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

Abstract

In this paper we extend a recent work \cite{HuyXuan2020} to study the forward asymptotically almost periodic (AAP-) mild solution of Navier-Stokes equation on the real hyperbolic manifold $\mathbb{H}^d(\mathbb{R})$ with dimension $d \geq 2$. Using the dispertive and smoothing estimates for Stokes equation \cite{Pi} we invoke the Massera-type principle to prove the existence and uniqueness of the AAP- mild solution for the Stokes equation in $L^p(\Gamma(TM)))$ space with $p>d$. We then establish the existence and uniqueness of the small AAP- mild solutions of the Navier-Stokes equation by using the fixed point argument. The asymptotic behaviour (exponential decay and stability) of these small solutions are also related. Our results extend also \cite{FaTa2013} for the forward asymptotic mild solution of the Navier-Stokes equation on the curved background.Comment: 21 page