Quasi-periodic Solutions for the Navier–Stokes Equation on Irrational Tori

This paper is concerned with the existence and stability of quasi-periodic solutions for the Navier–Stokes equation on irrational tori. We prove the persistence of quasi-periodic invariant tori under the small quasi-periodic external force. Furthermore, we obtain the orbital and asymptotic stability in the appropriate Sobolev space. Using the norm estimate of the heat propagator \(\textrm{e}^{t\Delta _{\alpha }}\) ${\text{e}}^{t{\mathrm{\Delta }}_{\alpha }}$ , we get the solutions convergence exponentially to the invariant tori as \(t\rightarrow \infty \) $t\to \infty$ , with the rate of convergence \(O(\textrm{e}^{-\rho t})\) $O({\text{e}}^{-\rho t})$ for any \(\rho \in (0,1)\) $\rho \in (0,1)$ .