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=\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*} 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 ($\alpha>\frac{ 3}{ 4}$) and the Beir${ \rm\tilde{ a}}$o da Veiga type criterion $(\alpha>\frac{ 1}{ 2})$. 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 $\frac{ 3}{ 4}<\alpha<\frac{ 5}{ 4}$, 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.
Categories: math.AP
