Suppression of blow-up in the 3D Patlak-Keller-Segel-Navier-Stokes system via non-parallel shear flows

In this paper, we consider the three-dimensional Patlak-Keller-Segel system coupled with the Navier-Stokes equations near the non-parallel shear flow $(Ay,0,Ay)$ in $\mathbb{T}\times \mathbb{R}\times \mathbb{T}$. 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 without any smallness restriction on the initial cell mass as long as the initial velocity satisfies $A^{\frac{2}{3}}\mathrm{\parallel }{u}_{\mathrm{i}\mathrm{n}}{\mathrm{\parallel }}_{H^2}\le C_0$, which seems to be the first result of studying the suppression effect of shear flows for the 3D Patlak-Keller-Segel-Navier-Stokes system. Moreover, it implies that the solutions of the 3D Navier-Stokes equations are global in time if the initial velocity satisfies $A^{\frac{2}{3}}\mathrm{\parallel }{v}_{\mathrm{i}\mathrm{n}}-(Ay,0,Ay){\mathrm{\parallel }}_{H^2}\le C_0,$ which also shows the transition threshold for the shear flow $(Ay,0,Ay)$ in $\mathbb{T}\times \mathbb{R}\times \mathbb{T}$.