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

Tatsuya Hosono

TL;DR

The paper addresses the global existence of solutions to the 2D fully parabolic Keller–Segel system at the critical mass $\|u_0\|_1=8\pi$ without symmetry or moment assumptions. It introduces a reconstructed Lyapunov functional $\mathcal{F}_m(t)$ to overcome regularity gaps in the critical regime and derives dissipation bounds that extend to the whole space. Through exterior-domain regularity and interior-domain entropy estimates, the authors obtain comprehensive a priori bounds and a continuation argument that precludes finite-time blow-up. Consequently, the work establishes global existence for general initial data at the critical mass and suggests a robust framework applicable to other chemotaxis systems on the whole space.

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.

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

TL;DR

The paper addresses the global existence of solutions to the 2D fully parabolic Keller–Segel system at the critical mass without symmetry or moment assumptions. It introduces a reconstructed Lyapunov functional to overcome regularity gaps in the critical regime and derives dissipation bounds that extend to the whole space. Through exterior-domain regularity and interior-domain entropy estimates, the authors obtain comprehensive a priori bounds and a continuation argument that precludes finite-time blow-up. Consequently, the work establishes global existence for general initial data at the critical mass and suggests a robust framework applicable to other chemotaxis systems on the whole space.

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.
Paper Structure (8 sections, 21 theorems, 238 equations)

This paper contains 8 sections, 21 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Local-in-time solutions
  4. Global existence of solutions with critical mass
  5. Refined Lyapunov functional
  6. Regularity estimates in exterior domains
  7. Regularity estimates in interior domains
  8. Global existence of solutions

Key Result

Theorem 1.1

For $(u_0,v_0)\in L_+^1(\mathbb R^2) \times L^1_+(\mathbb R^2)\cap \dot{H}^1(\mathbb R^2)$, let $(u,v)$ be the solution to eqn;KS on $(0,T)\times\mathbb R^2$. Suppose that $\|u_0\|_1=8\pi$. Then, the solution to eqn;KS exists globally in time.

Theorems & Definitions (25)

  • Definition
  • Theorem 1.1
  • Remark 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Proposition 3.1: local-in-time solution
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 4.1
  • ...and 15 more