Viscous approach for Linear Hyperbolic Systems with Discontinuous Coefficients

Fornet, Bruno

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, $C^{\infty }$ par morceaux. Notre second résultat montre que, pour l’opérateur ${𝔻}_t+a\left(x\right){𝔻}_x,$ avec $sign\left(xa\right(x\left)\right)>0$ (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.

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 ${𝔻}_t+a\left(x\right){𝔻}_x,$ with $sign\left(xa\right(x\left)\right)>0$ (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.

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