[Solution à petite viscosité de lois de conservations linéaires scalaires 1-D avec un coefficient discontinu]
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.
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.
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@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, Tome 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/
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