  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.
