Boundedness and stabilization of a three-dimensional parabolic-elliptic Keller-Segel-Stokes system
Authors: Zhang Pengmei , Zheng Jiashan 
Published: April 2022

<p style='text-indent:20px;'>This paper is concerned with the volume-filling effect on global solvability and stabilization in a parabolic-elliptic Keller-Segel-Stokes systems</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$\begin{align} \left\{ \begin{array}{l} n_t+u\cdot\nabla n = \Delta n-\nabla\cdot(nS(n)\nabla c),\quad x\in \Omega, t&gt;0,\\ u\cdot\nabla c = \Delta c-c+n,\quad x\in \Omega, t&gt;0,\\ u_t+\nabla P = \Delta u+n\nabla \phi,\quad x\in \Omega, t&gt;0,\\ \nabla\cdot u = 0,\quad x\in \Omega, t&gt;0\\ \end{array}\right. \end{align} \;\;\;\;\;\;\;\;\;\;\;\;(KSF)$ \end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with no-flux boundary conditions for <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ c $\end{document}</tex-math></inline-formula> as well as no-slip boundary condition for <inline-formula><tex-math id="M3">\begin{document}$ u $\end{document}</tex-math></inline-formula> in a bounded domain <inline-formula><tex-math id="M4">\begin{document}$ \Omega \subseteq \mathbb{R}^3 $\end{document}</tex-math></inline-formula> with smooth boundary. Here the nonnegative function <inline-formula><tex-math id="M5">\begin{document}$ S\in C^2(\bar{\Omega}) $\end{document}</tex-math></inline-formula> denotes the chemotactic sensitivity which fulfills</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ |S(n)|\leq C_S(1 + n)^{-\alpha} \; \; \; \; \text{for all}\; \; n\geq0 $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>with some <inline-formula><tex-math id="M6">\begin{document}$ C_S &gt; 0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \alpha&gt; 0 $\end{document}</tex-math></inline-formula>. Imposing no restriction on the size of the initial data, by seeking some new functionals and using the bootstrap arguments on the system, we establish the existence and boundedness of global classical solutions to parabolic-elliptic Keller-Segel-Stokes system under the assumption <inline-formula><tex-math id="M8">\begin{document}$ \alpha&gt; \frac{1}{2} $\end{document}</tex-math></inline-formula>. On the basis of this, we further prove that if the chemotactic coefficient <inline-formula><tex-math id="M9">\begin{document}$ C_S $\end{document}</tex-math></inline-formula> is appropriately small, the obtained solutions are shown to approach the spatially homogeneous steady state <inline-formula><tex-math id="M10">\begin{document}$ (\bar{n}_0, \bar{n}_0, 0) $\end{document}</tex-math></inline-formula> in the large time limit, where <inline-formula><tex-math id="M11">\begin{document}$ \bar{n}_0 = \frac{1}{|\Omega|}\int_{\Omega}n_0 $\end{document}</tex-math></inline-formula>, provided that merely <inline-formula><tex-math id="M12">\begin{document}$ n_0\not \equiv0 $\end{document}</tex-math></inline-formula> on <inline-formula><tex-math id="M13">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>.</p>