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, $M<\frac{4}{9}$) and $A\left(\Vert u_{2,0}(0){\Vert }_{L^2}+\Vert u_{3,0}(0){\Vert }_{L^2}\right)\le C_0$, 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.

arXiv commentarXiv admin note: substantial text overlap with arXiv:2401.15982