Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T19:50:12.585Z Has data issue: false hasContentIssue false
  • English
  • Français

Obstacle problems for scalar conservation laws

Published online by Cambridge University Press:  15 April 2002

Laurent Levi
Laurent Levi*
Affiliation:
University of Pau, CNRS, Laboratory of Applied Mathematics ERS 2055, I.P.R.A., Avenue de l'Université, 64000 Pau, France. ([email protected])
Get access

Abstract

In this paper we are interested in bilateral obstacle problems for quasilinear scalar conservation laws associated with Dirichlet boundary conditions. Firstly, we provide a suitable entropy formulation which ensures uniqueness. Then, we justify the existence of a solution through the method of penalization and by referring to the notion of entropy process solution due to specific properties of bounded sequences in L ∞. Lastly, we study the behaviour of this solution and its stability properties with respect to the associated obstacle functions.

Keywords

Obstacle problemconservation lawsentropy solution.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 35 , Issue 3 , May 2001 , pp. 575 - 593
Copyright
© EDP Sciences, SMAI, 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

J.M. Ball, A version of the fundamental theorem for young measures, in PDEs and continuum model of phase transition, Lect. Notes Phys. 344, Springer-Verlag, Berlin (1995) 241-259.
Bardos, C., LeRoux, A.Y. and Nedelec, J.C., First-order quasilinear equations with boundary conditions. Comm. Partial Differential Equations 4 (1979) 1017-1034. CrossRef
Benharbit, S., Chalabi, A. and Numerical, J.P. Vila viscosity and convergence of finite volume methods for conservation laws with boundary conditions. SIAM J. Numer. Anal. 32 (1995) 775-796. CrossRef
A. Bensoussan and J.L. Lions, Inéquations variationnelles non linéaires du premier et second ordre. C. R. Acad. Sci. Paris, Sér. A 276 (1973) 1411-1415.
P. Bia and M. Combarnous, Les méthodes thermiques de production des hydrocarbures, Chap. 1 : Transfert de chaleur et de masse. Revue de l'Institut français du pétrole (1975) 359-394.
Chainais-Hillairet, C., Finite volume schemes for a nonlinear hyperbolic equation. Convergence toward the entropy solution and error estimate. ESAIM: M2AN 33 (1999) 129-156. CrossRef
Champier, S., Gallouët, T. and Herbin, R., Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on triangular mesh. Numer. Math. 66 (1993) 139-157. CrossRef
Diperna, R.J., Measure-valued solutions to conservation laws. Arch. Rat. Mech. Anal. 88 (1985) 223-270. CrossRef
Eymard, R., Gallouët, T. and Herbin, R., Existence and uniqueness of the entropy solution to a nonlinear hyperbolic equation. Chin. Ann. Math. 16B (1995) 1-14.
G. Gagneux and M. Madaune-Tort, Analyse mathématique de modèles non linéaires de l'ingénierie pétrolière. Math. Appl. SMAI 22, Springer-Verlag, Berlin (1996).
Kröner, D. and Rokyta, M., Convergence of upwing finite volume schemes for scalar conservation laws in two dimensions. SIAM J. Numer. Anal. 31 (1994) 324-343. CrossRef
Kruskov, S.N., First-order quasilinear equations in several independent variables. Math. USSR Sb. 10 (1970) 217-243. CrossRef
L. Lévi and F. Peyroutet, A time-fractional step method for conservation law related obstacle problem. (Preprint 99/37. Laboratory of Applied Math., ERS 2055, Pau University.), Adv. Appl. Math. (to appear).
F. Otto, Conservation laws in bounded domains, uniqueness and existence via parabolic approximation, in Weak and measure-valued solutions to evolutionary PDE's, J. Malek, J. Necas, M. Rokyta and M. Ruzicka Eds., Chapman & Hall, London (1996) 95-143.
Szepessy, A., Measure solutions to scalar conservation laws with boundary conditions. Arch. Rat. Mech. Anal. 107 (1989) 181-193. CrossRef
Szepessy, A., Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary condition. ESAIM: M2AN 25 (1991) 749-782. CrossRef
L. Tartar, Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics. Heriot-Watt Symposium, R.J. Knops Ed., Res. Notes Math. 4, Pitman Press, New-York (1979).
G. Vallet, Dirichlet problem for nonlinear conservation law. Revista Matematica Complutense XIII (2000) 1-20.
Vignal, M.H., Convergence of a finite volume scheme for elliptic-hyperbolic system. RAIRO: Modél. Math. Anal. Numér. 30 (1996) 841-872.