Well-posedness for chemotaxis-fluid models in arbitrary dimensions

Abstract

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.