no. 1
The obstacle-mass constraint problem for hyperbolic conservation laws. Solvability
Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 1, pp. 221-248.

In this work we introduce the obstacle-mass constraint problem for a multidimensional scalar hyperbolic conservation law. We prove existence of an entropy solution to this problem by a penalization/viscosity method. The mass constraint introduces a nonlocal Lagrange multiplier in the penalized equation, giving rise to a nonlocal parabolic problem. We introduce a compatibility condition relating the initial datum and the obstacle function which ensures global in time existence of solution. This is not a smoothness condition, but relates to the propagation of the support of the initial datum.

DOI : 10.1016/j.anihpc.2015.11.003
Classification : 35L65, 35R35
Mots clés : Hyperbolic conservation law, Obstacle problem, Mass conservation, Nonlocal parabolic equation, Free boundary problem 
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     author = {Amorim, Paulo and Neves, Wladimir and Rodrigues, Jos\'e Francisco},
     title = {The obstacle-mass constraint problem for hyperbolic conservation laws. {Solvability}},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {221--248},
     publisher = {Elsevier},
     volume = {34},
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}
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Amorim, Paulo; Neves, Wladimir; Rodrigues, José Francisco. The obstacle-mass constraint problem for hyperbolic conservation laws. Solvability. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 1, pp. 221-248. doi : 10.1016/j.anihpc.2015.11.003. http://www.numdam.org/articles/10.1016/j.anihpc.2015.11.003/

[1] Barthélemy, L. Problème d'obstacle pour une équation quasi-linéaire du premier ordre, Ann. Fac. Sci. Toulouse, Volume 9 (1988), pp. 137–159 | DOI | Numdam | MR | Zbl

[2] Berthelin, F. Existence and weak stability for a pressureless model with unilateral constraint, Math. Models Methods Appl. Sci., Volume 12 (2002), pp. 249–272 | DOI | MR | Zbl

[3] Berthelin, F.; Bouchut, F. Weak solutions for a hyperbolic system with unilateral constraint and mass loss, Ann. Inst. Henri Poincaré, Volume 20 (2003) no. 6, pp. 975–997 | Numdam | MR | Zbl

[4] Caffarelli, L.; Lin, F. Nonlocal heat flows preserving the L2 energy, Discrete Contin. Dyn. Syst., Volume 23 (2009) no. 1&2 | MR | Zbl

[5] Chen, G.-Q.; Rascle, M. Initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws, Arch. Ration. Mech. Anal., Volume 153 (2000), pp. 205–220 | MR | Zbl

[6] Dafermos, C. Hyperbolic Conservation Laws in Continuum Physics, Grundlehren Math. Wissenschaften Series, vol. 325, Springer Verlag, 2000 | MR | Zbl

[7] Gagneux, G.; Lefevère, A.M.; Madaune-Tort, M. Une approche analytique d'un modèle black-oil des écoulements triphasiques compressibles en ingénierie pétrolière, J. Méc. Théor. Appl., Volume 6 (1987), pp. 547–569 | MR | Zbl

[8] Godlewski, E.; Raviart, P.-A. Hyperbolic Systems of Conservation Laws, Mathématiques & Applications (Paris), Ellipses, Paris, 1991 | MR | Zbl

[9] Kinderlehrer, D.; Stampacchia, G. An Introduction to Variational Inequalities and Their Applications, Academic Press, 1980 | MR | Zbl

[10] Lévi, L. Problèmes unilatéraux pour des équations non linéaires de réaction diffusion, Ann. Fac. Sci. Toulouse VI, Volume 4 (1995) no. 3, pp. 593–631 | Numdam | MR | Zbl

[11] Lévi, L. Obstacle problems for scalar conservation laws, ESAIM: Math. Model. Numer. Anal., Volume 35 (2001) no. 3, pp. 575–593 | DOI | Numdam | MR | Zbl

[12] Lévi, L. The singular limit of a bilateral obstacle problem for a class of degenerate parabolic–hyperbolic operators, Adv. Appl. Math., Volume 35 (2005), pp. 34–57 | DOI | MR | Zbl

[13] Mignot, F.; Puel, J.-P. Inéquations variationnelles et quasivariationnelles hyperboliques du premier ordre, J. Math. Pures Appl. (9), Volume 55 (1976), pp. 353–378 | MR | Zbl

[14] Panov, E.Y. Existence of strong traces for generalized solutions of multidimensional scalar conservation laws, J. Hyperbolic Differ. Equ., Volume 2 (2005) no. 4, pp. 885–908 | DOI | MR | Zbl

[15] Panov, E.Y. Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyperbolic Differ. Equ., Volume 4 (2007) no. 4, pp. 729–770 | DOI | MR | Zbl

[16] Rodrigues, J.-F. Obstacle Problems in Mathematical Physics, North-Holland Mathematics Studies, vol. 134, 1987 | MR | Zbl

[17] Rodrigues, J.-F. On the hyperbolic obstacle problem of first order, Chin. Ann. Math., Volume 23B (2002) no. 2, pp. 253–266 | MR | Zbl

[18] Rodrigues, J.-F. On hyperbolic variational inequalities of first order and some applications, Monatsh. Math., Volume 142 (2004) no. 1–2, pp. 157–177 | MR | Zbl

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