论文标题
2D Navier-Stokes方程的无粘性极限的KAM方法
A KAM approach to the inviscid limit for the 2D Navier-Stokes equations
论文作者
论文摘要
在本文中，我们调查了不可压缩的Navier-Stokes方程的时间 - 时间 - 周期$ {\ Mathbb t}^2 $，带有较小的时间 - Quasi-priodic periodic的外力。更准确地说，我们构建了强制Navier Stokes方程的解，从给定时间的Quasi periodic溶液分叉，不可压缩的Euler方程并承认粘度消失的粘度限制到后者均匀，并且在所有时间均匀地限制了外部扰动的大小。我们的证明是基于构建近似解决方案的构建，最多是$ o（ν^2）$的误差以及以这种新的近似解决方案开始的固定点参数。一个基本的步骤是证明在Euler方程的准周期溶液上线性化的Navier Stokes操作员的可逆性，相对于粘度参数，其较小的条件和估计值均匀。据我们所知，这是全局和统一的无粘性极限问题的第一个积极结果，这是奇异极限问题框架中的第一个KAM结果。
In this paper we investigate the inviscid limit $ν\to 0$ 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 $O(ν^2)$ 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.
