Nonlinear enhanced dissipation in viscous Burgers type equations

TL;DR

This paper demonstrates that solutions to the viscous Burgers equation can decay faster than the heat equation for certain initial data, due to nonlinear transport effects, with detailed asymptotic profiles and stability analysis.

Contribution

It introduces a new class of initial data leading to enhanced dissipation and constructs a time-dependent boundary layer profile for the viscous Burgers equation.

Findings

Solutions decay faster than heat equation for specific initial data

Constructed a stable, time-dependent boundary layer profile

Extended results to other convection-diffusion equations

Abstract

We construct a class of infinite mass functions for which solutions of the viscous Burgers equation decay at a better rate than solution of the heat equation for initial data in this class. In other words, we show an enhanced dissipation coming from a nonlinear transport term. We compute the asymptotic profile in this class for both equations. For the viscous Burgers equation, the main novelty is the construction and description of a time dependent profile with a boundary layer, which enhanced the dissipation. This profile will be stable up to a computable nonlinear correction depending on the perturbation. We also extend our results to other convection-diffusion equations.