Methods and Applications of Analysis
Volume 28 (2021)
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
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.
aggregation equation, mixing, fractional dissipation, suppression of blowup
2010 Mathematics Subject Classification
35A01, 35B45, 35Q92, 35R11
The authors’ research was supported by the National Natural Science Foundation of China 11831011, and by the Shanghai Science and Technology Innovation Action Plan (Grant No. 21JC1403600).
Received 19 September 2020
Accepted 25 June 2021
Published 10 June 2022