Contents Online
Communications in Mathematical Sciences
Volume 16 (2018)
Number 2
Aggregation equations with fractional diffusion: Preventing concentration by mixing
Pages: 333 – 361
DOI: https://dx.doi.org/10.4310/CMS.2018.v16.n2.a2
Authors
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
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