Suppression of blow-up in parabolic–parabolic Patlak–Keller–Segel via strictly monotone shear flows. (2nd July 2018)

• Record Type: Journal Article
• Authors: He, Siming
• Is Part Of: Nonlinearity. Volume 31:Number 8(2018:Aug.)
• Page Start: 3651
• Page End: 3688
• Publication Date: 2018-07-02
• Subjects: parabolic–parabolic Patlak–Keller Segel equation -- shear flow enhanced dissipation -- shear flow destabilizing effect; 35 -- 92; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aac1ce
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: In this paper we consider the parabolic–parabolic Patlak–Keller–Segel models in with advection by a large strictly monotone shear flow. Without the shear flow, the model is L 1 critical in two dimensions with critical mass : solutions with mass less than are global in time and there exist solutions with mass larger than 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.