Comptes Rendus
no. 11-12
Partial differential equations
Scalar conservation laws: Initial and boundary value problems revisited and saturated solutions
[Lois de conservation scalaires : retour sur les problèmes aux limites et solutions saturées]

Présenté par : 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.

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.

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.

Reçu le :
Accepté le :
Publié le :
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 
@article{CRMATH_2018__356_11-12_1167_0,
     author = {Pierre-Louis Lions and Panagiotis Souganidis},
     title = {Scalar conservation laws: {Initial} and boundary value problems revisited and saturated solutions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1167--1178},
     publisher = {Elsevier},
     volume = {356},
     number = {11-12},
     year = {2018},
     doi = {10.1016/j.crma.2018.09.010},
     language = {en},
}
TY  - JOUR
AU  - Pierre-Louis Lions
AU  - Panagiotis Souganidis
TI  - Scalar conservation laws: Initial and boundary value problems revisited and saturated solutions
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 1167
EP  - 1178
VL  - 356
IS  - 11-12
PB  - Elsevier
DO  - 10.1016/j.crma.2018.09.010
LA  - en
ID  - CRMATH_2018__356_11-12_1167_0
ER  - 
%0 Journal Article
%A Pierre-Louis Lions
%A Panagiotis Souganidis
%T Scalar conservation laws: Initial and boundary value problems revisited and saturated solutions
%J Comptes Rendus. Mathématique
%D 2018
%P 1167-1178
%V 356
%N 11-12
%I Elsevier
%R 10.1016/j.crma.2018.09.010
%G en
%F CRMATH_2018__356_11-12_1167_0
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/

[1] K. Ammar; P. Wittbold; J. Carrillo Scalar conservation laws with general boundary condition and continuous flux function, J. Differ. Equ., Volume 228 (2006) no. 1, pp. 111-139

[2] B. Andreianov; K. Sbihi Well-posedness of general boundary-value problems for scalar conservation laws, Trans. Amer. Math. Soc., Volume 367 (2015) no. 6, pp. 3763-3806

[3] C. Bardos; A.Y. le Roux; J.-C. Nédélec First order quasilinear equations with boundary conditions, Commun. Partial Differ. Equ., Volume 4 (1979) no. 9, pp. 1017-1034

[4] F. Dubois; P. LeFloch Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. Differ. Equ., Volume 71 (1988) no. 1, pp. 93-122

[5] S.N. Kruzkov First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), Volume 81 (1970) no. 123, pp. 228-255

[6] P.-L. Lions, P.E. Souganidis, Scalar conservation laws revisited, in preparation.

[7] F. Otto Initial-boundary value problem for a scalar conservation law, C. R. Acad. Sci. Paris, Ser. I, Volume 322 (1996) no. 8, pp. 729-734

[8] F. Otto L1-contraction and uniqueness for quasilinear elliptic–parabolic equations, J. Differ. Equ., Volume 131 (1996) no. 1, pp. 20-38

[9] E.Yu. Panov Existence of strong traces for generalized solutions of multidimensional scalar conservation laws, J. Hyperbolic Differ. Equ., Volume 2 (2005) no. 4, pp. 885-908

[10] A. Porretta; J. Vovelle L1 solutions to first order hyperbolic equations in bounded domains, Commun. Partial Differ. Equ., Volume 28 (2003) no. 1–2, pp. 381-408

[11] L. Silvestre Oscillation properties of scalar conservation laws, Aug 2017 http://adsabs.harvard.edu/abs/2017arXiv170803401S (arXiv e-print Provided by the SAO/NASA Astrophysics Data System) | arXiv

[12] A. Vasseur Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal., Volume 160 (2001) no. 3, pp. 181-193

Cité par Sources :

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

Commentaires - Politique