Math @ Duke

Publications [#246895] of JianGuo Liu
Papers Published
 Degond, P; Frouvelle, A; Liu, JG, Macroscopic limits and phase transition in a system of selfpropelled particles,
Journal of Nonlinear Science, vol. 23 no. 3
(2013),
pp. 427456, Springer Nature, ISSN 09388974 [doi]
(last updated on 2019/02/18)
Abstract: We investigate systems of selfpropelled particles with alignment interaction. Compared to previous work (Degond and Motsch, Math. Models Methods Appl. Sci. 18:11931215, 2008a; Frouvelle, Math. Models Methods Appl. Sci., 2012), the force acting on the particles is not normalized, and this modification gives rise to phase transitions from disordered states at low density to aligned states at high densities. This model is the spaceinhomogeneous extension of (Frouvelle and Liu, Dynamics in a kinetic model of oriented particles with phase transition, 2012), in which the existence and stability of the equilibrium states were investigated. When the density is lower than a threshold value, the dynamics is described by a nonlinear diffusion equation. By contrast, when the density is larger than this threshold value, the dynamics is described by a similar hydrodynamic model for selfalignment interactions as derived in (Degond and Motsch, Math. Models Methods Appl. Sci. 18:11931215, 2008a; Frouvelle, Math. Models Methods Appl. Sci., 2012). However, the modified normalization of the force gives rise to different convection speeds, and the resulting model may lose its hyperbolicity in some regions of the state space. © 2012 Springer Science+Business Media New York.


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