Suppression of blow-up in parabolic-parabolic Patlak-Keller-Segel via strictly monotone shear flows
© 2018 IOP Publishing Ltd & London Mathematical Society Printed in the UK. In this paper we consider the parabolic-parabolic Patlak-Keller-Segel models in T × ℝ with advection by a large strictly monotone 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 in time and there exist solutions with mass larger than 8π which blow up in finite time (Schweyer 2014 (arXiv:1403.4975)). 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 contrast with the parabolic-elliptic case (Bedrossian and He 2016 SIAM J. Math. Anal. 49 4722-66), the strong shear flow has destabilizing effect in addition to the enhanced dissipation effect, which makes the problem more difficult.
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