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

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.