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

Large time behaviour of a conservation law regularised by a Riesz-Feller operator: the sub-critical case