AbstractWe prove the existence of small amplitude, time-quasi-periodic solutions (invariant tori) for the incompressible Navier–Stokes equation on the d-dimensional torus $$\mathbb T^d$$ T d , with a small, quasi-periodic in time external force. We also show that they are orbitally and asymptotically stable in $$H^s$$ 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

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... ant torus for all times. Moreover, the solution converges asymptotically to the invariant torus for $$t \rightarrow + \infty $$ t → + ∞ , with an exponential rate of convergence $$O( e^{- \alpha t })$$ O ( e - α t ) for any arbitrary $$\alpha \in (0, 1)$$ α ∈ ( 0 , 1 ) .