Partial regularity of solutions to the 3D chemotaxis-Navier-Stokes equations at the first blow-up time

In this note, we investigate partial regularity of weak solutions of the three dimensional chemotaxis-Navier-Stokes equations, and obtain the $\frac53$-dimensional Hausdorff measure of the possible singular set is vanishing at the first blow-up time. The new ingredients are to establish certain type of local energy inequality and deal with the non-scaling invariant quantity of $n\ln n$, which seems to be the first description for the singular set of weak solutions of the chemotaxis-fluid model, which is motivated by Caffarelli-Kohn-Nirenberg's partial regularity theory \cite{CKN}

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