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Aggregation equations with fractional diffusion : preventing concentration by mixing
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Hopf, Katharina and Rodrigo Diez, José L. (2018) Aggregation equations with fractional diffusion : preventing concentration by mixing. Communications in Mathematical Sciences, 16 (2). pp. 333-361. doi:10.4310/CMS.2018.v16.n2.a2 ISSN 1539-6746.
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Official URL: http://dx.doi.org/10.4310/CMS.2018.v16.n2.a2
Abstract
We investigate a class of aggregation-diffusion equations with strongly singular kernels and weak (fractional) dissipation in the presence of an incompressible flow. Without the flow the equations are supercritical in the sense that the tendency to concentrate dominates the strength of diffusion and solutions emanating from sufficiently localised initial data may explode in finite time. The main purpose of this paper is to show that under suitable spectral conditions on the flow, which guarantee good mixing properties, for any regular initial datum the solution to the corresponding advection-aggregation-diffusion equation is global if the prescribed flow is sufficiently fast. This paper can be seen as a partial extension of [Kiselev & Xu, Arch. Rat. Mech. Anal., 222(2):1077-1112, 2016], and our arguments show in particular that the suppression mechanism for the classical 2D parabolic-elliptic Keller–Segel model devised by Kiselev and Xu also applies to the fractional Keller–Segel model (where Δ is replaced by −(−Δ)γ2) requiring only that γ>1. In addition, we remove the restriction to dimension d<4. As a by-product, a characterisation of the class of relaxation enhancing flows on the d-torus is extended to the case of fractional dissipation.
Item Type: | Journal Article | ||||||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||||
Library of Congress Subject Headings (LCSH): | Heat equation | ||||||||||||
Journal or Publication Title: | Communications in Mathematical Sciences | ||||||||||||
Publisher: | International Press | ||||||||||||
ISSN: | 1539-6746 | ||||||||||||
Official Date: | 14 May 2018 | ||||||||||||
Dates: |
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Volume: | 16 | ||||||||||||
Number: | 2 | ||||||||||||
Page Range: | pp. 333-361 | ||||||||||||
DOI: | 10.4310/CMS.2018.v16.n2.a2 | ||||||||||||
Status: | Peer Reviewed | ||||||||||||
Publication Status: | Published | ||||||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||||||
Date of first compliant deposit: | 20 October 2017 | ||||||||||||
Date of first compliant Open Access: | 15 May 2018 | ||||||||||||
RIOXX Funder/Project Grant: |
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