Mathematics > Analysis of PDEs
[Submitted on 6 Apr 2016 (v1), last revised 27 Jun 2016 (this version, v2)]
Title:The Sobolev stability threshold for 2D shear flows near Couette
View PDFAbstract: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 $\varepsilon \ll \nu^{1/2}$, where $\nu$ denotes the inverse Reynolds number, then the solution of the Navier-Stokes equation remains $\varepsilon$-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.
Submission history
From: Jacob Bedrossian [view email][v1] Wed, 6 Apr 2016 23:32:33 UTC (22 KB)
[v2] Mon, 27 Jun 2016 22:18:05 UTC (22 KB)
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