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On the stability threshold for the 3D Couette flow in Sobolev regularity. (English) Zbl 1366.35113

The periodic, plane Couette flow in the three-dimensional incompressible Navier-Stokes equations at high Reynolds number \(Re\) is studied in the framework of Sobolev regularity. The goal is to estimate how the stability threshold scales in \(Re\): the largest the initial perturbation can be while still resulting in a solution that does not transition away from Couette flow. In particular in this work it is proved that initial data which satisfies \(\|u\|_{H^\sigma}\leq \delta Re^{-3/2}\) for any \(\sigma> 9/2\) and some \(\delta=\delta(\sigma)\) depending only on \(\sigma\), is global in time, remains within \({\mathcal O}(Re^{-1/2})\) of the Couette flow in \(L^2\) for all time, and converges to the class of “2.5 dimensional” streamwise-independent solutions referred to as streaks for times \(t \gtrsim Re^{1/3}\). This is in a good agreement with numerical experiments performed by Reddy et. al.

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35B65 Smoothness and regularity of solutions to PDEs
35B35 Stability in context of PDEs
76E05 Parallel shear flows in hydrodynamic stability
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