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Analysis and Simulation of Nonlinear and Nonlocal Transport Equations

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Innovative Algorithms and Analysis

Part of the book series: Springer INdAM Series ((SINDAMS,volume 16))

Abstract

This article is devoted to the analysis of some nonlinear conservative transport equations, including the so-called aggregation equation with pointy potential, and numerical method devoted to its numerical simulation. Such a model describes the collective motion of individuals submitted to an attractive potential and can be written as a continuity transport equation with a velocity field computed through a self-consistent interaction potential. In the strongly attractive setting, L p solutions may blow up in finite time, then a theory of existence of weak measure solutions has been defined. In this approach, we show the existence of Filippov characteristics allowing to define solutions of the aggregation initial value problem as a pushforward measure. Then numerical analysis of an upwind type scheme is proposed allowing to recover the dynamics of aggregates beyond the blowup time.

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Author information

Authors and Affiliations

  1. Institut Camille Jordan UMR CNRS 5208, Université Lyon 1, 43 boulevard du 11 novembre 1918, 69622, Villeurbanne Cedex, France

    Frédéric Lagoutière

  2. LAGA - UMR 7539, Institut Galilée, Université Paris 13, 99, avenue Jean-Baptiste Clément, 93430, Villetaneuse, France

    Nicolas Vauchelet

Authors
  1. Frédéric Lagoutière
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  2. Nicolas Vauchelet
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Correspondence to Frédéric Lagoutière .

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Editors and Affiliations

  1. Istituto per le Applicazioni del Calcolo CNR, Rome, Italy

    Laurent Gosse

  2. Istituto per le Applicazioni del Calcolo CNR, Rome, Italy

    Roberto Natalini

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Lagoutière, F., Vauchelet, N. (2017). Analysis and Simulation of Nonlinear and Nonlocal Transport Equations. In: Gosse, L., Natalini, R. (eds) Innovative Algorithms and Analysis. Springer INdAM Series, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-319-49262-9_10

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