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

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 $A(\mathrm{\parallel }{u}_{2,0}(0){\mathrm{\parallel }}_{L^2}+\mathrm{\parallel }{u}_{3,0}(0){\mathrm{\parallel }}_{L^2})\le C_0$, 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.