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March  2011, 10(2): 625-638. doi: 10.3934/cpaa.2011.10.625

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, France

Received  May 2010 Revised  September 2010 Published  December 2010

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: dual formulation., p-Laplacian operator, subdifferential operator, numerical approximation of projection.
Mathematics Subject Classification: Primary: 35K55, 65M60; Secondary: 35B40, 65K1.
Citation: S. Dumont, Noureddine Igbida. On the collapsing sandpile problem. Communications on Pure & Applied Analysis, 2011, 10 (2) : 625-638. doi: 10.3934/cpaa.2011.10.625
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

Ph. Bénilan, M. G. Crandall and A. Pazy, "Evolution Equations Governed by Accretive Operators,", Preprint book., ().   Google Scholar

[5]

Ph. Bénilan, L. C. Evans and R. F. Gariepy, On some singular limits of homogeneous semigroups, J. Evol. Equ., 3 (2003), 203-214.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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,  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

N. Igbida, Equivalent formulations for Monge-Kantorovich equation,, {\em Submitted}., ().   Google Scholar

[16]

L. Prigozhin, Variational model of sandpile growth, Euro. J. Appl. Math., 7 (1996), 225-236. doi: doi:10.1017/S0956792500002321.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

Ph. Bénilan, M. G. Crandall and A. Pazy, "Evolution Equations Governed by Accretive Operators,", Preprint book., ().   Google Scholar

[5]

Ph. Bénilan, L. C. Evans and R. F. Gariepy, On some singular limits of homogeneous semigroups, J. Evol. Equ., 3 (2003), 203-214.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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,  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

N. Igbida, Equivalent formulations for Monge-Kantorovich equation,, {\em Submitted}., ().   Google Scholar

[16]

L. Prigozhin, Variational model of sandpile growth, Euro. J. Appl. Math., 7 (1996), 225-236. doi: doi:10.1017/S0956792500002321.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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