For 2D Navier-Stokes near Couette, ν^{1/2}-small perturbations in H^log_x L^2_y undergo enhanced dissipation at rate ν^{1/3} and inviscid damping, with L^2 initial data sufficient for any threshold exponent above 1/2.
Braz e Silva, Nonlinear stability for 2 dimensional plane Couette flow
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Enhanced dissipation for the 2D Couette flow in critical space
For 2D Navier-Stokes near Couette, ν^{1/2}-small perturbations in H^log_x L^2_y undergo enhanced dissipation at rate ν^{1/3} and inviscid damping, with L^2 initial data sufficient for any threshold exponent above 1/2.