Asymptotic properties of entropy solutions to fractal Burgers equation

TL;DR

This paper investigates the long-term behavior of solutions to a fractional Burgers equation, revealing a transition in asymptotic profiles at the critical exponent alpha=1, with self-similar solutions emerging for alpha=1 and nonlinearity becoming negligible for alpha<1.

Contribution

The study demonstrates how the asymptotic profile of solutions changes at alpha=1, introducing self-similar solutions for alpha=1 and showing nonlinearity's insignificance for alpha<1.

Findings

For alpha=1, a self-similar solution describes the large time behavior.

For alpha in (0,1), nonlinearity is negligible in asymptotics.

The asymptotic profile shifts from rarefaction waves to self-similar solutions at alpha=1.

Abstract

We study properties of solutions of the initial value problem for the nonlinear and nonlocal equation u_t+(-\partial^2_x)^{\alpha/2} u+uu_x=0 with alpha in (0,1], supplemented with an initial datum approaching the constant states u+/u- (u_-smaller than u_+) as x goes to +/-infty, respectively. It was shown by Karch, Miao & Xu (SIAM J. Math. Anal. 39 (2008), 1536--1549) that, for alpha in (1,2), the large time asymptotics of solutions is described by rarefaction waves. The goal of this paper is to show that the asymptotic profile of solutions changes for alpha \leq 1. If alpha=1, there exists a self-similar solution to the equation which describes the large time asymptotics of other solutions. In the case alpha \in (0,1), we show that the nonlinearity of the equation is negligible in the large time asymptotic expansion of solutions.