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Generalized solutions of a kinetic granular media equation by a gradient flow approach

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Abstract

We consider a one-dimensional kinetic model of granular media in the case where the interaction potential is quadratic. Taking advantage of a simple first integral, we can use a reformulation (equivalent to the initial kinetic model for classical solutions) which allows measure solutions. This reformulation has a Wasserstein gradient flow structure (on a possibly infinite product of spaces of measures) for a convex energy which enables us to prove global in time well-posedness.

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Acknowledgments

The authors are grateful to Yann Brenier, Reinhard Illner and Maxime Laborde for fruitful discussions about this work. M.A. acknowledges the support of NSERC through a Discovery Grant. G.C. gratefully acknowledges the hospitality of the Mathematics and Statistics Department at UVIC (Victoria, Canada), and the support from the CNRS, from the ANR, through the project ISOTACE (ANR-12- MONU-0013) and from INRIA through the action exploratoire MOKAPLAN.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Victoria, PO Box 3060 STN CSC, Victoria, BC, V8W 3R4, Canada

    Martial Agueh

  2. CEREMADE, UMR CNRS 7534, Université Paris IX Dauphine, Pl. de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Guillaume Carlier

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  1. Martial Agueh
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  2. Guillaume Carlier
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Correspondence to Guillaume Carlier.

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Communicated by L. Ambrosio.

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Agueh, M., Carlier, G. Generalized solutions of a kinetic granular media equation by a gradient flow approach. Calc. Var. 55, 37 (2016). https://doi.org/10.1007/s00526-016-0978-7

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  • DOI: https://doi.org/10.1007/s00526-016-0978-7

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