Global dynamics for the generalized chemotaxis-Navier-Stokes system in $\mathbb{R}^3$

Qingyou He, Ling-Yun Shou, Leyun Wu

Published 2024-07-05Version 1

We consider the Cauchy problem of the three-dimensional generalized chemotaxis-Navier-Stokes system \begin{eqnarray*} \begin{cases} \partial_t n+u\cdot \nabla n=\Delta n- \nabla \cdot (\chi(c)n \nabla c),\\ \partial_t c+u \cdot \nabla c=\Delta c-nf(c),\\ \partial_t u +u \cdot \nabla u+\nabla P=-(-\Delta)^\alpha u-n\nabla \phi,\\ \nabla \cdot u=0. \end{cases} \end{eqnarray*} First, we study the time extensibility criteria of strong solutions, including the Prodi-Serrin type criterion ($\alpha>\frac{3}{4}$) and the Beir${\rm\tilde{a}}$o da Veiga type criterion $(\alpha>\frac{1}{2})$. Furthermore, with Lions' dissipation exponent $\alpha\geq \frac{5}{4}$, we verify the global existence and uniqueness of strong solutions for arbitrarily large initial fluid velocity and oxygen concentration. These results reflect the influence of the generalized dissipation for the solutions of the coupled chemotaxis-fluid equations. Finally, in the scenario of weaker dissipation ($\frac{3}{4}<\alpha<\frac{5}{4}$), we establish uniform regularity estimates for global strong solutions and further obtain optimal time-decay rates under the mild condition that the initial $L^2$ energy is small. To our knowledge, this is the first result concerning the global existence and large-time behavior of strong solutions for the three-dimensional chemotaxis-Navier-Stokes equations with possibly large oscillations.