Stability Threshold of Nearly-Couette Shear Flows with Navier Boundary Conditions in 2D

Abstract

In this work, we prove a threshold theorem for the 2D Navier-Stokes equations posed on the periodic channel, \(\mathbb {T} \times [-1,1]\), supplemented with Navier boundary conditions \(\omega |_{y = \pm 1} = 0\). Initial datum is taken to be a perturbation of Couette in the following sense: the shear component of the perturbation is assumed small (in an appropriate Sobolev space) but importantly is independent of \(\nu \). On the other hand, the nonzero modes are assumed size \(O(\nu ^{\frac{1}{2}})\) in an anisotropic Sobolev space. For such datum, we prove nonlinear enhanced dissipation and inviscid damping for the resulting solution. The principal innovation is to capture quantitatively the inviscid damping, for which we introduce a new Singular Integral Operator which is a physical space analogue of the usual Fourier multipliers which are used to prove damping. We then include this SIO in the context of a nonlinear hypocoercivity framework.