Nonlinear enhanced dissipation in viscous Burgers type equations II

TL;DR

This paper investigates the long-term behavior of solutions to the viscous Burgers equation, demonstrating enhanced dissipation effects and the existence of global attractors for certain initial data classes.

Contribution

It provides a detailed analysis of the decay rates and global attractors for viscous Burgers equations using the Hopf-Cole transformation, highlighting enhanced dissipation phenomena.

Findings

Enhanced decay rates surpass heat equation benchmarks

Existence of global attractors for specific initial data classes

Long-time behavior characterized for infinite mass initial data

Abstract

In this follow up paper, we focus on the viscous Burgers equation. There, using the Hopf-Cole transformation, we compute the long time behavior of solutions for some classes of infinite mass initial datas. We show that an enhanced dissipation effect occurs generically, that is the decay rate in time is better than if we considered instead the heat equations for the same inital value. We also show the existence of a kind of global attractor per class.