Well-posedness for chemotaxis–fluid models in arbitrary dimensions*

Abstract

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 L 2 , N − 2 − 1 ( R N ) which collects divergence of vector-fields with components in the square Campanato space L 2 , N − 2 ( R N ) , N > 2 (and can be identified with the homogeneous Besov space B ˙ 22 − 1 ( R N ) 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.