Dissipation enhancement for a degenerated parabolic equation

TL;DR

This paper demonstrates that flow-induced mixing significantly accelerates the dissipation in nonlinear parabolic p-Laplacian equations, extending previous linear results to a nonlinear context with arbitrarily fast decay.

Contribution

It extends the dissipation enhancement results from linear advection-diffusion equations to nonlinear p-Laplacian equations, revealing the impact of flow on nonlinear dissipation rates.

Findings

Flow mixing enhances dissipation in p-Laplacian equations.

L^2 decay can be made arbitrarily fast due to flow effects.

The iteration structure of the equations underpins the dissipation enhancement.

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 into a non-linear setting.