On almost periodic solutions of chemotaxis-fluid systems

In this paper, we investigate the existence, uniqueness, and exponential stability of almost periodic mild solutions for the chemotaxis-Navier-Stokes system with a matrix-valued sensitivity on a real hyperbolic space H d ( R ) ( d ⩾ 2 ) H^d(R)\, (d 2) and on the whole-line time-axis R t R_t . First, we prove the existence and uniqueness of almost periodic mild solutions for the corresponding linear systems by using the dispersive estimates of the scalar and vectorial heat semigroups and the boundedness of the matrix-valued sensitivity. Then, we use the results of linear systems and fixed-point arguments, and we establish the well-posedness of almost periodic mild solutions for the chemotaxis-Navier-Stokes systems. Finally, we employ the Gronwall inequality to prove the exponential stability of such solutions. Our obtained results are also valid for such systems on a bounded domain with smooth boundary in the Euclidean space R d ( d ⩾ 2 ) R^d\, (d 2) .