Obstacle problems for scalar conservation laws
Levi, Laurent
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 35 (2001), p. 575-593 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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.

Publié le : 2001-01-01
Classification:  35L65,  35R35,  35L85 
@article{M2AN_2001__35_3_575_0,
     author = {Levi, Laurent},
     title = {Obstacle problems for scalar conservation laws},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {35},
     year = {2001},
     pages = {575-593},
     mrnumber = {1837085},
     zbl = {0990.35096},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2001__35_3_575_0}
}

Levi, Laurent. Obstacle problems for scalar conservation laws. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 35 (2001) pp. 575-593. http://gdmltest.u-ga.fr/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 0991.49500

[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 0418.35024

[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 0865.35082

[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 0264.49006

[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 0921.65071

[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 0801.65089

[8] R.J. Diperna, Measure-valued solutions to conservation laws. Arch. Rat. Mech. Anal. 88 (1985) 223-270. | Zbl 0616.35055

[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 0830.35077

[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 0842.35126

[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 0856.65104

[12] S.N. Kruskov, First-order quasilinear equations in several independent variables. Math. USSR Sb. 10 (1970) 217-243. | Zbl 0215.16203

[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 1867932 | Zbl 1005.35060

[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 0702.35155

[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 0751.65061

[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 584398 | Zbl 0437.35004

[18] G. Vallet, Dirichlet problem for nonlinear conservation law. Revista Matematica Complutense XIII (2000) 1-20. | Zbl 0979.35099

[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 0861.65084