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 $A^{ \frac{ 2}{ 3}}\|u_{ \rm in}\|_{ H^2}\leq C_0$, 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 $A^{ \frac{ 2}{ 3}}\|v_{ \rm in}-(Ay,0,Ay)\|_{ H^2}\leq C_0,$ which also shows the transition threshold for the shear flow $(Ay,0,Ay)$ in $\mathbb{ T}\times\mathbb{ R}\times\mathbb{ T}$.
Categories: math.AP
