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Nonradial stability of self-similar blowup to Keller-Segel equation in three dimensions

Zexing Li, Tao Zhou

TL;DR

The paper proves nonradial stability of a 3D self-similar blowup profile for the parabolic-elliptic Keller-Segel system by combining a self-similar renormalization with a detailed mode-stability analysis across angular momentum sectors. It introduces a wave-operator localization in the critical l=1 class to remove the sole unstable mode, and it establishes coercivity for higher angular modes (l≥3) and a GGMT-based treatment for l=2, together yielding a full spectral characterization. Building on a finite-codimensional nonlinear stability framework, the authors construct a stable manifold of codimension four and prove that perturbations remain close to the self-similar profile up to controlled blowup in renormalized variables, with explicit decay of the stable component and management of unstable directions. The approach is robust to nonlocal effects and does not rely on an explicit profile, suggesting potential applicability to other nonlocal PDE models and self-similar structures.

Abstract

In three dimensions, the parabolic-elliptic Keller-Segel system exhibits a rich variety of singularity formations. Notably, it admits an explicit self-similar blow-up solution whose radial stability, conjectured more than two decades ago in [Brenner-Constantin-Kadanoff-Schenkel-Venkataramani, 1999], was recently confirmed by [Glogić-Schörkhuber, 2024]. This paper aims to extend the radial stability to the nonradial setting, building on the finite-codimensional stability analysis in our previous work [Li-Zhou, 2024]. The main input is the mode stability of the linearized operator, whose nonlocal nature presents essential challenges for the spectral analysis. Besides a quantitative perturbative analysis for the high spherical classes, we adapt in the first spherical class the wave operator method of [Li-Wei-Zhang, 2020] for the fluid stability to localize the operator and remove the known unstable mode simultaneously. Our method provides localization beyond the partial mass variable and is independent of the explicit formula of the profile, so it potentially sheds light on other linear nonlocal problems.

Nonradial stability of self-similar blowup to Keller-Segel equation in three dimensions

TL;DR

The paper proves nonradial stability of a 3D self-similar blowup profile for the parabolic-elliptic Keller-Segel system by combining a self-similar renormalization with a detailed mode-stability analysis across angular momentum sectors. It introduces a wave-operator localization in the critical l=1 class to remove the sole unstable mode, and it establishes coercivity for higher angular modes (l≥3) and a GGMT-based treatment for l=2, together yielding a full spectral characterization. Building on a finite-codimensional nonlinear stability framework, the authors construct a stable manifold of codimension four and prove that perturbations remain close to the self-similar profile up to controlled blowup in renormalized variables, with explicit decay of the stable component and management of unstable directions. The approach is robust to nonlocal effects and does not rely on an explicit profile, suggesting potential applicability to other nonlocal PDE models and self-similar structures.

Abstract

In three dimensions, the parabolic-elliptic Keller-Segel system exhibits a rich variety of singularity formations. Notably, it admits an explicit self-similar blow-up solution whose radial stability, conjectured more than two decades ago in [Brenner-Constantin-Kadanoff-Schenkel-Venkataramani, 1999], was recently confirmed by [Glogić-Schörkhuber, 2024]. This paper aims to extend the radial stability to the nonradial setting, building on the finite-codimensional stability analysis in our previous work [Li-Zhou, 2024]. The main input is the mode stability of the linearized operator, whose nonlocal nature presents essential challenges for the spectral analysis. Besides a quantitative perturbative analysis for the high spherical classes, we adapt in the first spherical class the wave operator method of [Li-Wei-Zhang, 2020] for the fluid stability to localize the operator and remove the known unstable mode simultaneously. Our method provides localization beyond the partial mass variable and is independent of the explicit formula of the profile, so it potentially sheds light on other linear nonlocal problems.
Paper Structure (24 sections, 25 theorems, 277 equations)

This paper contains 24 sections, 25 theorems, 277 equations.

Table of Contents

  1. Introduction
  2. Background.
  3. Chemotaxis: global existence v.s. finite-time blowup.
  4. Singularity formation
  5. Main result
  6. Mode stability
  7. Sketch of proof.
  8. Notation
  9. Mode stability and linear theory
  10. Preliminaries
  11. Mode stability for $l \ge 3$
  12. Mode stability for $l = 2$
  13. Mode stability for $l = 1$.
  14. Proof of Theorem \ref{['thm: mode stability main']}
  15. Linear theory of $-{\mathcal{L}}$
  16. ...and 9 more sections

Key Result

Theorem 1.1

There exists $\delta \ll 1$ such that for any given $\varepsilon_0 \in H^2(\mathbb{R}^3)$ with $\| \varepsilon_0 \|_{H^2} \le \delta$, there exist $(\lambda_0,x_0) \in \mathbb{R}^+ \times \mathbb{R}^3$ with such that the initial data with $Q$ given in profile, generates a solution to equation, Keller-Segel blowing up at $T=\lambda_0^2$ with where for some $\tilde{\epsilon} >0$ independent of $

Theorems & Definitions (48)

  • Theorem 1.1: Stability of self-similar solution \ref{['profile']} to $3D$ Keller-Segel equation
  • Theorem 1.2: Mode stability
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 38 more