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Article

Li, Yanjiang ; Yu, Zhongqing ; Huang, Yumei
Global classical solutions in a self-consistent chemotaxis(-Navier)-Stokes system. (English). Czechoslovak Mathematical Journal, vol. 74 (2024), issue 1, pp. 153-175
MSC: 35K55, 35Q35, 35Q92, 92C17 | DOI: 10.21136/CMJ.2023.0570-22
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Keywords:
chemotaxis; Navier-Stokes system; self-consistent; global existence; boundedness
Summary:
The self-consistent chemotaxis-fluid system $$ \begin{cases} n_t+u\cdot \nabla n =\Delta n - \nabla \cdot (n\nabla c )+\nabla \cdot (n\nabla \phi ), &x\in \Omega ,\ t>0,\\ c_t +u\cdot \nabla c=\Delta c -nc,\quad &x\in \Omega ,\ t>0,\\ u_t+\kappa (u\cdot \nabla ) u+\nabla P=\Delta u - n\nabla \phi +n \nabla c,\qquad &x\in \Omega ,\ t>0,\\ \nabla \cdot u=0,\quad &x\in \Omega ,\ t>0, \end{cases} $$ is considered under no-flux boundary conditions for $n, c$ and the Dirichlet boundary condition for $u$ on a bounded smooth domain $ \Omega \subset \mathbb {R}^N$ $(N=2,3)$, $\kappa \in \lbrace 0,1 \rbrace $. The existence of global bounded classical solutions is proved under a smallness assumption on $\|c_{0}\|_{L^{\infty }(\Omega )}$. \endgraf Both the effect of gravity (potential force) on cells and the effect of the chemotactic force on fluid are considered here, and thus the coupling is stronger than the most studied chemotaxis-fluid systems. The literature on self-consistent chemotaxis-fluid systems of this type so far concentrates on the nonlinear cell diffusion as an additional dissipative mechanism. To the best of our knowledge, this is the first result on the boundedness of a self-consistent chemotaxis-fluid system with linear cell diffusion.
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