Nonlinear enhanced dissipation in viscous Burgers-type equations

Abstract

We construct a class of infinite mass functions for which solutions of the viscous Burgers equation decay at a better rate than the solution of the heat equation for the same 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 enhances 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.