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

On s’intéresse à des problèmes hyperboliques linéaires dont les coefficients sont discontinus au travers de l’hypersurface non-caractéristique {x d =0}. 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 par morceaux. Notre second résultat montre que, pour l’opérateur 𝔻 t +a(x)𝔻 x , avec sign(xa(x))>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.

We introduce small viscosity solutions of hyperbolic systems with discontinuous coefficients accross the fixed noncharacteristic hypersurface {x d =0}. 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(x)𝔻 x , with sign(xa(x))>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.

DOI : 10.5802/afst.1209
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
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     author = {Bruno Fornet},
     title = {Viscous approach for {Linear} {Hyperbolic} {Systems} with {Discontinuous} {Coefficients}},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {397--443},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {Ser. 6, 18},
     number = {2},
     year = {2009},
     doi = {10.5802/afst.1209},
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Bruno Fornet. Viscous approach for Linear Hyperbolic Systems with Discontinuous Coefficients. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 18 (2009) no. 2, pp. 397-443. doi : 10.5802/afst.1209. https://afst.centre-mersenne.org/articles/10.5802/afst.1209/

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