Periodic Solutions of Chemotaxis-Fluid Systems on Real Hyperbolic Spaces

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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:msup> <mml:mrow> <mml:mi mathvariant="double-struck">H</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:math> , with <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi>d</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:math> . 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.