Numerical Analysis
Small viscosity solution of linear scalar 1-D conservation laws with one discontinuity of the coefficient
Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 681-686.

While linear conservations laws have a classical well-defined solution for sufficiently regular coefficients, it is not the case when the coefficients are, for instance, discontinuous across a fixed hypersurface. In this case, another approach has to be proposed in order to answer the double concern of existence and uniqueness of a solution to the problem. We will focus mainly on showing such concerns can be solved by means of a small viscosity approach in 1-D scalar frameworks, in particular for expansive discontinuities of the coefficient. The obtained small viscosity solution is also the solution in the sense Bouchut and James or LeFloch for scalar equations.

Pour des coefficients suffisamment réguliers, les lois de conservations linéaires ont un sens classique bien établi. Cela cesse cependant d'être le cas lorsque les coefficients sont par exemple discontinus au travers d'une hypersurface fixée. Dans ce cas de figure, une autre approche doit être proposée pour répondre à la double préoccupation de l'existence et de l'unicité d'une solution au problème. Notre but va être principalement de montrer que, dans des cas scalaires 1-D, une approche à viscosité évanescente permet de répondre à ces préoccupations, en particulier dans le cas d'une discontinuité expansive du coefficient.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2008.03.029
Fornet, Bruno 1, 2

1 LATP, Université de Provence, 39, rue Joliot-Curie, 13453 Marseille cedex 13, France
2 LMRS, Université de Rouen, avenue de l'Université, BP 12, 76801 Saint Étienne du Rouvray, France
@article{CRMATH_2008__346_11-12_681_0,
     author = {Fornet, Bruno},
     title = {Small viscosity solution of linear scalar {1-D} conservation laws with one discontinuity of the coefficient},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {681--686},
     publisher = {Elsevier},
     volume = {346},
     number = {11-12},
     year = {2008},
     doi = {10.1016/j.crma.2008.03.029},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2008.03.029/}
}
TY  - JOUR
AU  - Fornet, Bruno
TI  - Small viscosity solution of linear scalar 1-D conservation laws with one discontinuity of the coefficient
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 681
EP  - 686
VL  - 346
IS  - 11-12
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2008.03.029/
DO  - 10.1016/j.crma.2008.03.029
LA  - en
ID  - CRMATH_2008__346_11-12_681_0
ER  - 
%0 Journal Article
%A Fornet, Bruno
%T Small viscosity solution of linear scalar 1-D conservation laws with one discontinuity of the coefficient
%J Comptes Rendus. Mathématique
%D 2008
%P 681-686
%V 346
%N 11-12
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2008.03.029/
%R 10.1016/j.crma.2008.03.029
%G en
%F CRMATH_2008__346_11-12_681_0
Fornet, Bruno. Small viscosity solution of linear scalar 1-D conservation laws with one discontinuity of the coefficient. Comptes Rendus. Mathématique, Volume 346 (2008) no. 11-12, pp. 681-686. doi : 10.1016/j.crma.2008.03.029. http://www.numdam.org/articles/10.1016/j.crma.2008.03.029/

[1] Bouchut, F.; James, F. One-dimensional transport equations with discontinuous coefficients, Nonlinear. Anal., Volume 32 (1998), pp. 891-933

[2] Bouchut, F.; James, F.; Mancini, S. Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), Volume IV (2005), pp. 1-25

[3] Dal Maso, G.; LeFloch, P.G.; Murat, F. Definition and weak stability of nonconservative products, J. Math. Pures Appl. (9), Volume 74 (1995) no. 6, pp. 483-548

[4] B. Fornet, The Cauchy problem for 1-D linear nonconservative hyperbolic systems with possibly expansive discontinuity of the coefficient: a viscous approach, J. Differential Equations, , in press | DOI

[5] B. Fornet, Viscous approach for linear hyperbolic systems with discontinuous coefficients, preprint, 2007

[6] B. Fornet, Small viscosity solution of linear scalar hyperbolic problems with discontinuous coefficients in several space dimensions, preprint, 2008

[7] Guès, O. Perturbations visqueuses de problèmes mixtes hyperboliques et couches limites, Ann. Inst. Fourier (Grenoble), Volume 45 (1995) no. 4, pp. 973-1006

[8] Guès, O.; Métivier, G.; Williams, M.; Zumbrun, K. Existence and stability of multidimensional shock fronts in the vanishing viscosity limit, Arch. Rat. Mech. Anal., Volume 175 (2004), pp. 151-244

[9] LeFloch, P.G. An existence and uniqueness result for two nonstrictly hyperbolic systems, Nonlinear Evolution Equations that Change Type, IMA Vol. Math. Appl., vol. 27, Springer, New York, 1990, pp. 126-138

[10] Métivier, G. Small Viscosity and Boundary Layer Methods: Theory, Stability Analysis, and Applications, Birkhäuser, 2003

[11] Poupaud, F.; Rascle, M. Measure solutions to the linear multidimensional transport equation with discontinuous coefficients, Comm. Differential Equations, Volume 22 (1997), pp. 337-358

[12] Sueur, F. Viscous approach of discontinuous solutions of semilinear symmetric hyperbolic systems, Ann. Inst. Fourier, Volume 56 (2006) no. 1, pp. 183-245

Cited by Sources: