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 {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.

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.

@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/

[1] Bachmann (F.).— Analysis of a scalar conservation law with a flux function with discontinuous coefficients, Adv. Diff. Eq. 11-12, p. 1317-1338 (2004). | MR 2099558 | Zbl 1102.35063

[2] Bachmann (F.), Vovelle (J.).— Existence and uniqueness of entropy solution of scalar conservation laws with a flux function involving discontinuous coefficients, C.P.D.E, 31, p. 371-395 (2006). | MR 2209759 | Zbl 1102.35064

[3] Bouchut (F.), James (F.).— One-dimensional transport equations with discontinuous coefficients, Nonlin. Anal., 32, p. 891-933 (1998). | MR 1618393 | Zbl 0989.35130

[4] Bouchut (F.), James (F.), Mancini (S.).— Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), IV, p. 1-25 (2005). | Numdam | MR 2165401 | Zbl pre05058795

[5] Chazarain (J.), Piriou (A.).— Introduction to the theory of linear partial differential equations. translated from the french, Studies in Mathematics and its Applications, 14 , North Holland Publishing Co., Amsterdam-New York,1982. | MR 678605 | Zbl 0487.35002

[6] Crasta (G.), LeFloch (P. G.).— Existence result for a class of nonconservative and nonstrictly hyperbolic systems, Commun. Pure Appl. Anal., 1(4), p. 513-530 (2002). | MR 1942282 | Zbl 1031.35096

[7] DiPerna (R.J.), Lions (P.-L.).— Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98, p. 511-547 (1989). | MR 1022305 | Zbl 0696.34049

[8] Fornet (B.).— Two Results concerning the Small Viscosity Solution of Linear Scalar Conservation Laws with Discontinuous Coefficients, HAL (2007).

[9] Fornet (B.).— The Cauchy problem for 1-D linear nonconservative hyperbolic systems with possibly expansive discontinuity of the coefficient: a viscous approach, Vol 245 pp. 2440-2476 (2008). | MR 2455771 | Zbl 1158.35061

[10] Gallouët (T.).— Hyperbolic equations and systems with discontinuous coefficients or source terms, 10 pages, Proceedings of Equadiff-11, Bratislava, Slovaquia (July 25-29, 2005).

[11] Guès (O.), Métivier (G.), Williams (M.), Zumbrun (K.).— Existence and stability of multidimensional shock fronts in the vanishing viscosity limit, Arch. Rat. Mech. Anal. 175 p. 151-244 (2004). | MR 2118476 | Zbl 1072.35122

[12] Guès (O.), Williams (M.).— Curved shocks as viscous limits: a boundary problem approach, Indiana Univ. Math. J., 51 (2002), 421-450. | MR 1909296 | Zbl 1046.35072

[13] Hayes (B. T.), LeFloch (P. G.).— Measure solutions to a strictly hyperbolic system of conservation laws Nonlinearity, 9(6), p. 1547-1563 (1996). | MR 1419460 | Zbl 0908.35075

[14] LeFloch (P. G.).— An existence and uniqueness result for two nonstrictly hyperbolic systems. In Nonlinear evolution 271 equations that change type, vol. 27 of IMA Vol. Math. Appl. (1990), 126-138. Springer, New York. | MR 1074190 | Zbl 0727.35083

[15] LeFloch (P. G.), Tzavaras (A.E.).— Representation of weak limits and definition of nonconservartive products, SIAM J. Math. Anal. 30, p. 1309-1342 (1999). | MR 1718304 | Zbl 0939.35115

[16] Métivier (G.).— Small Viscosity and Boundary Layer Methods : Theory, Stability Analysis, and Applications, Birkhauser (2003). | MR 2151414 | Zbl 1133.35001

[17] Métivier (G.), Zumbrun (K.).— Large Viscous Boundary Layers for Noncharacteristic Nonlinear Hyperbolic Problems, Mem. Amer. Math. Soc. 175, no. 826, vi+107 pp (2005). | MR 2130346 | Zbl 1074.35066

[18] Métivier (G.), Zumbrun (K.).— Symmetrizers and Continuity of Stable Subspaces for Parabolic-Hyperbolic Boundary Value Problems, Disc. Cont. Dyn. Syst., 11, p. 205-220 (2004). | MR 2073953 | Zbl 1102.35332

[19] Poupaud (F.), Rascle (M.).— Measure solutions to the linear multidimensional transport equation with discontinuous coefficients, Comm. Diff. Equ. 22, p. 337-358 (1997). | MR 1434148 | Zbl 0882.35026

[20] Rousset (F.).— Viscous approximation of strong shocks of systems of conservation laws, SIAM J. Math. Anal. 35 (2003), 492-519. | MR 2001110 | Zbl 1052.35128