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.

在本文中，我们研究了二维环面上不可压缩纳维-斯托克斯方程的时间准周期解的无粘极限\(\nu \rightarrow 0\) \ ({\mathbb {T}}^2\ )，具有小的时间准周期外力。更准确地说，我们构造了受迫纳维-斯托克斯方程的解，从不可压缩欧拉方程的给定时间准周期解分叉，并承认后者的粘度极限消失，所有时间一致且与外部扰动的大小无关。我们的证明基于近似解的构造，最高可达阶数\(O(\nu ^2)\)的误差，并基于从这个新的近似解开始的定点参数。一个基本步骤是证明线性纳维-斯托克斯算子在欧拉方程的准周期解中的可逆性，并且具有相对于粘度参数一致的小条件和估计值。据我们所知，这是全局且时间一致的无粘极限问题的第一个积极结果，也是奇异极限问题框架中的第一个 KAM 结果。