On the collapsing sandpile problem

S. Dumont; Noureddine Igbida

doi:10.3934/cpaa.2011.10.625

Article Contents

2011, Volume 10, Issue 2: 625-638. Doi: 10.3934/cpaa.2011.10.625

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On the collapsing sandpile problem

S. Dumont^1, and
Noureddine Igbida^2,

1.
LAMFA CNRS UMR 6140, Université de Picardie Jules Verne, 33 rue Saint-Leu 80039 Amiens cedex
2.
LAMFA, CNRS UMR 6140, Universite de Picardie Jules Verne, 33 rue Saint Leu, 80039 Amiens Cedex

Received: May 2010

Revised: September 2010

Published: March 2011

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Abstract

We are interested in the modeling of collapsing sandpiles. We use the collapsing model introduced by Evans, Feldman and Gariepy in [13], to provide a description of the phenomena in terms of a composition of projections onto interlocked convex sets around the set of stable sandpiles.

Keywords:

Mathematics Subject Classification: Primary: 35K55, 65M60; Secondary: 35B40, 65K10.

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References

[1]	G. Aronson, L. C. Evans and Y. Wu, Fast/Slow diffusion and growing sandpiles, J. Differential Equations, 131 (1996), 304-335.doi: doi:10.1006/jdeq.1996.0166.
[2]	P. Bak, C. Tang and K. Weisenfeld, Self-organized criticality, Phys. Rev. A, 38 (1988), 364-378.doi: doi:10.1103/PhysRevA.38.364.
[3]	J. W. Barrett and L. Prigozhin, Dual formulation in critical state problems, Interfaces and Free Boundaries, 8 (2006), 349-370.doi: doi:10.4171/IFB/147.
[4]	Ph. Bénilan, M. G. Crandall and A. Pazy, "Evolution Equations Governed by Accretive Operators," Preprint book.
[5]	Ph. Bénilan, L. C. Evans and R. F. Gariepy, On some singular limits of homogeneous semigroups, J. Evol. Equ., 3 (2003), 203-214.
[6]	J. P. Bouchaud, M. E. Cates, J. Ravi Prakash and S. F. Edwards, A model for the Dynamic of Sandpile Surfaces, J. Phys. I France, 4 (1994), 1383-1410.
[7]	G. Bouchitté, G. Buttazzo and P. Seppecher, Energies with respect to a Measure and Applications to Low Dimensional Structures, Calc. Var. Partial Differential Equations, 5 (1997), 37-54.
[8]	H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert (French), North-Holland Mathematics Studies, No. 5. Notas de Matemàtica (50). North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York, 1973.
[9]	S. Dumont and N. Igbida, On a Dual Formulation for the Growing Sandpile Problem, European Journal Applied Math., 20 (2009), 169-185.doi: doi:10.1017/S0956792508007754.
[10]	I. Ekeland and R. Témam, "Convex Analysis and Variational Problems," Classics in Applied Mathematics, 28. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1999.
[11]	L. C. Evans, Application of nonlinear semigroup theory to certain partial differential equations, Nonlinear evolution equations (Proc. Sympos., Univ. Wisconsin, Madison, Wis., 1977), pp. 163-188,
[12]	L. C. Evans, Partial differential equations and Monge-Kantorovich mass transfer, Current developments in mathematics, 1997 (Cambridge, MA), 65-126, Int. Press, Boston, MA, 1999.
[13]	L. C. Evans, M. Feldman and R. F. Gariepy, Fast/Slow diffusion and collapsing sandpiles, J. Differential Equations, 137 (1997), 166-209.doi: doi:10.1006/jdeq.1997.3243.
[14]	L. C. Evans and F. Rezakhanlou, A stochastic model for sandpiles and its continum limit, Comm. Math. Phys., 197 (1998), 325-345.doi: doi:10.1007/s002200050453.
[15]	N. Igbida, Equivalent formulations for Monge-Kantorovich equation, Submitted.
[16]	L. Prigozhin, Variational model of sandpile growth, Euro. J. Appl. Math., 7 (1996), 225-236.doi: doi:10.1017/S0956792500002321.
[17]	J. E. Roberts and J.-M. Thomas, "Mixed and Hybrid Methods," (P. G. Ciarlet and J. L. Lions eds.), Handbook of Numerical Analysis, vol. II, Finite Element Methods (Part 1), North-Holland, Amsterdam, 1991.
[18]	R. E. Showalter, "Monotone Operators in Banach Space and Nonlinear Partial Differential Equations," Mathematical Surveys and Monographs, 49, American Mathematical Society, Providence, RI, 1997.