Methods and Applications of Analysis

Volume 28 (2021)

Number 4

Special issue dedicated to Professor Ling Hsiao on the occasion of her 80th birthday, Part III

Guest editors: Qiangchang Ju (Institute of Applied Physics and Computational Mathematics, Beijing), Hailiang Li (Capital Normal University, Beijing), Tao Luo (City University of Hong Kong), and Zhouping Xin (Chinese University of Hong Kong)

Suppression of blowup by mixing in an aggregation equation with supercritical dissipation

Pages: 547 – 574

DOI: https://dx.doi.org/10.4310/MAA.2021.v28.n4.a7

Authors

Binbin Shi (School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai, China)

Weike Wang (School of Mathematical Science, CMA-Shanghai and Institute of Natural Science, Shanghai Jiao Tong University, Shanghai, China)

Abstract

In this paper, we consider the Cauchy problem for an aggregation equation with supercritical dissipation and an additional mixing mechanism of advection by an incompressible flow, where the attractive kernel is a non-negative radial decreasing kernel with a Lipschitz point at the origin. Without advection, the solution of equation blows up in finite time. Under a suitable mixing condition on the advection, we show the global existence of the solution with large initial data. Firstly, we study enhanced dissipation effect of mixing mechanism of advection by a linear equation with fractional dissipation. The main idea of proof is based on the Gearhart–Prüss type theorem. Next, we establish the $L^\infty$-criterion of solution and obtain the global $L^\infty$ estimate. We give a new proof, which is based on a new observation for mixing mechanism and the RAGE theorem. Finally, the nonlinear maximum principle on tours is applied to get the $L^\infty$ estimate of solution.

Keywords

aggregation equation, mixing, fractional dissipation, suppression of blowup

2010 Mathematics Subject Classification

35A01, 35B45, 35Q92, 35R11

Received 19 September 2020

Accepted 25 June 2021

Published 10 June 2022