Comptes Rendus
no. 11-12
Partial differential equations
Scalar conservation laws: Initial and boundary value problems revisited and saturated solutions

Presented by: Pierre-Louis Lions

Pierre-Louis Lions1, 2; Panagiotis Souganidis3
1 Collège de France, 11, place Marcelin-Berthelot, 75005 Paris, France
2 CEREMADE, Université Paris-Dauphine, place du Maréchal-de-Lattre-de-Tassigny, 75016 Paris, France
3 Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA
Comptes Rendus. Mathématique, Volume 356 (2018) no. 11-12, pp. 1167-1178.

We revisit the classical theory of multidimensional scalar conservation laws. We reformulate the notion of the classical Kruzkov entropy solutions and study some new properties as well as the well-posedness of the initial value problem with inhomogeneous fluxes and general initial data. We also consider Dirichlet boundary value problems. We put forward a new and transparent definition for solutions and give a simple proof for their well-posedness in domains with smooth boundaries. Finally, we introduce the notion of saturated solutions and show that it is well-posed.

Nous revenons sur la théorie classique des lois de conservation scalaires multidimensionnelles. Nous introduisons une notion nouvelle de sous- et de sur-solutions, qui est équivalente à la notion classique de solutions entropiques à la Kruzkov. Nous utilisons ensuite cette notion pour établir des propriétés nouvelles de ces solutions. Nous proposons également une formulation nouvelle et claire des solutions pour les problèmes aux limites et nous donnons une preuve simple du caractère bien posé de ces problèmes. Enfin, nous introduisons la notion de solutions saturées et montrons que de tels problèmes sont bien posés.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.09.010

Pierre-Louis Lions 1, 2; Panagiotis Souganidis 3

1 Collège de France, 11, place Marcelin-Berthelot, 75005 Paris, France
2 CEREMADE, Université Paris-Dauphine, place du Maréchal-de-Lattre-de-Tassigny, 75016 Paris, France
3 Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA 
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Pierre-Louis Lions; Panagiotis Souganidis. Scalar conservation laws: Initial and boundary value problems revisited and saturated solutions. Comptes Rendus. Mathématique, Volume 356 (2018) no. 11-12, pp. 1167-1178. doi : 10.1016/j.crma.2018.09.010. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.09.010/

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☆ Partially supported by the National Science Foundation grants DMS-1600129 and the Office of Naval Research grant N000141712095.

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