In this paper, a general method to obtain constructive proofs of existence of periodic orbits in the forced autonomous Navier–Stokes equations on the three-torus is proposed. After introducing a zero finding problem posed on a Banach space of geometrically decaying Fourier coefficients, a Newton–Kantorovich theorem is applied to obtain the (computer-assisted) proofs of existence. The required analytic estimates to verify the contractibility of the operator are presented in full generality and symmetries from the model are used to reduce the size of the problem to be solved. As applications, we present proofs of existence of spontaneous periodic orbits in the Navier–Stokes equations with Taylor–Green forcing.

本文提出了一种通用的方法来获得关于三重环的强迫自治Navier-Stokes方程中周期轨道的存在性的构造性证明。在引入几何衰减傅里叶系数的Banach空间上提出的零发现问题后，牛顿-坎托罗维奇定理被应用于获得（计算机辅助）存在的证明。全面概括了验证操作员可收缩性所需的分析估计，并且使用模型的对称性来减小要解决的问题的大小。作为应用程序，我们用泰勒-格林强迫给出了Navier-Stokes方程中自发周期性轨道的存在的证明。