Well-posedness for chemotaxis-fluid models in arbitrary dimensions

We study the Cauchy problem for the chemotaxis Navier-Stokes equations and the Keller-Segel-Navier-Stokes system. Local-in-time and global-in-time solutions satisfying fundamental properties such as mass conservation and nonnegativity preservation are constructed for low regularity data in $2$ and higher dimensions under suitable conditions. Our initial data classes involve a new scale of function space, that is \Y(\rn) which collects divergence of vector-fields with components in the square Campanato space \mathscr{L}_{2,N-2}(\rn), $N>2$ (and can be identified with the homogeneous Besov space \dot{B}^{-1}_{22}(\rn) when $N=2$) and are shown to be optimal in a certain sense. Moreover, uniqueness criterion for global solutions is obtained under certain limiting conditions

Global existence and decay rates to a 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: ∂tn + u · ∇n = ∆n − ∇ · (χ(c)n∇c) + ∇ · (n∇ϕ), ∂tc + u · ∇c = ∆c − nf(c), ∂tu + κ(u · ∇)u + ∇P = ∆u − n∇ϕ + χ(c)n∇c, ∇ · u = 0 in Rd × (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 due to the additional term χ(c)n∇c in the third equation. 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 ∥c0∥L∞ with the help of a new entropy functional inequality. Some further decay estimates are also obtained under some suitable growth restriction on the potential ∇ϕ 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