Methods and Applications of Analysis

Volume 20 (2013)

Number 2

Hydrodynamic models of self-organized dynamics: Derivation and existence theory

Pages: 89 – 114

DOI: http://dx.doi.org/10.4310/MAA.2013.v20.n2.a1

Authors

Pierre Degond (Institut de Mathématiques de Toulouse, Université de Toulouse, France)

Jian-Guo Liu (Department of Physics and Department of Mathematics, Duke University, Durham, North Carolina, U.S.A.)

Sebastien Motsch (Center of Scientific Computation and Mathematical Modeling (CSCAMM), University of Maryland, College Park, Md., U.S.A.)

Vladislav Panferov (Department of Mathematics, California State University at Northridge)

Abstract

This paper is concerned with the derivation and analysis of hydrodynamic models for systems of self-propelled particles subject to alignment interaction and attraction-repulsion. Introducing various scalings, the effects of the alignment and attraction-repulsion interactions give rise to a variety of hydrodynamic limits. For instance, local alignment produces a pressure term at the hydrodynamic limit whereas near-local alignment induces a viscosity term. Depending on the scalings, attraction-repulsion either yields an additional pressure term or a capillary force (also termed ‘Korteweg force’). The hydrodynamic limits are shown to be symmetrizable hyperbolic systems with viscosity terms. A local-in-time existence result is proved in the 2D case for the viscous model and in the 3D case for the inviscid model.

Keywords

self-propelled particles, alignment dynamics, hydrodynamic limit, diffusion correction, weakly non-local interaction, symmetrizable hyperbolic system, energy method, local wellposedness, capillary force, Korteweg force, attraction-repulsion potential

2010 Mathematics Subject Classification

35K55, 35L60, 35Qxx, 82C05, 82C22, 82C70, 92D50

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