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Quantitative blow-up suppression for the Patlak-Keller-Segel(-Navier-Stokes) system via Couette flow on $\mathbb{R}^2$

Yubo Chen, Wendong Wang, Guoxu Yang

TL;DR

The authors study the Patlak–Keller–Segel system on the plane with a Couette shear flow and prove global existence for large shear amplitude when the initial bacterial density has supercritical mass, using an anisotropic Y_{m,ε} norm to capture horizontal regularity and low-frequency suppression. They develop a multiplier-based energy method that combines enhanced dissipation and inviscid damping, yielding explicit bounds on the required amplitude A and enabling a bootstrap argument to close the estimates. The results extend to the PKS–NS system near Couette flow, providing explicit thresholds for global well-posedness in terms of the combined initial data $(\omega_{in}, n_{in}, |D_x|^{1/3}n_{in})$; special attention is given to the α=0 case with a particularly large explicit constant. Overall, the work demonstrates that strong mixing from Couette flow can suppress PKS blow-up in nonperiodic settings and supplies quantitative, fully explicit amplitude thresholds for practical verification and further study.

Abstract

It is well known that solutions to the Patlak--Keller--Segel system on $\mathbb{R}^2$ blow up in finite time if the initial mass exceeds $8π$. In this paper, we investigate the mixing effect induced by a Couette flow $(Ay, 0)$ with a quantitatively determined amplitude $A$, which suppresses bacterial aggregation. For the Patlak--Keller--Segel system advected by such a flow on $\mathbb{R}^2$, we prove that the solutions remain global in time even for large initial mass, provided the amplitude $A$ is sufficiently large. Specifically, global well-posedness holds if $A$ satisfies a lower bound of the form $C_* \left(\| \langle D_x\rangle^{m} \langle {D_x}^{-1}\rangle^εn_{\mathrm{in}} \|_{L^2}^2+1\right)^{9/2}$. A notable feature of our result is the explicit estimate of the sufficient constant, given by $C_* = 2,058,614$. Furthermore, for the coupled Patlak--Keller--Segel--Navier--Stokes system near the Couette flow, we establish an analogous global existence result, provided the amplitude is sufficiently large in form of $C_*\|(n_{\rm in}, |D_x|^{1/3} n_{\rm in},ω_{\rm in})\|_{Y_{m,ε}}^9$.

Quantitative blow-up suppression for the Patlak-Keller-Segel(-Navier-Stokes) system via Couette flow on $\mathbb{R}^2$

TL;DR

The authors study the Patlak–Keller–Segel system on the plane with a Couette shear flow and prove global existence for large shear amplitude when the initial bacterial density has supercritical mass, using an anisotropic Y_{m,ε} norm to capture horizontal regularity and low-frequency suppression. They develop a multiplier-based energy method that combines enhanced dissipation and inviscid damping, yielding explicit bounds on the required amplitude A and enabling a bootstrap argument to close the estimates. The results extend to the PKS–NS system near Couette flow, providing explicit thresholds for global well-posedness in terms of the combined initial data ; special attention is given to the α=0 case with a particularly large explicit constant. Overall, the work demonstrates that strong mixing from Couette flow can suppress PKS blow-up in nonperiodic settings and supplies quantitative, fully explicit amplitude thresholds for practical verification and further study.

Abstract

It is well known that solutions to the Patlak--Keller--Segel system on blow up in finite time if the initial mass exceeds . In this paper, we investigate the mixing effect induced by a Couette flow with a quantitatively determined amplitude , which suppresses bacterial aggregation. For the Patlak--Keller--Segel system advected by such a flow on , we prove that the solutions remain global in time even for large initial mass, provided the amplitude is sufficiently large. Specifically, global well-posedness holds if satisfies a lower bound of the form . A notable feature of our result is the explicit estimate of the sufficient constant, given by . Furthermore, for the coupled Patlak--Keller--Segel--Navier--Stokes system near the Couette flow, we establish an analogous global existence result, provided the amplitude is sufficiently large in form of .

Paper Structure

This paper contains 20 sections, 14 theorems, 198 equations.

Table of Contents

  1. Introduction
  2. Developments
  3. Stability mechanisms
  4. Main results
  5. Some notations and outlines
  6. Preliminaries and proofs of the main theorems
  7. Construction of the multipliers
  8. Elliptic estimates of $c$
  9. Space-time estimate
  10. Bootstrap argument
  11. Without coupling the fluid equations
  12. With coupling the fluid equations
  13. Proof of Proposition \ref{['main prop3']}
  14. Estimate of $n$
  15. Closing the energy
  16. ...and 5 more sections

Key Result

Theorem 1.1

For $0<\epsilon< 1/2<m$, assume that the initial data $n_{\mathrm{in}} \in L^\infty(\mathbb{R}^2) \cap L^1(\mathbb{R}^2)$ satisfying There exists a positive constant $\Lambda$ depending on $\epsilon, m,\alpha, \theta_1, \theta_2, \Xi$ and $a$ such that if the solutions to eq:main2 are global in time. Additionally, $\epsilon>1/3$ is required when $\alpha = 0.$ In fact, the value of $\Lambda$ can

Theorems & Definitions (35)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 25 more