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

Carlota Maria Cuesta; Xuban Diez-Izagirre

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 4, page 1057-1080
  • ISSN: 0011-4642

Abstract

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We study the large time behaviour of the solutions of a nonlocal regularisation of a scalar conservation law. This regularisation is given by a fractional derivative of order 1 + α , with α ∈ ( 0 , 1 ) , which is a Riesz-Feller operator. The nonlinear 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 + α , 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 L. I. Ignat and D. 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.

How to cite

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Cuesta, Carlota Maria, and Diez-Izagirre, Xuban. "Large time behaviour of a conservation law regularised by a Riesz-Feller operator: the sub-critical case." Czechoslovak Mathematical Journal 73.4 (2023): 1057-1080. <http://eudml.org/doc/299355>.

@article{Cuesta2023,
abstract = {We study the large time behaviour of the solutions of a nonlocal 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 nonlinear 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 L. I. Ignat and D. 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.},
author = {Cuesta, Carlota Maria, Diez-Izagirre, Xuban},
journal = {Czechoslovak Mathematical Journal},
keywords = {large time asymptotic; regularisation of conservation law; Riesz-Feller operator},
language = {eng},
number = {4},
pages = {1057-1080},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Large time behaviour of a conservation law regularised by a Riesz-Feller operator: the sub-critical case},
url = {http://eudml.org/doc/299355},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Cuesta, Carlota Maria
AU - Diez-Izagirre, Xuban
TI - Large time behaviour of a conservation law regularised by a Riesz-Feller operator: the sub-critical case
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 4
SP - 1057
EP - 1080
AB - We study the large time behaviour of the solutions of a nonlocal 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 nonlinear 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 L. I. Ignat and D. 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.
LA - eng
KW - large time asymptotic; regularisation of conservation law; Riesz-Feller operator
UR - http://eudml.org/doc/299355
ER -

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