Obstacle problems for scalar conservation laws
ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 3, pp. 575-593.

Ce travail a pour objet l’étude de problèmes d’obstacles bilatéraux pour des lois de conservation scalaires quasi-linénaires du premier ordre associées à des conditions aux limites de Dirichlet. On donne d’abord une formulation entropique qui garantit l’unicité. On justifie alors l’existence d’une solution par utilisation de la méthode de pénalisation et au moyen de la notion de processus entropique solution due aux propriétés des suites bornées dans L ∞ . Enfin, on étudie le comportement de cette solution et ses propriétés de stabilité en fonction des contraintes d’obstacle associées.

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.

Classification : 35L65, 35R35, 35L85
Mots-clés : obstacle problem, conservation laws, entropy solution 
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Levi, Laurent. Obstacle problems for scalar conservation laws. ESAIM: Modélisation mathématique et analyse numérique, Tome 35 (2001) no. 3, pp. 575-593. http://www.numdam.org/item/M2AN_2001__35_3_575_0/

[1] 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. | Zbl

[2] C. Bardos, A.Y. Leroux and J.C. Nedelec, First-order quasilinear equations with boundary conditions. Comm. Partial Differential Equations 4 (1979) 1017-1034. | Zbl

[3] S. Benharbit, A. Chalabi and J.P. Vila, Numerical viscosity and convergence of finite volume methods for conservation laws with boundary conditions. SIAM J. Numer. Anal. 32 (1995) 775-796. | Zbl

[4] 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. | Zbl

[5] 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.

[6] C. Chainais-Hillairet, Finite volume schemes for a nonlinear hyperbolic equation. Convergence toward the entropy solution and error estimate. ESAIM: M2AN 33 (1999) 129-156. | Numdam | Zbl

[7] S. Champier, T. Gallouët and R. Herbin, Convergence of an upstream finite volume scheme for a nonlinear hyperbolic equation on triangular mesh. Numer. Math. 66 (1993) 139-157. | Zbl

[8] R.J. Diperna, Measure-valued solutions to conservation laws. Arch. Rat. Mech. Anal. 88 (1985) 223-270. | Zbl

[9] R. Eymard, T. Gallouët and R. Herbin, Existence and uniqueness of the entropy solution to a nonlinear hyperbolic equation. Chin. Ann. Math. 16B (1995) 1-14. | Zbl

[10] G. Gagneux and M. Madaune-Tort, Analyse mathématique de modèles non linéaires de l'ingénierie pétrolière. Math. Appl. SMAI 22, Springer-Verlag, Berlin (1996). | Zbl

[11] D. Kröner and M. Rokyta, Convergence of upwing finite volume schemes for scalar conservation laws in two dimensions. SIAM J. Numer. Anal. 31 (1994) 324-343. | Zbl

[12] S.N. Kruskov, First-order quasilinear equations in several independent variables. Math. USSR Sb. 10 (1970) 217-243. | Zbl

[13] 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). | MR | Zbl

[14] 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.

[15] A. Szepessy, Measure solutions to scalar conservation laws with boundary conditions. Arch. Rat. Mech. Anal. 107 (1989) 181-193. | Zbl

[16] A. Szepessy, Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary condition. ESAIM: M2AN 25 (1991) 749-782. | Numdam | Zbl

[17] 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). | MR | Zbl

[18] G. Vallet, Dirichlet problem for nonlinear conservation law. Revista Matematica Complutense XIII (2000) 1-20. | Zbl

[19] M.H. Vignal, Convergence of a finite volume scheme for elliptic-hyperbolic system. RAIRO: Modél. Math. Anal. Numér. 30 (1996) 841-872. | Numdam | Zbl