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Transition threshold for the 3D Couette flow in Sobolev space. (English) Zbl 1513.35438

This paper is devoted to the stability problem for Couette flow in a simple three-dimensional domain \(\mathbb T\times \mathbb R\times \mathbb T\) for large Reynolds numbers. In the framework of Sobolev spaces, the authors use energy methods to study the stability threshold problem, i.e. to establish bounds on the size of perturbations of Couette flow (in terms of small viscosity \(\nu>0\)) that guarantee the global existence and convergence of solutions to essentially two-dimensional streak solutions for times \(t\gg \nu^{-1/3}\). The role of enhanced dissipation and inviscid damping is analyzed.

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
76E30 Nonlinear effects in hydrodynamic stability
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