Finite-time blowup for Keller-Segel-Navier-Stokes system in three dimensions

TL;DR

This paper constructs a smooth finite-time blowup solution for the 3D Keller-Segel-Navier-Stokes system, advancing understanding of chemotaxis-fluid models and their singularity formation.

Contribution

It introduces a novel method to directly construct blowup solutions and establishes non-radial stability, extending previous radial results to more general settings.

Findings

Constructed a smooth finite-time blowup solution in 3D

Established non-radial finite-codimensional stability

Developed a localization technique for solutions with finite mass

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-Rapha\"el-Rodnianski-Szeftel, 2022], which partially generalizes the radial stability result [Glogi\'c-Sch\"orkhuber, 2024] to the non-radial setting. Additionally, we introduce a robust localization argument to find blowup solutions with non-negative density and finite mass.