Suppression of Blow-Up in Patlak--Keller--Segel Via Shear Flows

In this paper we consider the parabolic-elliptic Patlak--Keller--Segel models in Tᵈ with d=2,3 with the additional effect of advection by a large shear flow. Without the shear flow, the model is L¹ critical in two dimensions with critical mass 8π; solutions with mass less than 8π are global and solutions with mass larger than 8 π with finite second moment all blow up in finite time. In three dimensions, the model is L3/2 critical and L¹ supercritical; there exist solutions with arbitrarily small mass which blow up in finite time arbitrarily fast. We show that the additional shear flow, if it is chosen sufficiently large, suppresses one dimension of the dynamics and hence can suppress blow-up. In two dimensions, the problem becomes effectively L¹ subcritical and so all solutions are global in time (if the shear flow is chosen large). In three dimensions, the problem is effectively L¹ critical, and solutions with mass less than 8π are global in time (and for all mass larger than 8π, there exists solutions which blow up in finite time).