Comptes Rendus
Numerical Analysis/Partial Differential Equations
Singular limits for the Riemann problem: general diffusion, relaxation, and boundary conditions
Comptes Rendus. Mathématique, Volume 344 (2007) no. 1, pp. 59-64.

We consider self-similar approximations of non-linear hyperbolic systems in one space dimension with Riemann initial data, especially the system tuε+A(uε)xuε=εtx(B(uε)xuε), with ε>0. We assume that the matrix A(u) is strictly hyperbolic and that the diffusion matrix satisfies |B(u)Id|1. No genuine non-linearity assumption is required. We show the existence of a smooth, self-similar solution uε=uε(x/t) which has bounded total variation, uniformly in the diffusion parameter ε>0. In the limit ε0, the functions uε converge towards a solution of the Riemann problem associated with the hyperbolic system. A similar result is established for the relaxation approximation tuε+xvε=0, tvε+a2B(u)xuε=(f(uε)vε)/(εt). We also cover the boundary-value problem in a half-space for the same regularizations.

Nous considérons les approximations auto-semblables d'un système hyperbolique non-linéaire à une dimension d'espace avec donnée initiale de type « problème de Riemann », en particulier le système tuε+A(uε)xuε=εtx(B(uε)xuε), avec ε>0. Nous supposons que la matrice A(u) est strictement hyperbolique et que la matrice de diffusion satisfait |B(u)Id|1. Aucune hypothèse de « vraie non-linéarité » n'est imposée. Nous démontrons que ce problème admet une solution régulière, auto-semblable uε=uε(x/t) de variation totale uniformément bornée par rapport au paramètre de diffusion ε>0. Lorsque ε0, les fonctions uε convergent vers une solution du problème de Riemann associé au système hyperbolique. Nous établissons aussi un résultat analogue pour les approximations par relaxation données par tuε+xvε=0, tvε+a2B(u)xuε=(f(uε)vε)/(εt). Ces résultats sont finalement étendus au problème de Riemann associé à ces mêmes régularisations et posé dans un demi-espace avec condition au bord.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2006.11.015
Kayyunnapara T. Joseph 1; Philippe G. LeFloch 2

1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
2 Laboratoire Jacques-Louis Lions & Centre national de la recherche scientifique, UMR 7598, Université Pierre et Marie Curie (Paris VI), 75252 Paris cedex 05, France
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Kayyunnapara T. Joseph; Philippe G. LeFloch. Singular limits for the Riemann problem: general diffusion, relaxation, and boundary conditions. Comptes Rendus. Mathématique, Volume 344 (2007) no. 1, pp. 59-64. doi : 10.1016/j.crma.2006.11.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.11.015/

[1] C.M. Dafermos Solution of the Riemann problem for a class of hyperbolic conservation laws by the viscosity method, Arch. Rational Mech. Anal., Volume 52 (1973), pp. 1-9

[2] C.M. Dafermos Admissible wave fans in nonlinear hyperbolic systems, Arch. Rational Mech. Anal., Volume 106 (1989), pp. 243-260

[3] G. Dal Maso; P.G. LeFloch; F. Murat Definition and weak stability of nonconservative products, J. Math. Pure Appl., Volume 74 (1995), pp. 483-548

[4] H.-T. Fan; M. Slemrod Dynamic flows with liquid/vapor phase transitions, Handbook of Mathematical Fluid Dynamics, vol. I, North-Holland, Amsterdam, 2002, pp. 373-420

[5] K.T. Joseph; P.G. LeFloch Boundary layers in weak solutions to hyperbolic conservation laws, Arch. Rational Mech. Anal., Volume 147 (1999), pp. 47-88

[6] K.T. Joseph; P.G. LeFloch Boundary layers in weak solutions of hyperbolic conservation laws II. Self-similar vanishing diffusion limits, Comm. Pure Appl. Anal., Volume 1 (2002), pp. 51-76

[7] K.T. Joseph; P.G. LeFloch Boundary layers in weak solutions of hyperbolic conservation laws III. Vanishing relaxation limits, Portugal. Math., Volume 59 (2002), pp. 453-494

[8] K.T. Joseph; P.G. LeFloch Singular limits for the Riemann problem: general diffusion, relaxation, and boundary conditions (O. Rozanova, ed.), Analytical Approaches to Multidimensional Balance Laws, Nova Publishers, 2005

[9] K.T. Joseph, P.G. LeFloch, Singular limits in phase dynamics with physical viscosity and capillarity, Proc. Roy. Soc. Edinburgh Sect. A, in press

[10] P.G. LeFloch, Shock waves for nonlinear hyperbolic systems in nonconservative form, Institute for Math. and its Appl., Minneapolis, Preprint # 593, 1989

[11] P.G. LeFloch Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Waves, Lecture Notes in Mathematics, ETH Zuerich, Birkhäuser, 2002

[12] P.G. LeFloch; A. Tzavaras Existence theory for the Riemann problem for nonconservative hyperbolic systems, C. R. Acad. Sci. Paris Ser. I, Volume 323 (1996), pp. 347-352

[13] M. Slemrod A limiting viscosity approach to the Riemann problem for materials exhibiting change of phase, Arch. Rational Mech. Anal., Volume 105 (1989), pp. 327-365

[14] A.E. Tzavaras Wave interactions and variation estimates for self-similar viscous limits in systems of conservation laws, Arch. Rational Mech. Anal., Volume 135 (1996), pp. 1-60

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