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

Abstract

We consider the chemotaxis-Navier-Stokes system with generalized fluid dissipation in ${\mathbb{R}}^3$: \begin{eqnarray*} \begin{cases} \partial_t n+u\cdot \nabla n=Δn- \nabla \cdot (χ(c)n \nabla c),\\ \partial_t c+u \cdot \nabla c=Δc-nf(c),\\ \partial_t u +u \cdot \nabla u+\nabla P=-(-Δ)^αu-n\nabla ϕ,\\ \nabla \cdot u=0, \end{cases} \end{eqnarray*} which describes the motion of swimming bacteria or bacillus subtilis suspended to water flows. First, we prove some blow-up criteria of strong solutions to the Cauchy problem, including the Prodi-Serrin type criterion ($\hat{I}\pm >\frac{3}{4}$) and the Beir$\tilde{\mathrm{a}}$o da Veiga type criterion $(\hat{I}\pm >\frac{1}{2})$. Then, we verify the global existence and uniqueness of strong solutions for arbitrarily large initial fluid velocity and bacteria density for $\hat{I}\pm \ge \frac{5}{4}$. Furthermore, in the scenario of $\frac{3}{4}<\hat{I}\pm <\frac{5}{4}$, we establish uniform regularity estimates and optimal time-decay rates of global solutions if the $L^2$-norm of initial data is small. To our knowledge, this is the first result concerning the global existence and large-time behavior of strong solutions for the chemotaxis-Navier-Stokes equations with possibly large oscillations.

39 pages