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Nonlinear Conservation Laws and Applications pp 169–179Cite as

Mathematical Aspects of A Model for Granular Flow

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Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 153))

Abstract

The model for granular flow being studied by the authors was proposed by Hadeler and Kuttler in [21]. In one space dimension, by a change of variable, the system can be written as a 2 × 2 hyperbolic system of balance laws. Various results are obtained for this system, under suitable assumptions on initial data which leads to a strictly hyperbolic system. For suitably small initial data, the solution remains smooth globally. Furthermore, the global existence of large BV solutions for Cauchy problem is established for initial data with small height of moving layer. Finally, at the slow erosion limit as the height of moving layer tends to zero, the slope of the mountain provides the unique entropy solution to a scalar integro’differential conservation law, implying that the profile of the standing layer depends only on the total mass of the avalanche flowing downhill. Various open problems and further research topics related to this model are discussed at the end of the paper.

AMS(MOS) subject classifications. Primary 35L45, 35L50, 35L60, 35L65; Secondary 35L40, 58J45.

The work of the second author is partially supported by NSF grant DMS-0908047.

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Authors and Affiliations

  1. Dipartimento di Matematica Pura ed Applicata, University of L’Aquila, L’Aquila, Italy

    Debora Amadori

  2. Department of Mathematics, Penn State University, University Park, PA, 16802, USA

    Wen Shen

Authors
  1. Debora Amadori
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  2. Wen Shen
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Corresponding author

Correspondence to Debora Amadori .

Editor information

Editors and Affiliations

  1. McAllister Bldg 201, University Park, 16802-6401, Pennsylvania, USA

    Alberto Bressan

  2. , Oxford Centre for Nonlinear PDE, Mathema, University of Oxford, St Giles 24–29, Oxford, OX1 3LB, Montserrat

    Gui-Qiang G. Chen

  3. , School of Mathematics, University of Minnesota, Church Street S.E, 206, Minneapolis, 55455, Minnesota, USA

    Marta Lewicka

  4. , Department of Mathematics, University of Pittsburgh, Pittsburgh, 15260, Pennsylvania, USA

    Dehua Wang

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Amadori, D., Shen, W. (2011). Mathematical Aspects of A Model for Granular Flow. In: Bressan, A., Chen, GQ., Lewicka, M., Wang, D. (eds) Nonlinear Conservation Laws and Applications. The IMA Volumes in Mathematics and its Applications, vol 153. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-9554-4_6

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