Quasi-periodic solutions for the incompressible Navier-Stokes equations with nonlocal diffusion

This paper studied the incompressible Navier-Stokes (NS) equations with nonlocal diffusion on ${\mathbb{T}}^d(d\ge 2)$. Driven by a time quasi-periodic force, the existence of time quasi-periodic solutions in the Sobolev space was established. The proof was based on the decomposition of the unknowns into the spatial average part and spatial oscillating one. The former were sought under the Diophantine non-resonance assumption, and the latter by the contraction mapping principle. Moreover, by constructing suitable time weighted function space and using the Banach fixed point theorem, the asymptotic stability of quasi-periodic solutions and the exponential decay of perturbation were proved.