Global solvability and stabilization in a chemotaxis(-Navier)-Stokes system with proliferation

Abstract

The following chemotaxis-fluid system $ \begin{equation*} \left\{ \begin{split} &n_t = \Delta n-\chi\nabla\cdot\left(n\nabla{c}\right)+nc, &\;\;x\in \Omega, t>0, \\ & c_t = \Delta c-{\bf u}\cdot\nabla c-nc, &\;\;x\in \Omega, t>0, \\ &{\bf u}_t+\kappa({\bf u}\cdot\nabla){\bf u}+\nabla{P} = \Delta{ {\bf u}}+n\nabla\phi , & \;\;x\in \Omega, \, t>0, \\ &\nabla\cdot {\bf u} = 0, &\;\; x\in \Omega, \, t>0\ \end{split} \right. \end{equation*} $ is considered under homogeneous Neumann boundary conditions in a bounded convex domain $ \Omega\subset \mathbb{R}^d (d\in \{2, 3\}) $ with smooth boundary. For $ d = 2 $, it is shown that the corresponding chemotaxis-Navier-Stokes system possesses a globally bounded classical solution which stabilizes toward a spatially homogeneous equilibrium in the sense that$ \begin{equation*} \begin{split} n(\cdot, t)\rightarrow n_\infty, \quad c(\cdot, t)\rightarrow 0 \text{ and } {\bf u}(\cdot, t)\rightarrow {\bf 0} \text{ in } L^\infty(\Omega) \end{split} \end{equation*} $as $ t\rightarrow \infty $, where $ n_\infty $ is a constant satisfying $ n_\infty\geq\frac{1}{|\Omega|}\int_{\Omega}n_0 $. For $ d = 3 $, it is seen that the corresponding chemotaxis-Stokes system possesses a globally defined weak solution.