The Sobolev stability threshold for 2D shear flows near Couette
Jacob Bedrossian, Vlad Vicol, Fei Wang
TL;DR
The paper analyzes nonlinear stability of 2D shear flows near Couette for the 2D Navier–Stokes equations on $\mathbb{T}\times\mathbb{R}$ in the high-Reynolds limit.It develops a near-Couette coordinate framework and a time-dependent Fourier multiplier to capture transient mixing and derive energy estimates that close under a Sobolev data size $\varepsilon \lesssim \nu^{1/2}$.The main contribution is a sharp stability threshold in $H^N$ that scales no worse than $\nu^{1/2}$, together with an enhanced-dissipation mechanism for nonzero modes and a description of transient gradient growth followed by relaxation to a decaying shear.The results extend known Couette stability to flows that are small perturbations of Couette, highlighting the role of mixing-enhanced dissipation in controlling nonlinear effects within Sobolev spaces.Techniques combine linear inviscid damping, a tailored Fourier multiplier, and careful coordinate changes to manage nonlinear resonances in Sobolev regularity.
Abstract
We consider the 2D Navier-Stokes equation on $\mathbb T \times \mathbb R$, with initial datum that is $\varepsilon$-close in $H^N$ to a shear flow $(U(y),0)$, where $\| U(y) - y\|_{H^{N+4}} \ll 1$ and $N>1$. We prove that if $\varepsilon \ll ν^{1/2}$, where $ν$ denotes the inverse Reynolds number, then the solution of the Navier-Stokes equation remains $\varepsilon$-close in $H^1$ to $(e^{t ν\partial_{yy}}U(y),0)$ for all $t>0$. Moreover, the solution converges to a decaying shear flow for times $t \gg ν^{-1/3}$ by a mixing-enhanced dissipation effect, and experiences a transient growth of gradients. In particular, this shows that the stability threshold in finite regularity scales no worse than $ν^{1/2}$ for 2D shear flows close to the Couette flow.