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

In this paper we study the forward asymptotically almost periodic (AAP-) mild solutions of Navier-Stokes equations on the real hyperbolic manifold $\mathcal{M}=\mathbb{H}^d(\mathbb{R})$ with dimension $d \geq 2$. 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 $L^p(\Gamma(T\mathcal{M})))$ space with $1<p\leq d$. Next, we establish the existence and uniqueness of the small AAP- mild solutions of the Navier-Stokes equations by using the fixed point argument and the results of inhomogeneous Stokes equations. The asymptotic behaviour (exponential decay and stability) of these small solutions are also related. This work, together with our recent work [P.T. Xuan, N.T. Van and B. Quoc, {\it On Asymptotically Almost Periodic Solution of Parabolic Equations on real hyperbolic Manifolds}, J. Math. Anal. Appl., Vol. 517, Iss. 1 (2023), pages 1-19], provide a full existence and asymptotic behaviour of AAP- mild solutions of Navier-Stokes equations in $L^p(\Gamma(T\mathcal{M})))$ spaces for all $p>1$.