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

Abstract

In this paper, we investigate the inviscid limit \(\nu \rightarrow 0\) for time-quasi-periodic solutions of the incompressible Navier–Stokes equations on the two-dimensional torus \({\mathbb {T}}^2\), 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 \(O(\nu ^2)\) and on a fixed point argument starting with this new approximate solution. A fundamental step is to prove the invertibility of the linearized Navier–Stokes operator at a quasi-periodic solution of the Euler equation, with smallness conditions and estimates which are uniform with respect to the viscosity parameter. To the best of our knowledge, this is the first positive result for the inviscid limit problem that is global and uniform in time and it is the first KAM result in the framework of the singular limit problems.