Communications in Mathematical Sciences

Volume 16 (2018)

Number 2

Aggregation equations with fractional diffusion: Preventing concentration by mixing

Pages: 333 – 361

DOI: http://dx.doi.org/10.4310/CMS.2018.v16.n2.a2

Authors

Katharina Hopf (Mathematics Institute, University of Warwick, Coventry, United Kingdom)

José L. Rodrigo (Mathematics Institute, University of Warwick, Coventry, United Kingdom)

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 $\Delta$ is replaced by $-{(-\Delta)}^{\frac{\gamma}{2}})$ requiring only that $\gamma \gt 1$. In addition, we remove the restriction to dimension $d \lt 4$. 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.

Keywords

preventing blowup, Keller–Segel, transport-diffusion, mixing, fractional dissipation

Full Text (PDF format)

K. Hopf is supported by MASDOC DTC at the University of Warwick, which is funded by the Engineering and Physical Sciences Research Council grant EP/HO23364/1. J. L. Rodrigo is partially supported by the European Research Council grant 616797.

Received 6 May 2017

Accepted 24 August 2017

Published 14 May 2018