New unilateral problems in stratigraphy
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 4, p. 765-784
This work deals with the study of some stratigraphic models for the formation of geological basins under a maximal erosion rate constrain. It leads to introduce differential inclusions of degenerated hyperbolic-parabolic type 0 t u-div{H( t u+E)u}, where H is the maximal monotonous graph of the Heaviside function and E is a given non-negative function. Firstly, we present the new and realistic models and an original mathematical formulation, taking into account the weather-limited rate constraint in the conservation law, with a unilateral constraint on the outflow boundary. Then, we give a study of the 1-D case with numerical illustrations.
DOI : https://doi.org/10.1051/m2an:2006029
Classification:  35K65,  35L80,  35Q35
Keywords: stratigraphic models, weather limited, degenerated parabolic-hyperbolic conservation laws
@article{M2AN_2006__40_4_765_0,
     author = {Antontsev, Stanislav N. and Gagneux, G\'erard and Luce, Robert and Vallet, Guy},
     title = {New unilateral problems in stratigraphy},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {40},
     number = {4},
     year = {2006},
     pages = {765-784},
     doi = {10.1051/m2an:2006029},
     zbl = {1133.35388},
     mrnumber = {2274777},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2006__40_4_765_0}
}
Antontsev, Stanislav N.; Gagneux, Gérard; Luce, Robert; Vallet, Guy. New unilateral problems in stratigraphy. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 40 (2006) no. 4, pp. 765-784. doi : 10.1051/m2an:2006029. http://www.numdam.org/item/M2AN_2006__40_4_765_0/

[1] S.N. Antontsev, D. Etienne, G. Gagneux and G. Vallet, New unilateral problems in stratigraphy. Publication interne du Laboratoire de Mathématiques Appliquées, UMR-CNRS 5142, Université de Pau, No. 05/13 (2005).

[2] S.N. Antontsev, G. Gagneux, R. Luce and G. Vallet, Weather limited constraint in stratigraphy. International conference “Tikhonov and Contemporary Mathematics”: section 5, Mathematical Geophysics, Moscow (2006) 7-8.

[3] S.N. Antontsev, G. Gagneux and G. Vallet, Analyse mathématique d'un modèle d'asservissement stratigraphique. Approche gravitationnelle d'un processus de sédimentation sous contrainte d'érosion maximale. Publication interne du Laboratoire de Mathématiques Appliquées, UMR-CNRS 5142, Université de Pau, No. 01/23 (2001).

[4] S.N. Antontsev, G. Gagneux and G. Vallet, On some stratigraphic control problems, Prikladnaya Mekhanika Tekhnicheskaja Fisika (Novosibirsk) 44 (2003) 85-94 (in Russian), and Journal of Applied Mechanics and Technical Physics (New York) 44 (2003) 821-828. | Zbl 1038.76039

[5] P. Bénilan, M.G. Crandall and A. Pazy, Bonnes solutions d'un problème d'évolution semi-linéaire. C. R. Acad. Sci. Paris Sér. I Math. (306) (1988) 527-530. | Zbl 0635.34013

[6] C. Cuesta, C.J. Van Duijn and J. Hulshof, Infiltration in porous media with dynamic capillary pressure: Travelling waves. Eur. J. Appl. Math. 11 (2000) 381-397. | Zbl 0970.76096

[7] D. Etienne, Contribution à l'analyse mathématique de modèles stratigraphiques. Thèse de l'Université de Pau, France (2004).

[8] R. Eymard and T. Gallouët, Analytical and numerical study of a model of erosion and sedimentation. SIAM J. Numer. Anal. 43 (2006) 2344-2370. | Zbl pre05029783

[9] R. Eymard, T. Gallouët, D. Granjeon, R. Masson and Q.H. Tran, Multi-lithology stratigraphic model under maximum erosion rate constraint. Internat. J. Numer. Methods Engrg. 60 (2004) 527-548. | Zbl 1098.76618

[10] G. Gagneux, R. Luce and G. Vallet, A non-standard free-boundary problem arising from stratigraphy. Publication interne du Laboratoire de Mathématiques Appliquées, UMR-CNRS 5142, Université de Pau, No. 04/33 (2004). | Zbl 1096.35069

[11] G. Gagneux and M. Madaune-Tort, Analyse mathématique de modèles non linéaires de l'ingénierie pétrolière. Mathématiques & Applications, Vol. 22 Springer, Paris (1996). | Zbl 0842.35126

[12] G. Gagneux and G. Vallet, Sur des problèmes d'asservissements stratigraphiques. ESAIM: COCV “A tribute to Jacques-Louis Lions” 8 (2002) 715-739. | Numdam | Zbl 1069.35033

[13] G. Gagneux and G. Vallet, A result of existence for an original convection-diffusion equation. Revista de la Real Academia de Ciencias, Serie A: Matemáticas (RACSAM) 99 (2005) 125-131. | Zbl 1090.35107

[14] T. Gallouët, Equations satisfaites par des limites de solutions approchées, conférence plénière, 34ème congrès d'analyse numérique, Anglet (Pyrénées Atlantiques), in Canum (2002) 87-96.

[15] V. Gervais and R. Masson, Mathematical and numerical analysis of a stratigraphic model. ESAIM: M2AN 38 (2004) 585-611. | Numdam | MR 2087725 | Zbl 1130.86315

[16] J.-L. Lions, Cours d'Analyse Numérique. Hermann, École Polytechnique, Paris (1973).

[17] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969). | MR 259693 | Zbl 0189.40603

[18] G. Vallet, Sur une loi de conservation issue de la géologie. C. R. Acad. Sci. Paris Sér. I Math. (337) (2003) 559-564. | Zbl 1038.86010