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

2025/06/10 by Arbon, Ryan, Bedrossian, Jacob
#Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2506.08769

Abstract

We characterize the behavior of stochastic Navier-Stokes on \mathbbT × [-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^1/2+ ΦdV_t + ν^2/3+α ΨdW_t, where ΦdV_t has spatial correlation in H₀³ and acts only on x-independent modes of the vorticity, while ΨdW_t 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. Letting ω solve the resulting perturbation equation, we split ω into the zero x-modes ω₀ and the non-zero x-modes ω_≠. We demonstrate 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.

Citations

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Related

• Stability threshold of nearly-Couette shear flows with Navier boundary conditions in 2D
• Improved stability threshold for 2D Navier-Stokes Couette flow in an infinite channel
• Quantitative Hydrodynamic Stability for Couette Flow on Unbounded Domains with Navier Boundary Conditions
• Stochastic Stabilization of Transient Axisymmetric Taylor-Couette Flow
• Nonlinear stability of 2-D Couette flow for the compressible Navier-Stokes equations at high Reynolds number