Almost Periodic Solutions of a Keller–Segel–Fluid System With Singular Data

We study the existence and exponential stability of almost periodic mild solutions for the Patlak–Keller–Segel–Navier–Stokes system on the real hyperbolic manifold in the framework of weak‐ spaces. First, we establish the dispersive estimates and Yamazaki‐type estimates for both scalar and vectorial heat semigroups on . Leveraging by these interpolation estimates, we obtain the global existence of mild solutions for the corresponding linear systems on the whole line time‐axis. Then, we prove a Massera‐type principle which guarantees the existence of almost periodic mild solutions for the linear systems. Next, we combine the linear results with fixed‐point arguments to obtain the existence of almost periodic mild solutions for the semilinear systems. Finally, we prove an exponential stability for such mild solutions.