# Viscous approach for Linear Hyperbolic Systems with Discontinuous Coefficients

Bruno Fornet[1]

• [1] LATP, Université de Provence, 39 rue Joliot-Curie, 13453 Marseille cedex 13, France LMRS, Université de Rouen, Avenue de l’Université, BP.12, 76801 Saint Etienne du Rouvray, France
• Volume: 18, Issue: 2, page 397-443
• ISSN: 0240-2963

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## Abstract

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We introduce small viscosity solutions of hyperbolic systems with discontinuous coefficients accross the fixed noncharacteristic hypersurface $\left\{{x}_{d}=0\right\}.$ Under a geometric stability assumption, our first result is obtained, in the multi-D framework, for piecewise smooth coefficients. For our second result, the considered operator is ${𝔻}_{t}+a\left(x\right){𝔻}_{x},$ with $sign\left(xa\left(x\right)\right)>0$ (expansive case not included in our first result), thus resulting in an infinity of weak solutions. Proving that this problem is uniformly Evans-stable, we show that our viscous approach successfully singles out a solution. Both results are new and incorporates a stability result as well as an asymptotic analysis of the convergence at any order, which results in an accurate boundary layer analysis.

## How to cite

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Fornet, Bruno. "Viscous approach for Linear Hyperbolic Systems with Discontinuous Coefficients." Annales de la faculté des sciences de Toulouse Mathématiques 18.2 (2009): 397-443. <http://eudml.org/doc/10112>.

@article{Fornet2009,
abstract = {We introduce small viscosity solutions of hyperbolic systems with discontinuous coefficients accross the fixed noncharacteristic hypersurface $\lbrace x_d=0\rbrace .$ Under a geometric stability assumption, our first result is obtained, in the multi-D framework, for piecewise smooth coefficients. For our second result, the considered operator is $\mathbb\{D\}_t+a(x)\mathbb\{D\}_x,$ with $sign(x a(x))&gt;0$ (expansive case not included in our first result), thus resulting in an infinity of weak solutions. Proving that this problem is uniformly Evans-stable, we show that our viscous approach successfully singles out a solution. Both results are new and incorporates a stability result as well as an asymptotic analysis of the convergence at any order, which results in an accurate boundary layer analysis.},
affiliation = {LATP, Université de Provence, 39 rue Joliot-Curie, 13453 Marseille cedex 13, France LMRS, Université de Rouen, Avenue de l’Université, BP.12, 76801 Saint Etienne du Rouvray, France},
author = {Fornet, Bruno},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {small viscosity solutions; geometric stability assumption; boundary layer analysis},
language = {eng},
month = {1},
number = {2},
pages = {397-443},
publisher = {Université Paul Sabatier, Toulouse},
title = {Viscous approach for Linear Hyperbolic Systems with Discontinuous Coefficients},
url = {http://eudml.org/doc/10112},
volume = {18},
year = {2009},
}

TY - JOUR
AU - Fornet, Bruno
TI - Viscous approach for Linear Hyperbolic Systems with Discontinuous Coefficients
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2009/1//
PB - Université Paul Sabatier, Toulouse
VL - 18
IS - 2
SP - 397
EP - 443
AB - We introduce small viscosity solutions of hyperbolic systems with discontinuous coefficients accross the fixed noncharacteristic hypersurface $\lbrace x_d=0\rbrace .$ Under a geometric stability assumption, our first result is obtained, in the multi-D framework, for piecewise smooth coefficients. For our second result, the considered operator is $\mathbb{D}_t+a(x)\mathbb{D}_x,$ with $sign(x a(x))&gt;0$ (expansive case not included in our first result), thus resulting in an infinity of weak solutions. Proving that this problem is uniformly Evans-stable, we show that our viscous approach successfully singles out a solution. Both results are new and incorporates a stability result as well as an asymptotic analysis of the convergence at any order, which results in an accurate boundary layer analysis.
LA - eng
KW - small viscosity solutions; geometric stability assumption; boundary layer analysis
UR - http://eudml.org/doc/10112
ER -

## References

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