Singular limits for the Riemann problem: general diffusion, relaxation, and boundary conditions

Kayyunnapara T. Joseph; Philippe G. LeFloch

doi:10.1016/j.crma.2006.11.015

Numerical Analysis/Partial Differential Equations

Singular limits for the Riemann problem: general diffusion, relaxation, and boundary conditions
[Limites singulières pour le problème de Riemann : diffusion, relaxation et conditions aux limites]

Présenté par : Olivier Pironneau

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

Comptes Rendus. Mathématique, Volume 344 (2007) no. 1, pp. 59-64.

Résumé
Abstract

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 $\partial_{t} u_{ε} + A (u_{ε}) \partial_{x} u_{ε} = ε t \partial_{x} (B (u_{ε}) \partial_{x} u_{ε})$ , 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 $u_{ε} = u_{ε} (x / t)$ de variation totale uniformément bornée par rapport au paramètre de diffusion $ε > 0$ . Lorsque $ε \to 0$ , les fonctions $u_{ε}$ 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 $\partial_{t} u^{ε} + \partial_{x} v^{ε} = 0$ , $\partial_{t} v^{ε} + a^{2} B (u) \partial_{x} u^{ε} = (f (u^{ε}) - v^{ε}) / (ε t)$ . Ces résultats sont finalement étendus au problème de Riemann associé à ces mêmes régularisations et posé dans un demi-espace avec condition au bord.

We consider self-similar approximations of non-linear hyperbolic systems in one space dimension with Riemann initial data, especially the system $\partial_{t} u_{ε} + A (u_{ε}) \partial_{x} u_{ε} = ε t \partial_{x} (B (u_{ε}) \partial_{x} u_{ε})$ , 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 $u_{ε} = u_{ε} (x / t)$ which has bounded total variation, uniformly in the diffusion parameter $ε > 0$ . In the limit $ε \to 0$ , the functions $u_{ε}$ converge towards a solution of the Riemann problem associated with the hyperbolic system. A similar result is established for the relaxation approximation $\partial_{t} u^{ε} + \partial_{x} v^{ε} = 0$ , $\partial_{t} v^{ε} + a^{2} B (u) \partial_{x} u^{ε} = (f (u^{ε}) - v^{ε}) / (ε t)$ . We also cover the boundary-value problem in a half-space for the same regularizations.

Reçu le : 2006-02-15
Accepté le : 2006-11-13
Publié le : 2006-12-15

DOI : 10.1016/j.crma.2006.11.015

Affiliations des auteurs :

Kayyunnapara T. Joseph ¹ ; Philippe G. LeFloch ²

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     title = {Singular limits for the {Riemann} problem: general diffusion, relaxation, and boundary conditions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {59--64},
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     doi = {10.1016/j.crma.2006.11.015},
     language = {en},
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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/

Bibliographie
Cité par

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[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

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