# 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.

## Key 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.