Viscous approach for Linear Hyperbolic Systems with Discontinuous Coefficients
Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 18 (2009) no. 2, p. 397-443

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.

On s’intéresse à des problèmes hyperboliques linéaires dont les coefficients sont discontinus au travers de l’hypersurface non-caractéristique $\left\{{x}_{d}=0\right\}.$ On prouve alors, sous une hypothèse de stabilité, la convergence, à la limite à viscosité évanescente, vers la solution d’un problème hyperbolique limite bien posé. Notre premier résultat concerne des systèmes multi-D, ${C}^{\infty }$ par morceaux. Notre second résultat montre que, pour l’opérateur ${𝔻}_{t}+a\left(x\right){𝔻}_{x},$ avec $sign\left(xa\left(x\right)\right)>0$ (cas exclu de notre premier résultat), notre critère de stabilité est satisfait, et qu’une unique solution à petite viscosité se dégage de notre approche. Nos deux résultats sont nouveaux et incluent une analyse asymptotique à tout ordre ainsi qu’un théorème de stabilité.

@article{AFST_2009_6_18_2_397_0,
author = {Fornet, Bruno},
title = {Viscous approach for Linear Hyperbolic Systems with Discontinuous Coefficients},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 18},
number = {2},
year = {2009},
pages = {397-443},
doi = {10.5802/afst.1209},
mrnumber = {2562832},
zbl = {1182.35030},
language = {en},
url = {http://www.numdam.org/item/AFST_2009_6_18_2_397_0}
}

Fornet, Bruno. Viscous approach for Linear Hyperbolic Systems with Discontinuous Coefficients. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 18 (2009) no. 2, pp. 397-443. doi : 10.5802/afst.1209. http://www.numdam.org/item/AFST_2009_6_18_2_397_0/

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