On asymptotically almost periodic solutions to the Navier-Stokes equations in hyperbolic manifolds

In this paper, we study the forward asymptotically almost periodic (AAP-) mild solutions of Navier-Stokes equations on the real hyperbolic manifold M = H-d(R) with dimension d ?. 2. Using the dispersive and smoothing estimates for the Stokes equation, we invoke the Massera-type principle to prove the existence and uniqueness of the AAP-mild solution for the inhomogeneous Stokes equations in L-p(G(TM))) space with 1 < p d. Next, we establish the existence and uniqueness of the small AAP-mild solutions of the Navier-Stokes equations using the fixed point argument, and the results of inhomogeneous Stokes equations. The asymptotic behaviour (exponential decay and stability) of these small solutions is also related. This work, together with our recent work (Xuan et al. in J Math Anal Appl 517(1):1-19, 2023), provides a full existence and asymptotic behaviour of AAP-mild solutions of Navier-Stokes equations in Lp(G(TM))) spaces for all p > 1.