Finite-Time Blowup for Keller–Segel–Navier–Stokes System in Three Dimensions

Abstract

While finite-time blowup solutions have been studied in depth for the Keller–Segel equation, a fundamental model describing chemotaxis, the existence of finite-time blowup solutions to chemotaxis-fluid models remains largely unexplored. To fill this gap in the literature, we use a quantitative method to directly construct a smooth finite-time blowup solution for the Keller–Segel–Navier–Stokes system with buoyancy in 3D. The heart of the proof is to establish the non-radial finite-codimensional stability of an explicit self-similar blowup solution to 3D Keller–Segel equation with the abstract semigroup tool from Merle et al. (Invent Math 227:247–413, 2022), which partially generalizes the radial stability result (Glogić and Schörkhuber in Arch Ration Mech Anal 248:4, 2024) to the non-radial setting. Additionally, we introduce a robust localization argument to find blowup solutions with non-negative density and finite mass.