Communications in Mathematical Sciences
Volume 21 (2023)
Dissipation enhancement for a degenerated parabolic equation
Pages: 173 – 193
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.
dissipation enhancement, mixing, degenerate diffusion
2010 Mathematics Subject Classification
35B27, 35B44, 35Q35, 76R05
Xiaoqian Xu is partially supported by the National Natural Science Foundation of China grant 12101278.
Received 26 June 2021
Received revised 23 December 2021
Accepted 19 April 2022
Published 27 December 2022