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Local Existence of Weak Solutions to Kinetic Models of Granular Media

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Abstract

We prove in any dimension \({d \geqq 1}\) a local in time existence of weak solutions to the Cauchy problem for the kinetic equation of granular media,

$$\partial_t f+v\cdot \nabla_x f = {div}_v[f(\nabla W *_v f)]$$

when the initial data are nonnegative, integrable and bounded functions with compact support in velocity, and the interaction potential \({W}\) is a \({C^2({\mathbb{R}}^d)}\) radially symmetric convex function. Our proof is constructive and relies on a splitting argument in position and velocity, where the spatially homogeneous equation is interpreted as the gradient flow of a convex interaction energy with respect to the quadratic Wasserstein distance. Our result generalizes the local existence result obtained by Benedetto et al. (RAIRO Modél Math Anal Numér 31(5):615–641, 1997) on the one-dimensional model of this equation for a cubic power-law interaction potential.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Victoria, PO Box 1700, STN CSC, Victoria, BC, V8W 2Y2, Canada

    Martial Agueh

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  1. Martial Agueh
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Correspondence to Martial Agueh.

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Communicated by D. Kinderlehrer

M. Agueh is supported by a grant from the Natural Science and Engineering Research Council of Canada (NSERC).

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Agueh, M. Local Existence of Weak Solutions to Kinetic Models of Granular Media. Arch Rational Mech Anal 221, 917–959 (2016). https://doi.org/10.1007/s00205-016-0975-1

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