We prove the existence of small amplitude, time-quasi-periodic solutions (invariant tori) for the incompressible Navier–Stokes equation on the d-dimensional torus \(\mathbb T^d\), with a small, quasi-periodic in time external force. We also show that they are orbitally and asymptotically stable in \(H^s\) (for s large enough). More precisely, for any initial datum which is close to the invariant torus, there exists a unique global in time solution which stays close to the invariant torus for all times. Moreover, the solution converges asymptotically to the invariant torus for \(t \rightarrow + \infty \), with an exponential rate of convergence \(O( e^{- \alpha t })\) for any arbitrary \(\alpha \in (0, 1)\).

我们证明了d维环（\ mathbb T ^ d \）上不可压缩的Navier–Stokes方程存在小幅度的准周期解（不变环面），且其时间小而准周期外部压力。我们还表明，他们以轨道和渐近稳定\（H，2S \） （对于小号足够大）。更准确地说，对于任何接近不变圆环的初始基准，存在唯一的全局时间解，该解始终在所有时间内都保持不变圆环。此外，解的渐近收敛于\（t \ rightarrow + \ infty \）的不变环面，其收敛速度为指数指数\（O（e ^ {-\ alpha t}）\）对于任何任意\（\ alpha \ in（0，1）\）。