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Suppression of chemotactic singularity by buoyancy

Zhongtian Hu, Alexander Kiselev, Yao Yao

TL;DR

This work addresses the question of whether active advection, via buoyancy-driven Darcy flow, can prevent finite-time blow-up in the Patlak-Keller-Segel system. The authors develop a framework combining energy methods, a potential energy functional $E(t)= frac{}{} ho x_2 ext{dx}$, and a partition of time into 'good' and 'bad' intervals to quantify energy exchange between diffusion, advection, and chemotaxis. Central to the argument are a mixing-driven, Nash-type estimate and sharp interval estimates that show a net decay of the potential energy on good intervals and controlled growth on bad intervals, yielding global regularity for any $g eq0$. The results suggest that active, buoyancy-driven advection can robustly regularize nonlinear chemotaxis models, with potential implications for other active-fluid systems and higher-dimensional settings.

Abstract

Chemotactic singularity formation in the context of the Patlak-Keller-Segel equation is an extensively studied phenomenon. In recent years, it has been shown that the presence of fluid advection can arrest the singularity formation given that the fluid flow possesses mixing or diffusion enhancing properties and its amplitude is sufficiently strong - this effect is conjectured to hold for more general classes of nonlinear PDEs. In this paper, we consider the Patlak-Keller-Segel equation coupled with a fluid flow that obeys Darcy's law for incompressible porous media via buoyancy force. We prove that in contrast with passive advection, this active fluid coupling is capable of suppressing singularity formation at arbitrary small coupling strength: namely, the system always has globally regular solutions.

Suppression of chemotactic singularity by buoyancy

TL;DR

This work addresses the question of whether active advection, via buoyancy-driven Darcy flow, can prevent finite-time blow-up in the Patlak-Keller-Segel system. The authors develop a framework combining energy methods, a potential energy functional , and a partition of time into 'good' and 'bad' intervals to quantify energy exchange between diffusion, advection, and chemotaxis. Central to the argument are a mixing-driven, Nash-type estimate and sharp interval estimates that show a net decay of the potential energy on good intervals and controlled growth on bad intervals, yielding global regularity for any . The results suggest that active, buoyancy-driven advection can robustly regularize nonlinear chemotaxis models, with potential implications for other active-fluid systems and higher-dimensional settings.

Abstract

Chemotactic singularity formation in the context of the Patlak-Keller-Segel equation is an extensively studied phenomenon. In recent years, it has been shown that the presence of fluid advection can arrest the singularity formation given that the fluid flow possesses mixing or diffusion enhancing properties and its amplitude is sufficiently strong - this effect is conjectured to hold for more general classes of nonlinear PDEs. In this paper, we consider the Patlak-Keller-Segel equation coupled with a fluid flow that obeys Darcy's law for incompressible porous media via buoyancy force. We prove that in contrast with passive advection, this active fluid coupling is capable of suppressing singularity formation at arbitrary small coupling strength: namely, the system always has globally regular solutions.
Paper Structure (12 sections, 20 theorems, 110 equations, 2 figures)

This paper contains 12 sections, 20 theorems, 110 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and conventions
  4. Functional spaces and eigenfunction expansions
  5. Local well-posedness and conditional regularity
  6. Energy estimates
  7. Partition into "good" and "bad" time intervals
  8. Main estimates
  9. Key lemmas
  10. Estimates on the good intervals
  11. Estimates on the bad intervals
  12. Proof of the main theorem

Key Result

Theorem 1.1

Let $g \ne 0.$ Then for every initial data $\rho_0 \in C^\infty(\Omega),$ the solution to the Patlak-Keller-Segel-IPM system eq:KSIPM is globally regular: $C^\infty$ in both space and time.

Figures (2)

  • Figure 1: The times $\{t_k\}$ are marked by the dotted vertical lines. On the $t$ axis, the good intervals are marked in blue color, and the bad intervals are marked in orange color.
  • Figure 2: An illustration of the cones in the definitions of $\mathcal{C}_1$ and $\mathcal{C}_2$ for some $N\gg 1$. Note that $\mathcal{C}_1$ and $\mathcal{C}_2$ are subsets of $\mathbb{Z}^2$, thus they only contain the integer points $k\in\mathbb{Z}^2$ that fall in the shaded regions.

Theorems & Definitions (44)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • proof
  • Proposition 4.1
  • ...and 34 more