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Finite-time blowup for Keller-Segel-Navier-Stokes system in three dimensions

Zexing Li, Tao Zhou

TL;DR

This work addresses finite-time blowup in a three-dimensional chemotaxis-fluid model (Keller-Segel-Navier-Stokes with buoyancy) by constructing a smooth blowup solution via a nonradial stability analysis of an explicit self-similar KS profile. The authors implement a self-similar renormalization and perform a detailed spectral analysis of the linearized operator ${\mathcal L}= {\mathcal L}_0+{\mathcal L}'$, obtaining a finite-dimensional unstable subspace and decay on the stable complement through a semigroup framework. A modified unstable space and a Brouwer fixed-point argument enable a finite-codimensional stability result, while a localization strategy converts the infinite-mass profile into a finite-mmass, nonnegative-density blowup solution. The density concentrates as $\rho(t,x) \sim \frac{1}{T-t} Q\left(\frac{x}{\sqrt{T-t}}\right)$ with a perturbation vanishing in $H^s$, whereas the fluid component blows up more slowly (e.g., logarithmically in time) due to buoyancy, illustrating how active flow interacts with chemotactic collapse in this system.

Abstract

While finite-time blowup solutions have been studied in depth for the Keller-Segel equation, a fundamental model describing chemotaxis, the existence of finite-time blowup solutions to chemotaxis-fluid models remains largely unexplored. To fill this gap in the literature, we use a quantitative method to directly construct a smooth finite-time blowup solution for the Keller-Segel-Navier-Stokes system with buoyancy in 3D. The heart of the proof is to establish the non-radial finite-codimensional stability of an explicit self-similar blowup solution to 3D Keller-Segel equation with the abstract semigroup tool from [Merle-Raphaël-Rodnianski-Szeftel, 2022], which partially generalizes the radial stability result [Glogić-Schörkhuber, 2024] to the non-radial setting. Additionally, we introduce a robust localization argument to find blowup solutions with non-negative density and finite mass.

Finite-time blowup for Keller-Segel-Navier-Stokes system in three dimensions

TL;DR

This work addresses finite-time blowup in a three-dimensional chemotaxis-fluid model (Keller-Segel-Navier-Stokes with buoyancy) by constructing a smooth blowup solution via a nonradial stability analysis of an explicit self-similar KS profile. The authors implement a self-similar renormalization and perform a detailed spectral analysis of the linearized operator , obtaining a finite-dimensional unstable subspace and decay on the stable complement through a semigroup framework. A modified unstable space and a Brouwer fixed-point argument enable a finite-codimensional stability result, while a localization strategy converts the infinite-mass profile into a finite-mmass, nonnegative-density blowup solution. The density concentrates as with a perturbation vanishing in , whereas the fluid component blows up more slowly (e.g., logarithmically in time) due to buoyancy, illustrating how active flow interacts with chemotactic collapse in this system.

Abstract

While finite-time blowup solutions have been studied in depth for the Keller-Segel equation, a fundamental model describing chemotaxis, the existence of finite-time blowup solutions to chemotaxis-fluid models remains largely unexplored. To fill this gap in the literature, we use a quantitative method to directly construct a smooth finite-time blowup solution for the Keller-Segel-Navier-Stokes system with buoyancy in 3D. The heart of the proof is to establish the non-radial finite-codimensional stability of an explicit self-similar blowup solution to 3D Keller-Segel equation with the abstract semigroup tool from [Merle-Raphaël-Rodnianski-Szeftel, 2022], which partially generalizes the radial stability result [Glogić-Schörkhuber, 2024] to the non-radial setting. Additionally, we introduce a robust localization argument to find blowup solutions with non-negative density and finite mass.
Paper Structure (22 sections, 18 theorems, 184 equations)

This paper contains 22 sections, 18 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Chemotaxis and singularity formation.
  4. Suppression of blowup by fluids
  5. Main result
  6. Structure of the paper
  7. Notations
  8. Linear theory in the self-similar coordinate
  9. Self-similar coordinate and renormalization
  10. Linear theory of ${\mathcal{L}}_0$
  11. Perturbed maximal dissipativity of $-{\mathcal{L}}$
  12. Spectral properties of $-{\mathcal{L}}$
  13. Modified unstable space
  14. Existence of Stable Trajectory
  15. A priori estimates
  16. ...and 7 more sections

Key Result

Theorem 1.1

For any integer $s \ge 3$ and any divergence-free vector field $u_0 \in H_\sigma^\infty(\mathbb{R}^3)$ fixed, there exists non-negative $\rho_0 \in C^\infty_0(\mathbb{R}^3)$, such that the smooth solution to equation: NS-KS velocity form with initial data $(\rho_0, u_0)$ blows up at some time $t=T<\ where $Q$ is given by Q: definition and $\mathop{\rm lim}_{t\to T^-}\| \varepsilon(t) \|_{H^s(\math

Theorems & Definitions (36)

  • Theorem 1.1: Existence of smooth finite-time blowup solution with nonnegative density and finite mass
  • Lemma 2.1: Self-adjointness and semigroup of ${\mathcal{L}}_0$ in $L^2_\omega$
  • proof
  • Lemma 2.2: Closedness and smoothing resolvent estimate for ${\mathcal{L}}_0$ in $H^k$
  • proof
  • Proposition 2.3: Perturbed maximal dissipativity of $-{\mathcal{L}}$
  • Lemma 2.5: Compactness of ${\mathcal{L}}_1', {\mathcal{L}}_2'$, ${\mathcal{L}}_3'$ and ${\mathcal{C}}_3'$
  • proof
  • Lemma 2.6: Almost coercivity for ${\mathcal{L}}'$
  • proof
  • ...and 26 more