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Blow up analysis for Keller-Segel system

Hua Chen, Jian-Meng Li, Kelei Wang

TL;DR

This work develops a comprehensive blow-up theory for the 2D parabolic-elliptic Keller-Segel system, establishing measure-valued limits μ_t with mass quantization into 8π-atoms and a dynamical law for their motion. Using localization, symmetrization, and ε-regularity, the authors prove convergence of smooth solutions to μ_t = ∑_{j=1}^{N(t)} 8π δ_{q_j(t)} + ρ, with the atoms moving according to a gradient-flow-type ODE driven by inter-particle and diffusion interactions. They analyze first time singularities, ancient solutions, and entire solutions, showing that blow-up limits are governed by a renormalized energy W whose critical points determine the asymptotic multi-bubble configurations; whole-space limits reduce to Liouville-type bubbles and admit explicit classification. The results yield a precise micro-to-macro picture of singularity formation, including mass quantization, dynamics of singularities, and large-scale structure, with extensions to boundary blow-up points. Overall, the paper provides a rigorous, quantized, and energetically organized framework for understanding blow-up phenomena in Keller-Segel systems and related parabolic-elliptic models.

Abstract

In this paper we develop a blow up theory for the parabolic-elliptic Keller-Segel system, which can be viewed as a parabolic counterpart to the Liouville equation. This theory is applied to the study of first time singularities, ancient solutions and entire solutions, leading to a description of the blow-up limit in the first problem, and the large scale structure in the other two problems.

Blow up analysis for Keller-Segel system

TL;DR

This work develops a comprehensive blow-up theory for the 2D parabolic-elliptic Keller-Segel system, establishing measure-valued limits μ_t with mass quantization into 8π-atoms and a dynamical law for their motion. Using localization, symmetrization, and ε-regularity, the authors prove convergence of smooth solutions to μ_t = ∑_{j=1}^{N(t)} 8π δ_{q_j(t)} + ρ, with the atoms moving according to a gradient-flow-type ODE driven by inter-particle and diffusion interactions. They analyze first time singularities, ancient solutions, and entire solutions, showing that blow-up limits are governed by a renormalized energy W whose critical points determine the asymptotic multi-bubble configurations; whole-space limits reduce to Liouville-type bubbles and admit explicit classification. The results yield a precise micro-to-macro picture of singularity formation, including mass quantization, dynamics of singularities, and large-scale structure, with extensions to boundary blow-up points. Overall, the paper provides a rigorous, quantized, and energetically organized framework for understanding blow-up phenomena in Keller-Segel systems and related parabolic-elliptic models.

Abstract

In this paper we develop a blow up theory for the parabolic-elliptic Keller-Segel system, which can be viewed as a parabolic counterpart to the Liouville equation. This theory is applied to the study of first time singularities, ancient solutions and entire solutions, leading to a description of the blow-up limit in the first problem, and the large scale structure in the other two problems.
Paper Structure (19 sections, 37 theorems, 173 equations)

This paper contains 19 sections, 37 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Blow up analysis
  3. First time singularity
  4. Ancient solutions
  5. Entire solutions
  6. Setting
  7. Two basic tools
  8. Convergence of solutions
  9. The $8\pi$ phenomena
  10. Local structure of blow up locus
  11. Two technical lemmas
  12. Proof of Proposition \ref{['prop local structure']}
  13. Limit equations
  14. Blow-up limits: Proof of Theorem \ref{['thm first time singularity']}
  15. Preliminary estimates
  16. ...and 4 more sections

Key Result

Theorem 1.1

Suppose $u_i$ is a sequence of smooth, positive solutions of the Keller-Segel system eqn in the unit parabolic cylinder $Q_1:=B_1\times(-1,1)\subset\mathbb{R}^2\times\mathbb{R}$, satisfying Then the followings hold.

Theorems & Definitions (73)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3: Suzuki
  • Theorem 1.4
  • Remark 1.5
  • Corollary 1.6
  • Definition 1.7
  • Theorem 1.8
  • Definition 1.9
  • Theorem 1.10
  • ...and 63 more