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Sharp upperbounds for the number of large amplitude limit cycles in polynomial Lienard systems
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 |
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.
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,'' 3rd 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,'' 3rd 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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