Suppression of blow-up in Patlak-Keller-Segel system coupled with linearized Navier-Stokes equations via the 3D Couette flow

TL;DR

This paper demonstrates that strong shear flow in a coupled Patlak-Keller-Segel and Navier-Stokes system with non-slip boundaries can prevent finite-time blow-up, ensuring global solutions for small initial cell mass.

Contribution

It is the first to show suppression of blow-up in the 3D Patlak-Keller-Segel-Navier-Stokes system via Couette flow with non-slip boundary conditions.

Findings

Strong shear flow prevents blow-up for small initial mass.

Solutions are global in time under certain shear strength and initial conditions.

First analysis of Couette flow's stabilizing effect in this coupled system.

Abstract

It is known that finite-time blow-up in the 3D Patlak-Keller-Segel system may occur for arbitrarily small values of the initial mass. It's interesting whether one can prevent the finite-time blow-up via the stabilizing effect of the moving fluid. Consider the three-dimensional Patlak-Keller-Segel system coupled with the linearized Navier-Stokes equations near the Couette flow $(\ Ay, 0, 0 \ )$ in a finite channel $\mathbb{T}\times\mathbb{I}\times\mathbb{T}$ with $ \mathbb{T}=[0,2\pi) $ and $ \mathbb{I}=[-1,1] $, with the non-slip boundary condition, and we show that if the shear flow is sufficiently strong (A is large enough), then the solutions to Patlak-Keller-Segel-Navier-Stokes system are global in time as long as the initial cell mass is sufficiently small (for example, $M<\frac49$) and $ A\left(\|u_{2,0}(0)\|_{L^{2}}+\|u_{3,0}(0)\|_{L^{2}} \right)\leq C_{0} $, which seems to be the first result of considering the suppression effect of Couette flow in the 3D Patlak-Keller-Segel-Navier-Stokes model, and also the first time considering the non-slip boundary condition.