Global existence for the fully parabolic Keller--Segel system with critical mass on the plane

TL;DR

This paper proves the global existence of solutions for the two-dimensional fully parabolic Keller--Segel system at critical mass without symmetry or moment restrictions, using a novel Lyapunov functional and refined estimates.

Contribution

It introduces a new Lyapunov functional and regularity estimates to establish global solutions for the Keller--Segel system at critical mass without symmetry assumptions.

Findings

Global existence for general initial data at critical mass

Construction of a reconstructed Lyapunov functional

Refined regularity estimates for dissipative terms

Abstract

We study the global existence of solutions to the Cauchy problem for the two-dimensional fully parabolic Keller--Segel system at the critical mass. It is known that global-in-time existence holds for initial data with critical mass under radial symmetry or suitable moment conditions, whereas the behavior of general solutions in the critical regime remains delicate. In this paper, we establish global-in-time existence for general initial data with critical mass, without imposing any symmetry or moment assumptions. The proof relies on the construction of a reconstructed Lyapunov functional, combined with refined regularity estimates for the associated dissipative terms, which enable us to control the solution dynamics in the critical regime.