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

Article de recherche - Équations aux dérivées partielles

A note on contractive semi-groups on a 1:1 junction for scalar conservation laws and Hamilton–Jacobi equations
[À propos des semi-groupes contractants sur des jonctions 1 :1 pour des lois de conservation scalaires et des équations de Hamilton–Jacobi]

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 (VO)
Résumé

We characterize the continuous semi-groups on $L^{1} (R)$ which coincide with a scalar conservation law $ρ_{t} + (f (ρ))_{x} = 0$ in $R_{+} \times (R ∖ {0})$ and are $L^{1}$ -contracting. In a symmetric way, we characterize the continuous semi-groups on $W^{1, \infty} (R)$ which coincide with a Hamilton–Jacobi equation $u_{t} + H (u_{x}) = 0$ in $R_{+} \times (R ∖ {0})$ and are $L^{\infty}$ -contracting. These questions appear in traffic flow models for instance.

Nous caractérisons les semi-groupes continus sur $L^{1} (R)$ qui coïncident avec une loi de conservation scalaire $ρ_{t} + (f (ρ))_{x} = 0$ dans $R_{+} \times (R ∖ {0})$ et sont $L^{1}$ -contractants. De façon symétrique, nous caractérisons les semi-groupes sur $W^{1, \infty} (R)$ qui coïncident avec l’équation Hamilton–Jacobi $u_{t} + H (u_{x}) = 0$ dans $R_{+} \times (R ∖ {0})$ et sont $L^{\infty}$ -contractants. Ces équations apparaissent notamment dans des modèles de trafic.

Reçu le : 2024-11-21
Révisé le : 2025-01-31
Accepté le : 2025-01-31
Publié le : 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

Affiliations des auteurs :

Pierre Cardaliaguet ¹

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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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},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.727/

Bibliographie
Cité par

[1] A. Adimurthi; Rajib Dutta; Shyam Sundar Ghoshal; G. D. Veerappa Gowda Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux, Commun. Pure Appl. Math., Volume 64 (2011) no. 1, pp. 84-115 | DOI | MR | Zbl

[2] Boris Andreianov; Kenneth Hvistendahl Karlsen; Nils Henrik Risebro A theory of $L^{1}$ -dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., Volume 201 (2011) no. 1, pp. 27-86 | DOI | MR | Zbl

[3] Emmanuel Audusse; Benoît Perthame Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies, Proc. R. Soc. Edinb., Sect. A, Math., Volume 135 (2005) no. 2, pp. 253-265 | DOI | MR | Zbl

[4] Paolo Baiti; Helge Kristian Jenssen Well-posedness for a class of $2 \times 2$ conservation laws with $L^{\infty}$ data, J. Differ. Equations, Volume 140 (1997) no. 1, pp. 161-185 | DOI | MR | Zbl

[5] Raimund Bürger; Kenneth Hvistendahl Karlsen; John D. Towers An Engquist–Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections, SIAM J. Numer. Anal., Volume 47 (2009) no. 3, pp. 1684-1712 | DOI | MR | Zbl

[6] Pierre Cardaliaguet; Nicolas Forcadel; Theo Girard; Régis Monneau Conservation law and Hamilton–Jacobi equations on a junction: the convex case (2024) | HAL

[7] Pierre Cardaliaguet; Nicolas Forcadel; Régis Monneau A class of germs arising from homogenization in traffic flow on junctions (2023) (To appear in J. Hyperbolic Differ. Equ.) | arXiv

[8] Michael G. Crandall; Luc Tartar Some relations between nonexpansive and order preserving mappings, Proc. Am. Math. Soc., Volume 78 (1980) no. 3, pp. 385-390 | DOI | MR | Zbl

[9] Nicolas Forcadel; Cyril Imbert; Régis Monneau Germs for scalar conservation laws: the Hamilton–Jacobi equation point of view (2024) | arXiv

[10] Mauro Garavello; Roberto Natalini; Benedetto Piccoli; Andrea Terracina Conservation laws with discontinuous flux, Netw. Heterog. Media, Volume 2 (2007) no. 1, pp. 159-179 | DOI | MR | Zbl

[11] Cyril Imbert; Régis Monneau Flux-limited solutions for quasi-convex Hamilton–Jacobi equations on networks, Ann. Sci. Éc. Norm. Supér. (4), Volume 50 (2017) no. 2, pp. 357-448 | DOI | Numdam | MR | Zbl

[12] Régis Monneau Strictly convex Hamilton–Jacobi equations: strong trace of the gradient (2023) | HAL

[13] Evgueni Yu. Panov Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyperbolic Differ. Equ., Volume 4 (2007) no. 4, pp. 729-770 | DOI | MR | Zbl

[14] John D. Towers Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., Volume 38 (2000) no. 2, pp. 681-698 | DOI | MR | Zbl

[15] John D. Towers A difference scheme for conservation laws with a discontinuous flux: the nonconvex case, SIAM J. Numer. Anal., Volume 39 (2001) no. 4, pp. 1197-1218 | DOI | MR | Zbl

[16] Alexis Vasseur Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., Volume 160 (2001) no. 3, pp. 181-193 | DOI | MR | Zbl

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