# Obstacle problems for scalar conservation laws

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

- Volume: 35, Issue: 3, page 575-593
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topLevi, Laurent. "Obstacle problems for scalar conservation laws." ESAIM: Mathematical Modelling and Numerical Analysis 35.3 (2010): 575-593. <http://eudml.org/doc/197535>.

@article{Levi2010,

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.
},

author = {Levi, Laurent},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Obstacle problem; conservation laws; entropy solution.; entropy solution; bilateral obstacle problem; Dirichlet boundary conditions; penalization method},

language = {eng},

month = {3},

number = {3},

pages = {575-593},

publisher = {EDP Sciences},

title = {Obstacle problems for scalar conservation laws},

url = {http://eudml.org/doc/197535},

volume = {35},

year = {2010},

}

TY - JOUR

AU - Levi, Laurent

TI - Obstacle problems for scalar conservation laws

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2010/3//

PB - EDP Sciences

VL - 35

IS - 3

SP - 575

EP - 593

AB -
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.

LA - eng

KW - Obstacle problem; conservation laws; entropy solution.; entropy solution; bilateral obstacle problem; Dirichlet boundary conditions; penalization method

UR - http://eudml.org/doc/197535

ER -

## References

top- 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.
- C. Bardos, A.Y. LeRoux and J.C. Nedelec, First-order quasilinear equations with boundary conditions. Comm. Partial Differential Equations4 (1979) 1017-1034. Zbl0418.35024
- 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. Zbl0865.35082
- A. Bensoussan and J.L. Lions, Inéquations variationnelles non linéaires du premier et second ordre. C. R. Acad. Sci. Paris, Sér. A276 (1973) 1411-1415. Zbl0264.49006
- 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.
- C. Chainais-Hillairet, Finite volume schemes for a nonlinear hyperbolic equation. Convergence toward the entropy solution and error estimate. ESAIM: M2AN33 (1999) 129-156.
- 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. Zbl0801.65089
- R.J. Diperna, Measure-valued solutions to conservation laws. Arch. Rat. Mech. Anal.88 (1985) 223-270. Zbl0616.35055
- 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. Zbl0830.35077
- G. Gagneux and M. Madaune-Tort, Analyse mathématique de modèles non linéaires de l'ingénierie pétrolière. Math. Appl. SMAI22, Springer-Verlag, Berlin (1996).
- 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. Zbl0856.65104
- S.N. Kruskov, First-order quasilinear equations in several independent variables. Math. USSR Sb.10 (1970) 217-243.
- 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). Zbl1005.35060
- 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.
- A. Szepessy, Measure solutions to scalar conservation laws with boundary conditions. Arch. Rat. Mech. Anal.107 (1989) 181-193. Zbl0702.35155
- A. Szepessy, Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary condition. ESAIM: M2AN25 (1991) 749-782. Zbl0751.65061
- 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). Zbl0437.35004
- G. Vallet, Dirichlet problem for nonlinear conservation law. Revista Matematica ComplutenseXIII (2000) 1-20. Zbl0979.35099
- M.H. Vignal, Convergence of a finite volume scheme for elliptic-hyperbolic system. RAIRO: Modél. Math. Anal. Numér.30 (1996) 841-872. Zbl0861.65084

## Citations in EuDML Documents

top- Florent Berthelin, Numerical flux-splitting for a class of hyperbolic systems with unilateral constraint
- Florent Berthelin, Numerical flux-splitting for a class of hyperbolic systems with unilateral constraint
- F Berthelin, F Bouchut, Weak solutions for a hyperbolic system with unilateral constraint and mass loss

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.