Hyperbolic systems and vanishing viscosity

Frédéric Rousset

Séminaire Bourbaki (2002-2003)

  • Volume: 45, page 231-250
  • ISSN: 0303-1179

Abstract

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In this talk we will present the works of S. Bianchini and A. Bressan on the Cauchy problem for viscous perturbations ∂ t u ε + ∂ x f ( u ε ) = ε ∂ x x u ε of one-dimensional strictly hyperbolic systems ∂ t u + ∂ x f ( u ) = 0 . They have shown global existence ( t ≥ 0 ), uniqueness and stability and they have justified the limit when ε goes to zero for initial data with small total variation. Their analysis also shows that the solutions of the hyperbolic system obtained by this method coincide with the solutions obtained by other types of approximations.

How to cite

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Rousset, Frédéric. "Systèmes hyperboliques et viscosité évanescente." Séminaire Bourbaki 45 (2002-2003): 231-250. <http://eudml.org/doc/252131>.

@article{Rousset2002-2003,
abstract = {Le but de l’exposé est de présenter les résultats obtenus par S. Bianchini et A. Bressan sur le problème de Cauchy pour des perturbations visqueuses $ \partial _t u^\varepsilon + \partial _x f(u^\varepsilon ) = \varepsilon \partial _\{xx\} u^\varepsilon $ de systèmes strictement hyperboliques $\partial _t u + \partial _x f(u) =0$ en une dimension d’espace. Ils ont en particulier montré l’existence globale ($t\ge 0$), l’unicité et la stabilité des solutions et justifié la convergence quand $\varepsilon $ tend vers zéro pour des données initiales à petite variation totale. Leur analyse montre aussi que les solutions du système hyperbolique ainsi obtenues coïncident avec les solutions provenant d’autres types d’approximations.},
author = {Rousset, Frédéric},
journal = {Séminaire Bourbaki},
keywords = {hyperbolic systems; vanishing viscosity method},
language = {fre},
pages = {231-250},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {Systèmes hyperboliques et viscosité évanescente},
url = {http://eudml.org/doc/252131},
volume = {45},
year = {2002-2003},
}

TY - JOUR
AU - Rousset, Frédéric
TI - Systèmes hyperboliques et viscosité évanescente
JO - Séminaire Bourbaki
PY - 2002-2003
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 45
SP - 231
EP - 250
AB - Le but de l’exposé est de présenter les résultats obtenus par S. Bianchini et A. Bressan sur le problème de Cauchy pour des perturbations visqueuses $ \partial _t u^\varepsilon + \partial _x f(u^\varepsilon ) = \varepsilon \partial _{xx} u^\varepsilon $ de systèmes strictement hyperboliques $\partial _t u + \partial _x f(u) =0$ en une dimension d’espace. Ils ont en particulier montré l’existence globale ($t\ge 0$), l’unicité et la stabilité des solutions et justifié la convergence quand $\varepsilon $ tend vers zéro pour des données initiales à petite variation totale. Leur analyse montre aussi que les solutions du système hyperbolique ainsi obtenues coïncident avec les solutions provenant d’autres types d’approximations.
LA - fre
KW - hyperbolic systems; vanishing viscosity method
UR - http://eudml.org/doc/252131
ER -

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