Front tracking approximations for slow erosion

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May 2012, 32(5): 1481-1502. doi: 10.3934/dcds.2012.32.1481

Front tracking approximations for slow erosion

Debora Amadori ^1, and Wen Shen ^2,

1.	Dipartimento di Matematica Pura & Applicata, University of L’Aquila, Italy
2.	Mathematics Department, Penn State University, University Park, PA 16802, United States

Received December 2010 Revised May 2011 Published January 2012

Abstract
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In this paper we study an integro-differential equation describing slow erosion, in a model of granular flow. In this equation the flux is non local and depends on $x$, $t$. We define approximate solutions by using a front tracking technique, adapted to this special equation. Convergence of the approximate solutions is established by means of suitable a priori estimates. In turn, these yield the global existence of entropy solutions in BV. Such entropy solutions are shown to be unique.
We also prove the continuous dependence on initial data and on the erosion function, for the approximate as well as for the exact solutions. This establishes the well-posedness of the Cauchy problem.

Keywords: existence and uniqueness of BV solutions., Granular flow, front tracking, slow erosion, conservation laws.

Mathematics Subject Classification: Primary: 35L65; Secondary: 35L67, 35Q70, 35L60, 35L0.

Citation: Debora Amadori, Wen Shen. Front tracking approximations for slow erosion. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1481-1502. doi: 10.3934/dcds.2012.32.1481

References:

[1]	D. Amadori and W. Shen, Global existence of large BV solutions in a model of granular flow, Comm. Partial Differential Equations, 34 (2009), 1003-1040. doi: 10.1080/03605300902892279.
[2]	D. Amadori and W. Shen, The slow erosion limit in a model of granular flow, Arch. Ration. Mech. Anal., 199 (2011), 1-31. doi: 10.1007/s00205-010-0313-y.
[3]	D. Amadori and W. Shen, An integro-differential conservation law arising in a model of granular flow, J. Hyperbolic Differ. Equ., to appear.
[4]	T. Boutreux and P.-G. Gennes, Surface flows of granular mixtures, I. General principles and minimal model, J. Phys. I France, 6 (1996), 1295-1304.
[5]	A. Bressan, "Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem," Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000.
[6]	A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z.
[7]	A. Bressan, P. Zhang and Y. Zheng, Asymptotic variational wave equations, Arch. Ration. Mech. Anal., 183 (2007), 163-185. doi: 10.1007/s00205-006-0014-8.
[8]	R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.
[9]	P. Cannarsa and P. Cardaliaguet, Representation of equilibrium solutions to the table problem for growing sandpiles, J. Eur. Math. Soc. (JEMS), 6 (2004), 435-464.
[10]	P. Cannarsa, P. Cardaliaguet, G. Crasta and E. Giorgieri, A boundary value problem for a PDE model in mass transfer theory: representation of solutions and applications, Calc. Var. PDE, 24 (2005), 431-457. doi: 10.1007/s00526-005-0328-7.
[11]	R. Colombo, G. Guerra and F. Monti, Modelling the dynamics of granular matter, IMA Journal of Applied Mathematics (2011), 1-17.
[12]	R. Colombo, M. Mercier and M. Rosini, Stability and total variation estimates on general scalar balance laws, Commun. Math. Sci., 7 (2009), 37-65.
[13]	C. M. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics,'' 3^rd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325, Springer-Verlag, Berlin, 2010.
[14]	J. Duran, "Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials,'' Springer-Verlag, 2000.
[15]	M. Falcone and S. Finzi Vita, A finite-difference approximation of a two-layer system for growing sandpiles, SIAM J. Sci. Comput., 28 (2006), 1120-1132. doi: 10.1137/050629410.
[16]	K. P. Hadeler and C. Kuttler, Dynamical models for granular matter, Granular Matter, 2 (1999), 9-18. doi: 10.1007/s100350050029.
[17]	K. H. Karlsen and N. H. Risebro, On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients, Discrete Contin. Dyn. Syst., 9 (2003), 1081-1104. doi: 10.3934/dcds.2003.9.1081.
[18]	C. Klingenberg and N. H. Risebro, Convex conservation laws with discontinuous coefficients. Existence, uniqueness and asymptotic behavior, Comm. Partial Differential Equations, 20 (1995), 1959-1990. doi: 10.1080/03605309508821159.
[19]	O. A. Oleinik, Discontinuous solutions of non-linear differential equations, Usp. Mat. Nauk, 12 (1957), 3-73, Translation in AMS Translations, Ser. II, 26, 95-172.
[20]	S. B. Savage and K. Hutter, The dynamics of avalanches of granular materials from initiation to runout. I. Analysis, Acta Mech., 86 (1991), 201-223. doi: 10.1007/BF01175958.
[21]	D. Serre, "Systems of Conservation Laws. 1. Hyperbolicity, Entropies, Shock Waves,'' Translated from the 1996 French original by I. N. Sneddon, Cambridge University Press, Cambridge, 1999.
[22]	W. Shen, On the shape of avalanches, J. Math. Anal. Appl., 339 (2008), 828-838. doi: 10.1016/j.jmaa.2007.07.036.
[23]	W. Shen and T. Y. Zhang, Erosion profile by a global model for granular flow, Arch. Ration. Mech. Anal., to appear.

