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Axisymmetric type II blowup solutions to the three dimensional Keller-Segel system

Thomas Y. Hou, Van Tien Nguyen, Peicong Song

Abstract

We construct axisymmetric solutions to the three-dimensional parabolic-elliptic Keller-Segel system that blows up in finite time. In particular, the singularity is of type II, which admits locally a leading order profile of the rescaled stationary solution of the two-dimensional system. Additionally, mass concentration occurs along a one-dimensional ring in the plane. In the analysis, we rely on an approximate solution of the eigenproblem associated with the linearized operator around the stationary solution as well as the modulation dynamics to control the perturbation function and derive the accurate blowup rate.

Axisymmetric type II blowup solutions to the three dimensional Keller-Segel system

Abstract

We construct axisymmetric solutions to the three-dimensional parabolic-elliptic Keller-Segel system that blows up in finite time. In particular, the singularity is of type II, which admits locally a leading order profile of the rescaled stationary solution of the two-dimensional system. Additionally, mass concentration occurs along a one-dimensional ring in the plane. In the analysis, we rely on an approximate solution of the eigenproblem associated with the linearized operator around the stationary solution as well as the modulation dynamics to control the perturbation function and derive the accurate blowup rate.

Paper Structure

This paper contains 21 sections, 19 theorems, 477 equations.

Table of Contents

  1. Introduction
  2. Setting of the Problem
  3. Previous Results
  4. Statement of the Result
  5. Strategy of the Proof
  6. Properties of the linearized operator
  7. Two approximate eigenfunctions
  8. Coercivity of the Linearized Operator
  9. Properties of the Operator $\mathscr{M}^\zeta_\nu$
  10. Adapted Inner Product and Coercivity
  11. Construction of Blowup Solutions
  12. Decomposition of the Solution and Formulation of the Linearized Problem
  13. Bootstrap Regime and Modulation Equations
  14. Energy Estimates
  15. $L^2$ Inner Estimate
  16. ...and 6 more sections

Key Result

Theorem 1

For any $T>0$, there exists initial data $u_0$ in the function space $\mathcal{E}$ and $R_0>0$, such that the following holds for the associate solution to Introduction:3dks. It blows up at finite time $T$ according to the dynamic such that:

Theorems & Definitions (37)

  • Theorem 1: Axisymetric type II blow up for the $3$D Keller-Segel system
  • Proposition 1: Two approximate eigenfunctions
  • proof : Proof of Proposition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Proposition 2: Coercivity estimate
  • proof
  • Corollary 1
  • ...and 27 more