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Global Weak Solutions for Kolmogorov–Vicsek Type Equations with Orientational Interactions

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Abstract

We study the global existence and uniqueness of weak solutions to kinetic Kolmogorov–Vicsek models that can be considered as non-local, non-linear, Fokker–Planck type equations describing the dynamics of individuals with orientational interactions. This model is derived from the discrete Couzin–Vicsek algorithm as mean-field limit (Bolley et al., Appl Math Lett, 25:339–343, 2012; Degond et al., Math Models Methods Appl Sci 18:1193–1215, 2008), which governs the interactions of stochastic agents moving with a velocity of constant magnitude, that is, the corresponding velocity space for these types of Kolmogorov–Vicsek models is the unit sphere. Our analysis for L p estimates and compactness properties take advantage of the orientational interaction property, meaning that the velocity space is a compact manifold.

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Authors and Affiliations

  1. Department of Mathematics and ICES, The University of Texas at Austin, Austin, TX78712, USA

    Irene M. Gamba & Moon-Jin Kang

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  1. Irene M. Gamba
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  2. Moon-Jin Kang
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Correspondence to Irene M. Gamba.

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Communicated by D. Kinderlehrer

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Gamba, I.M., Kang, MJ. Global Weak Solutions for Kolmogorov–Vicsek Type Equations with Orientational Interactions. Arch Rational Mech Anal 222, 317–342 (2016). https://doi.org/10.1007/s00205-016-1002-2

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