Well-posedness for chemotaxis–fluid models in arbitrary dimensions* *The author acknowledges the support of the DAAD through the program ‘Graduate School Scholarship Programme, 2018’ (Number 57395813) and the Hausdorff Centre for Mathematics at Bonn.

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 which collects divergence of vector-fields with components in the square Campanato space , N > 2 (and can be identified with the homogeneous Besov space 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.