Communications in Mathematical Sciences

Volume 21 (2023)

Number 1

Dissipation enhancement for a degenerated parabolic equation

Pages: 173 – 193

DOI: https://dx.doi.org/10.4310/CMS.2023.v21.n1.a8

Authors

Yu Feng (Beijing International Center for Mathematical Research, Peking University, Beijing, China)

Bingyang Hu (Department of Mathematics, Purdue University, West Lafayette, Indiana, U.S.A.)

Xiaoqian Xu (Zu Chongzhi Center for Mathematics and Computational Sciences, Duke Kunshan University, Kunshan, China)

Abstract

In this paper, we quantitatively consider the enhanced-dissipation effect of the advection term to the parabolic $p$-Laplacian equations. More precisely, we show the mixing property of flow for the passive scalar enhances the dissipation process of the $p$-Laplacian in the sense of $L^2$ decay, that is, the $L^2$ decay can be arbitrarily fast. The main ingredient of our argument is to understand the underlying iteration structure inherited from the parabolic $p$-Laplacian equations. This extends the dissipation enhancement result of the advection diffusion equation by Yuanyuan Feng and Gautam Iyer to a non-linear setting.

Keywords

dissipation enhancement, mixing, degenerate diffusion

2010 Mathematics Subject Classification

35B27, 35B44, 35Q35, 76R05

Received 26 June 2021

Received revised 23 December 2021

Accepted 19 April 2022

Published 27 December 2022