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Dissipation enhancement for a degenerated parabolic equation

Yu Feng, Bingyang Hu, Xiaoqian Xu

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.

Dissipation enhancement for a degenerated parabolic equation

Abstract

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

Paper Structure

This paper contains 6 sections, 10 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Preliminaries
  4. Dissipation enhancement for evolution $p$-Laplacian advection equations
  5. The Strongly Mixing Case.
  6. The Weakly Mixing Case.

Key Result

theorem 1

Let $0<\alpha \le 1$, $\beta>0$, $p>2$, $\theta_{0, 0} \in L^2_0(\mathbb{T}^d)$ and $h: [0, \infty) \to (0, \infty)$ be a strictly decreasing function that vanishes at infinity. If $u$ is strongly $\alpha, \beta$ mixing with rate function $h$, then where $C$ is an absolute constant that only depends on $h$, $\|\nabla u\|_\infty$, $p$, $\|\theta_{0, 0}\|_2$, the strongly mixing condition, any dime

Theorems & Definitions (29)

  • remark 1
  • definition 1
  • remark 2
  • definition 2
  • remark 3
  • theorem 1
  • corollary 1
  • remark 4
  • theorem 2
  • corollary 2
  • ...and 19 more