A KAM Approach to the Inviscid Limit for the 2D Navier-Stokes Equations.

In this paper, we investigate the inviscid limit for time-quasi-periodic solutions of the incompressible Navier-Stokes equations on the two-dimensional torus , with a small time-quasi-periodic external force. More precisely, we construct solutions of the forced Navier-Stokes equation, bifurcating from a given time quasi-periodic solution of the incompressible Euler equations and admitting vanishing viscosity limit to the latter, uniformly for all times and independently of the size of the external perturbation. Our proof is based on the construction of an approximate solution, up to an error of order and on a fixed point argument starting with this new approximate solution.

Space Quasi-Periodic Steady Euler Flows Close to the Inviscid Couette Flow.

We prove the existence of steady stream functions, solutions for the Euler equation in a vorticity-stream function formulation in the two dimensional channel . These solutions bifurcate from a prescribed shear equilibrium near the Couette flow, whose profile induces finitely many modes of oscillations in the horizontal direction for the linearized problem. Using a Nash-Moser implicit function iterative scheme, near such equilibrium we construct small amplitude, space reversible stream functions, slightly deforming the linear solutions and retaining the horizontal quasi-periodic structure.