Non-uniqueness of weak solutions for the fractal Burgers equation

TL;DR

This paper demonstrates that for the fractional Burgers equation with diffusion order less than one, the uniqueness of weak solutions can fail, highlighting limitations of existing entropy solution frameworks.

Contribution

It shows that in the case of fractional diffusion less than order one, the uniqueness of entropy solutions does not necessarily hold, extending the understanding of solution behavior.

Findings

Weak solutions may not be unique for fractional diffusion order less than one.

Entropy solutions' uniqueness can fail in the fractional Burgers equation.

The paper clarifies limitations of entropy solution frameworks for certain fractional PDEs.

Abstract

The notion of Kruzhkov entropy solution was extended by the first author in 2007 to conservation laws with a fractional laplacian diffusion term; this notion led to well-posedness for the Cauchy problem in the $L^\infty$-framework. In the present paper, we further motivate the introduction of entropy solutions, showing that in the case of fractional diffusion of order strictly less than one, uniqueness of a weak solution may fail.