A note on contractive semi-groups on a 1:1 junction for scalar conservation laws and Hamilton–Jacobi equations

Pierre Cardaliaguet

doi:10.5802/crmath.727

Research article - Partial differential equations

A note on contractive semi-groups on a 1:1 junction for scalar conservation laws and Hamilton–Jacobi equations

Pierre Cardaliaguet ¹

¹ Université Paris-Dauphine, PSL Research University, Ceremade, Place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 499-510

Abstract (Original language)
Résumé

We characterize the continuous semi-groups on $L^1(\mathbb{R})$ which coincide with a scalar conservation law $\rho _t+(f(\rho ))_x=0$ in $\mathbb{R}_+\times (\mathbb{R}\setminus \lbrace 0\rbrace )$ and are $L^1$-contracting. In a symmetric way, we characterize the continuous semi-groups on $W^{1,\infty }(\mathbb{R})$ which coincide with a Hamilton–Jacobi equation $u_t+H(u_x)=0$ in $\mathbb{R}_+\times (\mathbb{R}\setminus \lbrace 0\rbrace )$ and are $L^\infty $-contracting. These questions appear in traffic flow models for instance.

Nous caractérisons les semi-groupes continus sur $L^1(\mathbb{R})$ qui coïncident avec une loi de conservation scalaire $\rho _t+(f(\rho ))_x=0$ dans $\mathbb{R}_+\times (\mathbb{R}\setminus \lbrace 0\rbrace )$ et sont $L^1$-contractants. De façon symétrique, nous caractérisons les semi-groupes sur $W^{1,\infty }(\mathbb{R})$ qui coïncident avec l’équation Hamilton–Jacobi $u_t+H(u_x)=0$ dans $\mathbb{R}_+\times (\mathbb{R}\setminus \lbrace 0\rbrace )$ et sont $L^\infty $-contractants. Ces équations apparaissent notamment dans des modèles de trafic.

Received: 2024-11-21
Revised: 2025-01-31
Accepted: 2025-01-31
Published online: 2025-05-26

DOI: 10.5802/crmath.727

Keywords: Scalar conservation laws, traffic flow models, Hamilton–Jacobi equations, flux limiter, discontinuous fluxes
Mots-clés : Lois de conservation scalaire, modèles de trafic, équations de Hamilton–Jacobi, limiteur de flux, flux discontinus

Author's affiliations:

Pierre Cardaliaguet ¹

¹ Université Paris-Dauphine, PSL Research University, Ceremade, Place du Maréchal de Lattre de Tassigny, 75775 Paris cedex 16, France

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Pierre Cardaliaguet},
     title = {A note on contractive semi-groups on a 1:1 junction for scalar conservation laws and {Hamilton{\textendash}Jacobi} equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {499--510},
     year = {2025},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     doi = {10.5802/crmath.727},
     language = {en},
}

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Pierre Cardaliaguet. A note on contractive semi-groups on a 1:1 junction for scalar conservation laws and Hamilton–Jacobi equations. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 499-510. doi: 10.5802/crmath.727

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