [4]. More specifically, we write win (x, y) = w E + Win (y) + ωin (x, y). We view Win as being more regular than ωin , and small in a ν-independent manner, while ωin will be small relative to ν. We let W be the unique solution to the following stochastic heat equation:  dW(t) = ν∂ y2 W(t)dt + ν 5/6 dVt , W(0) = Win ∈ H03 ([−1, 1]). The shear flow U corresponding to 1 + W is given by the Biot-Savart law.  1 G(y, y  )W(t, y  )dy  , U(t, y):=y + ∂ y −1 Arch. Rational Mech. Anal. (2026) 250:31 Page 3 of 69 31 −1 on [−1, 1] with homogeneous Dirichlet where G is the Green’s function for ∂ yy boundary conditions. We write w as w(t, x, y) = 1 + W(t, y) + ω(t, x, y) and arrive at the following perturbation equation for ω: ⎧   −1 2/3+α dW , ⎪ t ⎨dω = νω − U∂x ω + U ∂x  ω − u · ∇ω dt + ν ⊥ −1 (1.2) ∇ · u = 0, u = ∇  ω, ⎪ ⎩ ω(t, x, y = ±1) = 0. This is a stochastic version of the nearly-Couette system in [4], with the randomness entering through the forcing dWt as well as the background flow U. We extract the zero and non-zero x-modes of ω as ω0 (t, y):= T ω(t, x, y)dx and ω= :=ω − ω0 . In general, we set P= f := f − T f (x)dx. We wish to characterize the behavior of w0 = 1 + W + ω0 , v0 = (U, 0) + u 0 as ν → 0. More precisely, we seek to establish stability criteria on long time scales of order O(ν γ −1 ). The stability of the Couette flow in the deterministic setting, i.e  ≡ 0,  ≡ 0, has been well studied, with papers examining the stability properties of the Couette flow with various domains, boundary conditions, and regularities; see e.g. [1,4,5,7,9,21,34–36,42,43,48] and the references therein. One of the key findings of these works is the enhanced dissipation of ω= . For example, [4] found that for the nearly-Couette system (1.2) with  ≡ 0,  ≡ 0, that if Win and ωin are sufficiently small in suitable topologies, then the non-zero frequencies ω= decay at the enhanced rate of exp(−ν 1/3 t). This contrasts with ω0 , which decays in line with the heat equation like exp(−νt). Hence we may expect (1.2) to behave as a fast-slow system in the inviscid limit of ν → 0. The object of this paper is (...truncated)