Global existence and decay rates to a self-consistent chemotaxis-fluid system

作者：José A. Carrillo, Yingping Peng, Zhaoyin Xiang · 发表于：Discrete and Continuous Dynamical Systems · 年份：2023 · DOI：10.3934/dcds.2023098 · 被引用次数：10 · 研究领域：Mathematical Biology Tumor Growth、Gene Regulatory Network Analysis、Molecular Communication and Nanonetworks

In this paper, we investigate a chemotaxis-fluid system involving both the effect of potential force on cells and the effect of chemotactic force on fluid:$ \begin{equation*} \left\{ \begin{split} \partial_t n + {\bf{u}}\cdot\nabla n & = \Delta n - \nabla\cdot\left(\chi(c)n\nabla c\right) + \nabla\cdot(n\nabla\phi), \\ \partial_t c + {\bf{u}}\cdot\nabla c & = \Delta c - nf(c), \\ \partial_t{\bf{u}} + \kappa({\bf{u}}\cdot\nabla){\bf{u}} + \nabla P & = \Delta{\bf{u}} - n\nabla\phi + \chi(c)n\nabla c, \\ \nabla\cdot{\bf{u}} & = 0 \end{split} \right. \end{equation*} $in $ \mathbb R^d\times(0,T)\, (d = 2,3) $. One of the novelties and difficulties here is that the coupling in this model is stronger and more nonlinear than the most-studied chemotaxis-fluid model due to the additional term $ \chi(c)n\nabla c $ in the third equation. We will first establish several extensibility criteria of classical solutions, which ensure us to extend the local solutions to global ones in the three dimensional chemotaxis-Stokes case and in the two dimensional chemotaxis-Navier-Stokes version under suitable smallness assumption on $ \|c_0\|_{L^{\infty}} $ with the help of a new entropy functional inequality. Some further decay estimates are also obtained under some suitable growth restriction on the potential $ \nabla \phi $ at infinity. As a byproduct of the entropy functional inequality, we also establish the global-in-time existence of weak solutions to the three dimensional chemo...