An Oleinik inequality for a class of heterogeneous balance laws

Rida Harb; Stéphane Junca; Rida Harb; Stéphane Junca

doi:10.3934/nhm.2026002

Networks and Heterogeneous Media

2026, Volume 21, Issue 1: 22-54. doi: 10.3934/nhm.2026002

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Research article

An Oleinik inequality for a class of heterogeneous balance laws

Rida Harb ,
Stéphane Junca ^,

Université Côte d'Azur, LJAD, Parc Valrose, F-06108 Nice, France

Received: 05 November 2025 Revised: 28 December 2025 Accepted: 05 January 2026 Published: 13 January 2026

Although an Oleinik inequality is not expected for general heterogeneous scalar balance laws, there is a favorable case, when the source term $ g(x) $ is smooth and nondecreasing. In this setting, we are able to obtain an Oleinik-type inequality and, consequently, a smoothing effect.
- balance laws,
- entropy solution,
- strictly convex flux,
- Oleinik inequality,
- wave interaction,
- generalized Riemann problem,
- regularizing effect
Citation: Rida Harb, Stéphane Junca. An Oleinik inequality for a class of heterogeneous balance laws[J]. Networks and Heterogeneous Media, 2026, 21(1): 22-54. doi: 10.3934/nhm.2026002

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Abstract

Although an Oleinik inequality is not expected for general heterogeneous scalar balance laws, there is a favorable case, when the source term $ g(x) $ is smooth and nondecreasing. In this setting, we are able to obtain an Oleinik-type inequality and, consequently, a smoothing effect.

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