show all references

References:

[1]	D. Amadori and W. Shen, Global existence of large BV solutions in a model of granular flow, Comm. Partial Differential Equations, 34 (2009), 1003-1040. doi: 10.1080/03605300902892279.
[2]	D. Amadori and W. Shen, The slow erosion limit in a model of granular flow, Arch. Ration. Mech. Anal., 199 (2011), 1-31. doi: 10.1007/s00205-010-0313-y.
[3]	D. Amadori and W. Shen, An integro-differential conservation law arising in a model of granular flow, J. Hyperbolic Differ. Equ., to appear.
[4]	T. Boutreux and P.-G. Gennes, Surface flows of granular mixtures, I. General principles and minimal model, J. Phys. I France, 6 (1996), 1295-1304.
[5]	A. Bressan, "Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem," Oxford Lecture Series in Mathematics and its Applications, 20, Oxford University Press, Oxford, 2000.
[6]	A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation, Arch. Ration. Mech. Anal., 183 (2007), 215-239. doi: 10.1007/s00205-006-0010-z.
[7]	A. Bressan, P. Zhang and Y. Zheng, Asymptotic variational wave equations, Arch. Ration. Mech. Anal., 183 (2007), 163-185. doi: 10.1007/s00205-006-0014-8.
[8]	R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons, Phys. Rev. Lett., 71 (1993), 1661-1664. doi: 10.1103/PhysRevLett.71.1661.
[9]	P. Cannarsa and P. Cardaliaguet, Representation of equilibrium solutions to the table problem for growing sandpiles, J. Eur. Math. Soc. (JEMS), 6 (2004), 435-464.
[10]	P. Cannarsa, P. Cardaliaguet, G. Crasta and E. Giorgieri, A boundary value problem for a PDE model in mass transfer theory: representation of solutions and applications, Calc. Var. PDE, 24 (2005), 431-457. doi: 10.1007/s00526-005-0328-7.
[11]	R. Colombo, G. Guerra and F. Monti, Modelling the dynamics of granular matter, IMA Journal of Applied Mathematics (2011), 1-17.
[12]	R. Colombo, M. Mercier and M. Rosini, Stability and total variation estimates on general scalar balance laws, Commun. Math. Sci., 7 (2009), 37-65.
[13]	C. M. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics,'' 3^rd edition, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325, Springer-Verlag, Berlin, 2010.
[14]	J. Duran, "Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials,'' Springer-Verlag, 2000.
[15]	M. Falcone and S. Finzi Vita, A finite-difference approximation of a two-layer system for growing sandpiles, SIAM J. Sci. Comput., 28 (2006), 1120-1132. doi: 10.1137/050629410.
[16]	K. P. Hadeler and C. Kuttler, Dynamical models for granular matter, Granular Matter, 2 (1999), 9-18. doi: 10.1007/s100350050029.
[17]	K. H. Karlsen and N. H. Risebro, On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients, Discrete Contin. Dyn. Syst., 9 (2003), 1081-1104. doi: 10.3934/dcds.2003.9.1081.
[18]	C. Klingenberg and N. H. Risebro, Convex conservation laws with discontinuous coefficients. Existence, uniqueness and asymptotic behavior, Comm. Partial Differential Equations, 20 (1995), 1959-1990. doi: 10.1080/03605309508821159.
[19]	O. A. Oleinik, Discontinuous solutions of non-linear differential equations, Usp. Mat. Nauk, 12 (1957), 3-73, Translation in AMS Translations, Ser. II, 26, 95-172.
[20]	S. B. Savage and K. Hutter, The dynamics of avalanches of granular materials from initiation to runout. I. Analysis, Acta Mech., 86 (1991), 201-223. doi: 10.1007/BF01175958.
[21]	D. Serre, "Systems of Conservation Laws. 1. Hyperbolicity, Entropies, Shock Waves,'' Translated from the 1996 French original by I. N. Sneddon, Cambridge University Press, Cambridge, 1999.
[22]	W. Shen, On the shape of avalanches, J. Math. Anal. Appl., 339 (2008), 828-838. doi: 10.1016/j.jmaa.2007.07.036.
[23]	W. Shen and T. Y. Zhang, Erosion profile by a global model for granular flow, Arch. Ration. Mech. Anal., to appear.

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