The Navier–Stokes Equation with Time Quasi-Periodic External Force: Existence and Stability of Quasi-Periodic Solutions

We prove the existence of small amplitude, time-quasi-periodic solutions (invariant tori) for the incompressible Navier-Stokes equation on the d-dimensional torus 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 ->+infinity, with an exponential rate of convergence O(e(-alpha t)) for any arbitrary alpha is an element of(0, 1).