Suppression of blow-up for the 3D Patlak-Keller-Segel-Navier-Stokes system via the Couette flow

TL;DR

This paper demonstrates that strong Couette flow can prevent finite-time blow-up in the 3D Patlak-Keller-Segel-Navier-Stokes system for small initial cell mass, ensuring global solutions.

Contribution

It proves for the first time that the stabilizing effect of moving Navier-Stokes flows can prevent blow-up in the system with sub-critical initial mass.

Findings

Couette flow stabilizes the system when sufficiently strong

Global existence of solutions under certain initial conditions

Blow-up can be suppressed below a critical mass threshold

Abstract

As is well known, for the 3D Patlak-Keller-Segel system, regardless of whether they are parabolic-elliptic or parabolic-parabolic forms, finite-time blow-up may occur for arbitrarily small values of the initial mass. In this paper, it is proved for the first time that one can prevent the finite-time blow-up when the initial mass is less than a certain critical threshold via the stabilizing effect of the moving Navier-Stokes flows. In details, we investigate the nonlinear stability of the Couette flow $(Ay, 0, 0)$ in the Patlak-Keller-Segel-Navier-Stokes system and show that if the Couette flow is sufficiently strong (A is large enough), then the solutions for Patlak-Keller-Segel-Navier-Stokes system are global in time provided that the initial velocity is sufficiently small and the initial cell mass is less than $\frac{24}{5} \pi^2$.