Lipschitz semigroup for an integro–differential equation for slow erosion

Colombo, Rinaldo; Guerra, Graziano; Shen, Wen

doi:10.1090/S0033-569X-2012-01309-2

Authors: Rinaldo M. Colombo, Graziano Guerra and Wen Shen
Journal: Quart. Appl. Math. 70 (2012), 539-578
MSC (2010): Primary 35L65.; Secondary 35L67, 35Q70, 35L60, 35L03
DOI: https://doi.org/10.1090/S0033-569X-2012-01309-2
Published electronically: May 18, 2012
MathSciNet review: 2986134
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Abstract: In this paper we study an integro-differential equation describing granular flow dynamics with slow erosion. This nonlinear partial differential equation is a conservation law where the flux contains an integral term. Through a generalized wave front tracking algorithm, approximate solutions are constructed and shown to converge strongly to a Lipschitz semigroup.

References

D. Amadori and W. Shen, Global existence of large BV solutions in a model of granular flow, Comm. Partial Differential Equations 34 (2009), no. 7–9, 1003–1040.

Debora Amadori and Wen Shen, A hyperbolic model of granular flow, Nonlinear partial differential equations and hyperbolic wave phenomena, Contemp. Math., vol. 526, Amer. Math. Soc., Providence, RI, 2010, pp. 1–18. MR 2731985, DOI https://doi.org/10.1090/conm/526/10374

---, Mathematical aspects of a model for granular flow, Nonlinear Conservation Laws and Applications, IMA Volumes in Mathematics and its Applications, vol. 153, Springer, 2011, pp. 169–180.

Debora Amadori and Wen Shen, The slow erosion limit in a model of granular flow, Arch. Ration. Mech. Anal. 199 (2011), no. 1, 1–31. MR 2754335, DOI https://doi.org/10.1007/s00205-010-0313-y

---, Front tracing approximations for slow erosion, Dis. Cont. Dyn. Sys. (To appear).

---, An integro-differential conservation law arising in a model of granular flow, J. Hyp. Diff. Eq. (To appear).

L. Ambrosio, N. Fusco, and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. MR 2003a:49002

Alberto Bressan, Hyperbolic systems of conservation laws, Oxford Lecture Series in Mathematics and its Applications, vol. 20, Oxford University Press, Oxford, 2000. The one-dimensional Cauchy problem. MR 1816648

A. Cattani, R.M. Colombo, and G. Guerra, A hyperbolic model for granular flow, Zeitschrift für Angewandte Mathematik und Mechanik (2011), To appear.

Gui-Qiang Chen and Cleopatra Christoforou, Solutions for a nonlocal conservation law with fading memory, Proc. Amer. Math. Soc. 135 (2007), no. 12, 3905–3915. MR 2341940, DOI https://doi.org/10.1090/S0002-9939-07-08942-3

Cleopatra Christoforou, Systems of hyperbolic conservation laws with memory, J. Hyperbolic Differ. Equ. 4 (2007), no. 3, 435–478. MR 2339804, DOI https://doi.org/10.1142/S0219891607001215

C. Christoforou, Nonlocal conservation laws with memory, Hyperbolic problems: theory, numerics, applications, Springer, Berlin, 2008, pp. 381–388. MR 2549169, DOI https://doi.org/10.1007/978-3-540-75712-2_34

Rinaldo M. Colombo and Graziano Guerra, Hyperbolic balance laws with a non local source, Comm. Partial Differential Equations 32 (2007), no. 10-12, 1917–1939. MR 2372493, DOI https://doi.org/10.1080/03605300701318849

R.M. Colombo, G. Guerra, and F. Monti, Modelling the dynamics of granular matter, IMA Journal of Applied Mathematics (2011).

Constantine M. Dafermos, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl. 38 (1972), 33–41. MR 303068, DOI https://doi.org/10.1016/0022-247X%2872%2990114-X

K.P. Hadeler and C. Kuttler, Dynamical models for granular matter, Granular Matter 2 (1999), 9–18.

Philip Hartman, Ordinary differential equations, 2nd ed., Birkhäuser, Boston, Mass., 1982. MR 658490

D. Li and T. Li, Shock formation in a traffic flow model with Arrhenius look–ahead dynamics, Networks and Heterogeneous Media 6 (2011), no. 4, 681–694.

S. B. Savage and K. Hutter, The motion of a finite mass of granular material down a rough incline, J. Fluid Mech. 199 (1989), 177–215. MR 985199, DOI https://doi.org/10.1017/S0022112089000340

S. B. Savage and K. Hutter, The dynamics of avalanches of granular materials from initiation to runout. I. Analysis, Acta Mech. 86 (1991), no. 1-4, 201–223. MR 1093945, DOI https://doi.org/10.1007/BF01175958

Wen Shen, On the shape of avalanches, J. Math. Anal. Appl. 339 (2008), no. 2, 828–838. MR 2375239, DOI https://doi.org/10.1016/j.jmaa.2007.07.036

W. Shen and T.Y. Zhang, Erosion profile by a global model for granular flow, Arch. Ration. Mech. Anal. (To appear).