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We study the stability threshold of the two-dimensional Couette flow in Sobolev spaces at high Reynolds number Re. We prove that if the initial vorticity ₈₍ satisfies \|₈₍- (-1) \|₇^ Re^-1/3, then the solution of the two-dimensional Navier–Stokes equation approaches some shear flow which is also close to Couette flow for time t Re^1/3 by a mixing-enhanced dissipation effect, and then converges back to Couette flow when t +.
Masmoudi et al. (Mon,) studied this question.