Young Researchers Workshop: Current trends in kinetic theory

Suppression of blow-up in chemotaxis through fluid flow

Siming He

Duke University

Abstract:

We study the parabolic-elliptic Patlak-Keller-Segel models in $T^d$, with $d=2,3$ with the additional effect of advection by a large shear flow. Without the shear flow, the model is $L^1$ critical in two dimensions with critical mass 8 ? ; solutions with mass less than 8 ? are global and solutions with mass larger than 8 ? with finite second moment, all blow up in finite time. In three dimensions, the model is $L^1$ supercritical; there exists 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. Joint work with Jacob Bedrossian.