Systèmes hyperboliques et viscosité évanescente
[Hyperbolic systems and vanishing viscosity]
Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 918, pp. 231-250.

In this talk we will present the works of S. Bianchini and A. Bressan on the Cauchy problem for viscous perturbations ${\partial }_{t}{u}^{\epsilon }+{\partial }_{x}f\left({u}^{\epsilon }\right)=\epsilon {\partial }_{xx}{u}^{\epsilon }$ of one-dimensional strictly hyperbolic systems ${\partial }_{t}u+{\partial }_{x}f\left(u\right)=0$. They have shown global existence ($t\ge 0$), uniqueness and stability and they have justified the limit when $\epsilon$ 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.

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 ${\partial }_{t}{u}^{\epsilon }+{\partial }_{x}f\left({u}^{\epsilon }\right)=\epsilon {\partial }_{xx}{u}^{\epsilon }$ de systèmes strictement hyperboliques ${\partial }_{t}u+{\partial }_{x}f\left(u\right)=0$ 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 $\epsilon$ 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.

Classification: 35F20,  35F25,  35B25,  35B35
Keywords: hyperbolic systems, vanishing viscosity method
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Rousset, Frédéric. Systèmes hyperboliques et viscosité évanescente, in Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Talk no. 918, pp. 231-250. http://www.numdam.org/item/SB_2002-2003__45__231_0/`

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