American Institute of Mathematical Sciences

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

On the collapsing sandpile problem

 1 LAMFA CNRS UMR 6140, Université de Picardie Jules Verne, 33 rue Saint-Leu 80039 Amiens cedex 2 LAMFA, CNRS UMR 6140, Universite 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.
Citation: S. Dumont, Noureddine Igbida. On the collapsing sandpile problem. Communications on Pure and 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. [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.

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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.
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