Communications in Mathematical Sciences

Volume 18 (2020)

Number 2

Optimal critical mass for the two-dimensional Keller–Segel model with rotational flux terms

Pages: 379 – 394



Elio Espejo (School of Mathematical Sciences, University of Nottingham, Ningbo, China)

Hao Wu (School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai, China; and Key Laboratory of Mathematics for Nonlinear Sciences (Fudan University), Ministry of Education, Shanghai, China)


Our aim is to show that several important systems of partial differential equations arising in mathematical biology, fluid dynamics and electrokinetics can be approached within a single model, namely, a Keller–Segel-type system with rotational flux terms. In particular, we establish sharp conditions on the optimal critical mass for having global existence and finite time blow-up of solutions in two spatial dimensions. Our results imply that the rotated chemotactic response can delay or even avoid the blow-up. The key observation is that for any angle of rotation $\alpha \in (- \pi , \pi ]$, the resulting PDE system preserves a dissipative energy structure. Inspired by this property, we also provide an alternative derivation of the general system via an energetic variational approach.


chemotaxis, rotational flux, critical mass, blow-up, global existence, dissipative energy structure

2010 Mathematics Subject Classification

35B40, 35K57, 92C15, 92C17

Received 3 July 2019

Accepted 7 October 2019

Published 20 June 2022