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 \(d \geqslant 2\). 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 \(1<p\leqslant d\). 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 \(L^p(\Gamma (T\mathcal {M})))\) spaces for all \(p>1\).

在本文中，我们研究了维数为 \( d \geqslant 2\)的实双曲流形\(\mathcal {M}=\mathbb {H}^d(\mathbb {R})\)上纳维-斯托克斯方程的前向渐近几乎周期(AAP-) 温和解。利用 Stokes 方程的色散和平滑估计，我们调用 Massera 型原理来证明在\(L^p(\Gamma (T\mathcal {M})))\)空间中具有\(1<p\leqslant d\)的非齐次 Stokes 方程的 AAP-mild 解的存在性和唯一性。接下来，我们利用不动点参数建立了纳维-斯托克斯方程的小 AAP-温和解的存在性和唯一性，以及非齐次斯托克斯方程的结果。这些小解的渐近行为（指数衰减和稳定性）也是相关的。这项工作与我们最近的工作（Xuan et al. in J Math Anal Appl 517(1):1–19, 2023）一起，为所有\(p>1\)提供了\(L^p(\Gamma (T\mathcal {M})))\)空间中纳维-斯托克斯方程的 AAP 温和解的完整存在性和渐近行为。