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

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 + $\mathbf{u}$\cdot\nabla n & = Δn - \nabla\cdot\left(χ©n\nabla c\right) + \nabla\cdot(n\nablaφ), \ \partial_t c + $\mathbf{u}$\cdot\nabla c &= Δc - nf©, \ \partial_t$\mathbf{u}$ + κ($\mathbf{u}$\cdot\nabla)$\mathbf{u}$ + \nabla P & = Δ$\mathbf{u}$ - n\nablaφ+ χ©n\nabla c, \ \nabla\cdot$\mathbf{u}$ &= 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. 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 $\mathrm{\parallel }{c}_0{\mathrm{\parallel }}_{L^{\mathrm{\infty }}}$ with the help of a new entropy functional inequality. Some further decay estimates are also obtained under some suitable growth restriction on the potential $\mathrm{\nabla }\phi$ 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 chemotaxis-Navier-Stokes system. To the best of our knowledge, this seems to be the first work addressing the global well-posedness and decay property of solutions to the Cauchy problem of self-consistent chemotaxis-fluid system.