Mild solutions to the Cauchy problem for time-space fractional Keller-Segel-Navier-Stokes system

This paper investigates the Cauchy problem of time-space fractional Keller-Segel-Navier-Stokes system in \({\mathbb {R}}^d~(d\ge 2)\), which describes both memory effect and Lévy process of the system. The local and global existence of mild solutions are obtained by the \(L^p-L^q\) estimates of Mittag-Leffler operators combined with Banach fixed point theorem and Banach implicit function theorem, respectively. Furthermore, some properties are established, such as mass conservation, decay estimates, stability and self-similarity of mild solutions.