Vol. 3, No. 3, 2015

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Stationary solutions of Keller–Segel-type crowd motion and herding models: Multiplicity and dynamical stability

Jean Dolbeault, Gaspard Jankowiak and Peter Markowich

Vol. 3 (2015), No. 3, 211–242
DOI: 10.2140/memocs.2015.3.211

In this paper we study two models for crowd motion and herding. Each of the models is of Keller–Segel type and involves two parabolic equations, one for the evolution of the density and one for the evolution of a mean field potential. We classify all radial stationary solutions, prove multiplicity results, and establish some qualitative properties of these solutions, which are characterized as critical points of an energy functional. A notion of variational stability is associated with such solutions.

Dynamical stability in the neighborhood of a stationary solution is also studied in terms of the spectral properties of the linearized evolution operator. For one of the two models, we exhibit a Lyapunov functional which allows us to make the link between the two notions of stability. Even in that case, for certain values of the mass parameter, with all other parameters taken in an appropriate range, we find that two dynamically stable stationary solutions exist. We further discuss the qualitative properties of the solutions using theoretical methods and numerical computations.

crowd motion, herding, continuum model, Lyapunov functional, variational methods, dynamical stability, non-self-adjoint evolution operators
Mathematical Subject Classification 2010
Primary: 35J20, 35K40, 35Q91
Received: 7 May 2013
Revised: 26 July 2013
Accepted: 11 September 2013
Published: 11 October 2015

Communicated by Roberto Natalini
Jean Dolbeault
Université Paris-Dauphine
Place de Lattre de Tassigny
75775 Paris Cedex 16
Gaspard Jankowiak
Université Paris-Dauphine
Place de Lattre de Tassigny
75775 Paris Cedex 16
Peter Markowich
Department of Applied Mathematics & Theoretical Physics
Centre for Mathematical Sciences
University of Cambridge
Wilberforce Road
United Kingdom