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arXiv · 1710.10315

Suppression of blow-up in Parabolic-Parabolic Patlak-Keller-Segel via strictly monotone shear flows

Siming He
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Abstract

In this paper we consider the parabolic-parabolic Patlak-Keller-Segel models in $\mathbb{T}\times\mathbb{R}$ 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 $8π$: solutions with mass less than $8π$ are global in time and there exist solutions with mass larger than $8 π$ which blow up in finite time \cite{Schweyer14}. 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 \cite{BedrossianHe16}, the strong shear flow has destabilizing effect in addition to the enhanced dissipation effect, which make the problem more difficult.

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Siming He. 2017-10-27. Suppression of blow-up in Parabolic-Parabolic Patlak-Keller-Segel via strictly monotone shear flows. https://doi.org/10.1088/1361-6544%2Faac1ce

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