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Stability threshold of the 2D Couette flow in Sobolev spaces
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abstract
We study the stability threshold of the 2D Couette flow in Sobolev spaces at high Reynolds number $Re$. We prove that if the initial vorticity $\Omega_{in}$ satisfies $\|\Omega_{in}-(-1)\|_{H^{\sigma}}\leq \epsilon Re^{-1/3}$, then the solution of the 2D Navier-Stokes equation approaches to some shear flow which is also close to Couette flow for time $t\gg Re^{1/3}$ by a mixing-enhanced dissipation effect and then converges back to Couette flow when $t\to +\infty$.
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The stability threshold for 3D MHD equations around Couette with rationally aligned magnetic field
For 3D MHD with a rationally aligned background magnetic field, the sharp stability threshold around Couette flow is gamma=1, with inviscid damping and a nu^{-1/3} magnetic amplification.
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