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In this paper, we consider the Cauchy problem for the three dimensionalparabolic-elliptic Keller-Segel equation with a large non-shear incompressibleflow. Without advection, there exist solution with arbitrarily mass which blowup in finite time. Firstly, we introduce a three dimensional non-shearincompressible flow and study the enhanced dissipation of such flows byresolvent estimate method. Next, we show that the enhanced dissipation of suchflow can suppress blow-up of solution to three dimensional parabolic-ellipticKeller-Segel equation and establish global classical solution with largeinitial data.

Enhanced dissipation and blow-up suppression for the three dimensional Keller-Segel equation with a non-shear incompressible flow