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

TL;DR

This paper investigates the long-term behavior of solutions to a non-local regularized scalar conservation law involving a Riesz-Feller operator, demonstrating that in the sub-critical case, solutions tend toward the entropy solution of the original law.

Contribution

It extends existing results on fractional regularizations by analyzing the non-symmetric Riesz-Feller operator and establishing large time behavior in the sub-critical regime.

Findings

Solutions converge to the entropy solution in the sub-critical case

The analysis applies to general Riesz-Feller operators

The proof adapts methods from local and fractional Laplacian cases

Abstract

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.