A KAM approach to the inviscid limit for the 2D Navier-Stokes equations

TL;DR

This paper proves the inviscid limit for 2D Navier-Stokes equations with time-quasi-periodic solutions on a torus, using a KAM approach to construct solutions that converge uniformly to Euler solutions as viscosity vanishes.

Contribution

It introduces a novel KAM-based method to establish the inviscid limit for 2D Navier-Stokes equations with quasi-periodic forcing, ensuring uniform convergence over time.

Findings

Constructed solutions with vanishing viscosity limit to Euler solutions.

Proved invertibility of the linearized Navier-Stokes operator at quasi-periodic solutions.

First global, uniform-in-time KAM result for the inviscid limit problem.

Abstract

In this paper we investigate the inviscid limit $\nu \to 0$ 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.