# American Institute of Mathematical Sciences

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

## Front tracking approximations for slow erosion

 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

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.
Citation: Debora Amadori, Wen Shen. Front tracking approximations for slow erosion. Discrete & Continuous Dynamical Systems - A, 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.  doi: 10.1080/03605300902892279.  Google Scholar [2] D. Amadori and W. Shen, The slow erosion limit in a model of granular flow,, Arch. Ration. Mech. Anal., 199 (2011), 1.  doi: 10.1007/s00205-010-0313-y.  Google Scholar [3] D. Amadori and W. Shen, An integro-differential conservation law arising in a model of granular flow,, J. Hyperbolic Differ. Equ., ().   Google Scholar [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.   Google Scholar [5] A. Bressan, "Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem,", Oxford Lecture Series in Mathematics and its Applications, 20 (2000).   Google Scholar [6] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215.  doi: 10.1007/s00205-006-0010-z.  Google Scholar [7] A. Bressan, P. Zhang and Y. Zheng, Asymptotic variational wave equations,, Arch. Ration. Mech. Anal., 183 (2007), 163.  doi: 10.1007/s00205-006-0014-8.  Google Scholar [8] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar [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.   Google Scholar [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.  doi: 10.1007/s00526-005-0328-7.  Google Scholar [11] R. Colombo, G. Guerra and F. Monti, Modelling the dynamics of granular matter,, IMA Journal of Applied Mathematics (2011), (2011), 1.   Google Scholar [12] R. Colombo, M. Mercier and M. Rosini, Stability and total variation estimates on general scalar balance laws,, Commun. Math. Sci., 7 (2009), 37.   Google Scholar [13] C. M. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics,'', 3rd edition, 325 (2010).   Google Scholar [14] J. Duran, "Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials,'', Springer-Verlag, (2000).   Google Scholar [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.  doi: 10.1137/050629410.  Google Scholar [16] K. P. Hadeler and C. Kuttler, Dynamical models for granular matter,, Granular Matter, 2 (1999), 9.  doi: 10.1007/s100350050029.  Google Scholar [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.  doi: 10.3934/dcds.2003.9.1081.  Google Scholar [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.  doi: 10.1080/03605309508821159.  Google Scholar [19] O. A. Oleinik, Discontinuous solutions of non-linear differential equations,, Usp. Mat. Nauk, 12 (1957), 3.   Google Scholar [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.  doi: 10.1007/BF01175958.  Google Scholar [21] D. Serre, "Systems of Conservation Laws. 1. Hyperbolicity, Entropies, Shock Waves,'', Translated from the 1996 French original by I. N. Sneddon, (1996).   Google Scholar [22] W. Shen, On the shape of avalanches,, J. Math. Anal. Appl., 339 (2008), 828.  doi: 10.1016/j.jmaa.2007.07.036.  Google Scholar [23] W. Shen and T. Y. Zhang, Erosion profile by a global model for granular flow,, Arch. Ration. Mech. Anal., ().   Google Scholar

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##### 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.  doi: 10.1080/03605300902892279.  Google Scholar [2] D. Amadori and W. Shen, The slow erosion limit in a model of granular flow,, Arch. Ration. Mech. Anal., 199 (2011), 1.  doi: 10.1007/s00205-010-0313-y.  Google Scholar [3] D. Amadori and W. Shen, An integro-differential conservation law arising in a model of granular flow,, J. Hyperbolic Differ. Equ., ().   Google Scholar [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.   Google Scholar [5] A. Bressan, "Hyperbolic Systems of Conservation Laws. The One Dimensional Cauchy Problem,", Oxford Lecture Series in Mathematics and its Applications, 20 (2000).   Google Scholar [6] A. Bressan and A. Constantin, Global conservative solutions of the Camassa-Holm equation,, Arch. Ration. Mech. Anal., 183 (2007), 215.  doi: 10.1007/s00205-006-0010-z.  Google Scholar [7] A. Bressan, P. Zhang and Y. Zheng, Asymptotic variational wave equations,, Arch. Ration. Mech. Anal., 183 (2007), 163.  doi: 10.1007/s00205-006-0014-8.  Google Scholar [8] R. Camassa and D. Holm, An integrable shallow water equation with peaked solitons,, Phys. Rev. Lett., 71 (1993), 1661.  doi: 10.1103/PhysRevLett.71.1661.  Google Scholar [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.   Google Scholar [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.  doi: 10.1007/s00526-005-0328-7.  Google Scholar [11] R. Colombo, G. Guerra and F. Monti, Modelling the dynamics of granular matter,, IMA Journal of Applied Mathematics (2011), (2011), 1.   Google Scholar [12] R. Colombo, M. Mercier and M. Rosini, Stability and total variation estimates on general scalar balance laws,, Commun. Math. Sci., 7 (2009), 37.   Google Scholar [13] C. M. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics,'', 3rd edition, 325 (2010).   Google Scholar [14] J. Duran, "Sands, Powders, and Grains: An Introduction to the Physics of Granular Materials,'', Springer-Verlag, (2000).   Google Scholar [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.  doi: 10.1137/050629410.  Google Scholar [16] K. P. Hadeler and C. Kuttler, Dynamical models for granular matter,, Granular Matter, 2 (1999), 9.  doi: 10.1007/s100350050029.  Google Scholar [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.  doi: 10.3934/dcds.2003.9.1081.  Google Scholar [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.  doi: 10.1080/03605309508821159.  Google Scholar [19] O. A. Oleinik, Discontinuous solutions of non-linear differential equations,, Usp. Mat. Nauk, 12 (1957), 3.   Google Scholar [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.  doi: 10.1007/BF01175958.  Google Scholar [21] D. Serre, "Systems of Conservation Laws. 1. Hyperbolicity, Entropies, Shock Waves,'', Translated from the 1996 French original by I. N. Sneddon, (1996).   Google Scholar [22] W. Shen, On the shape of avalanches,, J. Math. Anal. Appl., 339 (2008), 828.  doi: 10.1016/j.jmaa.2007.07.036.  Google Scholar [23] W. Shen and T. Y. Zhang, Erosion profile by a global model for granular flow,, Arch. Ration. Mech. Anal., ().   Google Scholar
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