The Sobolev Stability Threshold for 2D Shear Flows Near Couette |
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Authors: | Jacob?Bedrossian Email authorView authors OrcID profile" target="_blank">Vlad?VicolEmail authorView authors OrcID profile Fei?Wang |
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Institution: | 1.Department of Mathematics,University of Maryland,College Park,USA;2.Department of Mathematics,Princeton University,Princeton,USA;3.Department of Mathematics,University of Southern California,Los Angeles,USA |
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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 \(\Vert U(y) - y\Vert _{H^{N+4}} \ll 1\) 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. |
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