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

Abstract

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 \(\textrm{e}^{t\Delta _{\alpha }}\), we get the solutions convergence exponentially to the invariant tori as \(t\rightarrow \infty \), with the rate of convergence \(O(\textrm{e}^{-\rho t})\) for any \(\rho \in (0,1)\).