The Sobolev stability threshold for 2D shear flows near Couette

Abstract

We consider the 2D Navier-Stokes equation on $\mathbb{T}\times \mathbb{R}$, with initial datum that is $\epsilon$-close in $H^N$ to a shear flow $(U(y),0)$, where $\Vert U(y)-y{\Vert }_{H^{N+4}}\ll 1$ and $N>1$. We prove that if $\epsilon \ll {\nu }^{1/2}$, where $\nu$ denotes the inverse Reynolds number, then the solution of the Navier-Stokes equation remains $\epsilon$-close in $H^1$ to $(e^{t\nu {\partial }_{yy}}U(y),0)$ for all $t>0$. Moreover, the solution converges to a decaying shear flow for times $t\gg {\nu }^{-1/3}$ 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 ${\nu }^{1/2}$ for 2D shear flows close to the Couette flow.

References in corpus (2)

• Enhanced dissipation and inviscid damping in the inviscid limit of the Navier-Stokes equations near the 2D Couette flow
• Invariant measures for passive scalars in the small noise inviscid limit

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