Various regularity estimates for the Keller-Segel-Navier-Stokes system in Besov spaces

Abstract

We show the local well-posedness for the Keller-Segel-Navier-Stokes system with initial data in the scaling invariant Besov spaces, where the solution exists globally in time if the initial data is sufficiently small. We also reveal that the solution belongs to the Lorentz spaces in time direction, while the solution is smooth in space and time. Moreover, we obtain the maximal regularity estimates of solutions under the certain conditions. We further show that the solution has the additional regularities if the initial data has higher regularities. This result implies that global solutions decay as the limit t→∞ in the same norm of the space of the initial data. Our results on the Lorentz regularity estimates are based on the strategy by Kozono-Shimizu (2019) [26].