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

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