Analysis Seminar

In our upcoming work, we consider small (and smooth) perturbations of the 2D Couette flow in a channel with Navier boundary conditions for the Navier-Stokes equations. We prove that if the data is small enough (independent of Reynolds number Re) then the vorticity converges to the inviscid Euler dynamics uniformly on time-scales t < Re^{1/3-} (the optimal time-scale) and that the inviscid damping observed in the Euler equations holds uniformly in time and Reynolds number. To our knowledge, this is the first long-time inviscid limit for the Navier-Stokes equations in the presence of a solid boundary, even with Navier-type boundary conditions. The methods are a merging of the Fourier-analysis methods developed in previous work with N. Masmoudi (with the extensions+improvements provided by Hao Jia and Alex Ionescu) together with new energy methods employing a spatially-dependent Gevrey regularity measured with variable-coefficient vector fields that depend on the solution itself in highly nonlinear way. The methods may be useful for other problems in fluid mechanics and moreover, we believe they are an important stepping stone to a better understanding of the Dirichlet boundary condition case. Joint work with Siming He, Sameer Iyer, and Fei Wang.