A dispersive estimate for the multidimensional Burgers equation

TL;DR

This paper proves that solutions to the multidimensional Burgers equation exhibit polynomial decay in their maximum norm over time, with decay rate depending solely on the initial data's $L^1$ norm.

Contribution

It establishes a new dispersive estimate for entropy solutions of the multidimensional Burgers equation, demonstrating decay in the $L^ abla$ norm over time.

Findings

Polynomial decay of $L^ abla$ norm of solutions

Decay rate depends only on initial $L^1$ norm

Results apply to entropy solutions of the multidimensional Burgers equation

Abstract

We study the multi-dimensional Burgers equation $u_t + u u_{x_1} + \dots + u^d u_{x_d} = 0$. We prove that the $L^\infty$ norm of entropy solutions of this equation decays polynomially as $t \to \infty$ in terms of the $L^1$ norm of the initial data only.