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

Abstract We consider the chemotaxis-Navier–Stokes system with generalised fluid dissipation in $\mathbb{R}^3$ : \begin{eqnarray*} \begin{cases} \partial _t n+u\cdot \nabla n=\Delta n- \nabla \cdot (\chi (c)n \nabla c),\\[5pt] \partial _t c+u \cdot \nabla c=\Delta c-nf(c),\\[5pt] \partial _t u +u \cdot \nabla u+\nabla P=-(\!-\Delta )^\alpha u-n\nabla \phi, \\[5pt] \nabla \cdot u=0, \end{cases} \end{eqnarray*} which models the motion of swimming bacteria in water flows. First, we prove blow-up criteria of strong solutions to the Cauchy problem, including the Prodi-Serrin-type criterion for $\alpha \gt \frac {3}{4}$ and the Beir $\tilde {\textrm {a}}$ o da Veiga-type criterion for $\alpha \gt \frac {1}{2}$ . Then, we verify the global existence and uniqueness of strong solutions for arbitrarily large initial fluid velocity and bacteria density for $\alpha \geq \frac {5}{4}$ . Furthermore, in the scenario of $\frac {3}{4}\lt \alpha \lt \frac {5}{4}$ , we establish uniform regularity estimates and optimal time-decay rates of global solutions if only the $L^2$ -norm of initial data is small. To our knowledge, this work provides the first result concerning the global existence and large-time behaviour of strong solutions for the chemotaxis-Navier–Stokes equations with possibly large oscillations.