Abstract
Boundary conditions for multidimensional scalar conservation laws are obtained in the context of hydrodynamic limits from a kinetic point of view. The initial boundary value kinetic problem is well posed since inward and outward characteristics of the domain can be distinguished. The convergence of the first momentum of the distribution function to an entropy solution of the conservation law is established. Boundary conditions are obtained. The equivalence with the Bardos, Leroux, and Nedelec conditions is studied.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
C. Bardos, A. Y. Leroux, and J. C. Nedelec, First order quasi linear equations with boundary conditions, Comm. PDE 4(9):1017–1034 (1979).
Y. Brenier, Résolution d'équations d'évolution quasilinéaires en N dimensions d'espace è l'aide d'équations en dimension N+1 d'espace, JDE, 375–390 (1983).
J. F. Bourgeat, P. Le Tallec, B. Perthame, and Y. Qiu, Coupling Boltzmann and Euler equations without overlapping, Contemp. Math. 157:377–398 (1994).
F. Dubois and P. Le Floch, Boundary conditions for nonlinear hyperbolic systems of conservation laws, JDE 4(9):93–122 (1988).
T. Gallouet and J. M. Masella, Un schéma de Godounov approché, C.R.A.S. Sér. I 323:77–84 (1996).
Y. Giga and T. Miyakawa, A kinetic construction of global solutions of first order quasi linear equations, Duke Math. J. 50:505–515 (1983).
F. Golse, Boundary and interface layers for kinetic models, Lecture Notes, GDR SPARCH, Oléron (1997).
F. Golse, Shock profiles for the Perthame-Tadmore kinetic model, Technical report.
S. N. Kruskov, First order quasilinear equations with several independent variables, Math. Sbornik. 81(123):228–255; Math. USSR Sbornik 10:217–243 (1970).
F. A. Howes, Multidimensional initial-boundary value problems with strong linearities, Arch. Rat. Mech. Anal. 91(2):153–168 (1986).
P. L. Lions, B. Perthame, and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7(1):169–191 (1994).
A. Omrane and J. P. Vila, On two kinetic approaches for scalar conservation laws, Preprint (1995).
B. Perthame and E. Tadmore, A kinetic equation with kinetic entropy functions for scalar conservation laws, Comm. Math. Phys. 136:501–517 (1991).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Nouri, A., Omrane, A. & Vila, J.P. Boundary Conditions for Scalar Conservation Laws from a Kinetic Point of View. Journal of Statistical Physics 94, 779–804 (1999). https://doi.org/10.1023/A:1004574814876
Issue Date:
DOI: https://doi.org/10.1023/A:1004574814876