  We study the large time behaviour of the solutions of a non-local
regularisation of a scalar conservation law. This regularisation is given by a
fractional derivative of order $1+\alpha$, with $\alpha\in(0,1)$, which is a
Riesz-Feller operator. The non-linear flux is given by the locally Lipschitz
function $|u|^{q-1}u/q$ for $q>1$. We show that in the sub-critical case, $1<q
< 1 +\alpha$, the large time behaviour is governed by the unique entropy
solution of the scalar conservation law. Our proof adapts the proofs of the
analogous results for the local case (where the regularisation is the
Laplacian) and, more closely, the ones for the regularisation given by the
fractional Laplacian with order larger than one, see Ignat and Stan (2018). The
main difference is that our operator is not symmetric and its Fourier symbol is
not real. We can also adapt the proof and obtain similar results for general
Riesz-Feller operators.
