In this article, we investigate the existence, uniqueness, and exponential stability of periodic mild solutions to the chemotaxis–Navier–Stokes equations on the real hyperbolic space , with . First, we establish the well-posedness of mild solutions for the corresponding linear systems by employing dispersive and smoothing estimates for the scalar and vectorial heat semigroups. Then, by combining a Messera-type principle with the ergodic method, we derive the existence of periodic mild solutions to the linear systems. Utilizing these linear results and fixed-point arguments, we further prove the well-posedness of periodic mild solutions for the chemotaxis–Navier–Stokes system. Finally, the exponential stability of such solutions is obtained via the Gronwall inequality.
CaoX. (2015). Global bounded solutions of the higher-dimensional Keller–Segel system under smallness conditions in optimal spaces. Discrete and Continuous Dynamical Systems, 35(5), 1891–1904.
2.
ChaeM.KangK.LeeJ. (2013). Existence of smooth solutions to coupled chemotaxis-fluid equations. Discrete and Continuous Dynamical Systems, 33, 2271–2297.
3.
ChaeM.KangK.LeeJ. (2014). Global existence and temporal decay in Keller–Segel models coupled to fluid equations. Communications in Partial Differential Equations, 39, 1205–1235.
4.
ChoeH. J.LkhagvasurenB. (2017). Global existence result for chemotaxis Navier–Stokes equations in the critical Besov spaces. Journal of Mathematical Analysis and Applications, 446(2), 1415–1426.
5.
DuanR.LiX.XiangZ. (2017). Global existence and large time behavior for a two-dimensional chemotaxis–Navier–Stokes system. Journal of Differential Equations, 263, 6284–6316.
6.
DuanR.LorzA.MarkowichP. (2010). Global solutions to the coupled chemotaxis-fluid equations. Communications in Partial Differential Equations, 35, 1635–1673.
7.
FerreiraL. C. F.LimaD. P. A. (2025). Global solutions for a 2D chemotaxis-fluid system with large measures as initial density and vorticity. Discrete and Continuous Dynamical Systems-Series B, 30(3), 843–873.
8.
FerreiraL. C. F.PostigoM. (2019). Global well-posedness and asymptotic behavior in Besov–Morrey spaces for chemotaxis–Navier–Stokes fluids. Journal of Mathematical Physics, 60, 061502.
9.
HillesdonA. J.PedleyT. J.KesslerJ. O. (1995). The development of concentration gradients in a suspension of chemotaxis bacteria. Bulletin of Mathematical Biology, 57, 299–344.
10.
JiangJ. (2019). Global stability of homogeneous steady states in scaling-invariant spaces for a Keller–Segel–Navier–Stokes system. Journal of Differential Equations, 267(2), 659–692.
11.
JiangJ. (2020). Global stability of Keller–Segel systems in critical Lebesgue spaces. Discrete and Continuous Dynamical Systems, 40(1), 609–634.
12.
KellerE. F.SegelL. A. (1970). Initiation of slime mold aggregation viewed as an instability. Journal of Theoretical Biology, 26(3), 399–415.
13.
KlausK. (2020). Time-periodic solutions to bidomain, chemotaxis-fluid, and Q-tensor models [Ph.D. Thesis]. Technische Universität Darmstadt. https://doi.org/10.25534/tuprints-00013505
14.
KozonoH.MiuraM.SugiyamaY. (2016). Existence and uniqueness theorem on mild solutions to the Keller–Segel system coupled with the Navier–Stokes fluid. Journal of Functional Analysis, 270, 1663–1683.
15.
KozonoH.MiuraM.SugiyamaY. (2019). Time global existence and finite time blow-up criterion for solutions to the Keller–Segel system coupled with the Navier–Stokes fluid. Journal of Differential Equations, 267(9), 5410–5492.
16.
LohouéN. (2006). Estimation des projecteurs de De Rham–Hodge de certaines variétés riemanniennes non compactes. Mathematische Nachrichten, 279(3), 272–298.
PierfeliceV. (2017). The incompressible Navier–Stokes equations on non-compact manifolds. Journal of Geometric Analysis, 27(1), 577–617.
19.
TuvalI.CisnerosL.DombrowskiC.WolgemuthC. W.KesslerJ. O.GoldsteinR. E. (2005). Bacterial swimming and oxygen transport near contact lines. Proceedings of the National Academy of Sciences of the United States of America, 102, 2277–2282.
20.
WatanabeK. (2022). Strong time-periodic solutions to chemotaxis–Navier–Stokes equations on bounded domains. Discrete and Continuous Dynamical Systems, 42(11), 5577–5590.
21.
WinklerM. (2012). Global large-data solutions in a chemotaxis–(Navier–)Stokes system modeling cellular swimming in fluid drops. Communications in Partial Differential Equations, 37, 319–351.
22.
XuanP. T.LoanN. T. (2024). Almost periodic solutions of the parabolic–elliptic Keller–Segel system on the whole space. Archiv der Mathematik, 123, 431–446.
23.
XuanP. T.NgocT. T.VanN. T.TrungH. (2024). On periodic solution for the Boussinesq system on real hyperbolic manifolds. Dynamical Systems, 40(1), 102–127.
24.
XuanP. T.Nguyen ThiV. A.ThuyT. V.LoanN. T. (2024). Periodic solutions of the parabolic–elliptic Keller–Segel system on whole spaces. Mathematische Nachrichten, 297, 3003–3023.
25.
XuanP. T.VanN. T. (2023). On asymptotically almost periodic solutions to the Navier–Stokes equations in hyperbolic manifolds. Journal of Fixed Point Theory and Applications, 25(71), 32.
26.
XuanP. T.VanN. T.QuocB. (2023). On asymptotically almost periodic solution of parabolic equations on real hyperbolic manifolds. Journal of Mathematical Analysis and Applications, 517(1), 1–19.
27.
XuanP. T.VanN. T.ThuyT. V. (2024). On asymptotically almost periodic solutions of the parabolic–elliptic Keller–Segel system on real hyperbolic manifolds. Dynamical Systems, 39(4), 665–682.
28.
YuH.WangW.ZhengS. (2018). Global classical solutions to the Keller–Segel–Navier–Stokes system with matrix-valued sensitivity. Journal of Mathematical Analysis and Applications, 461, 1748–1770.
29.
ZhangQ. (2014). Local well-posedness for the chemotaxis–Navier–Stokes equations in Besov spaces. Nonlinear Analysis: Real World Applications, 17, 89–100.
