Strong time-periodic solutions to chemotaxis–Navier–Stokes equations on bounded domains

Abstract

<p style='text-indent:20px;'>Consider the chemotaxis–Navier–Stokes equations on a bounded convex domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega \subset \mathbb{R}^3 $\end{document}</tex-math></inline-formula>, where the boundary <inline-formula><tex-math id="M2">\begin{document}$ \partial \Omega $\end{document}</tex-math></inline-formula> of <inline-formula><tex-math id="M3">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is not necessarily smooth. It is shown that this system admits a unique strong <inline-formula><tex-math id="M4">\begin{document}$ 2 \pi $\end{document}</tex-math></inline-formula>-periodic solution provided that given <inline-formula><tex-math id="M5">\begin{document}$ 2 \pi $\end{document}</tex-math></inline-formula>-periodic forcing functions are sufficiently small in their natural norm. The result may extend to general cases <inline-formula><tex-math id="M6">\begin{document}$ \Omega \subset \mathbb{R}^d $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ d \ge 2 $\end{document}</tex-math></inline-formula>, if one additionally assumes that <inline-formula><tex-math id="M8">\begin{document}$ \partial \Omega $\end{document}</tex-math></inline-formula> is of class <inline-formula><tex-math id="M9">\begin{document}$ C^3 $\end{document}</tex-math></inline-formula>. The nonnegativity of solutions is also discussed.</p>