Stability Analysis of Flock and Mill rings for 2nd Order Models in Swarming

G. Albi, D. Balagué, J. A. Carrillo, J. von Brecht

We study the linear stability of flock and mill ring solutions of two individual based models for biological swarming. The individuals interact via a nonlocal interaction potential that is repulsive in the short range and attractive in the long range. We relate the instability of the flock rings with the instability of the ring solution of the first order model. We observe that repulsive-attractive interactions lead to new configurations for the flock rings such as clustering and fattening formation. Finally, we numerically explore mill patterns arising from this kind of interactions together with the asymptotic speed of the system.

arXiv Fri, 19 Apr 2013
SIAM J. Appl. Math., 74(3), 794–818, 2014

Complementary Materials on Giacomo's web page.

Dimensionality of Local Minimizers of the Interaction Energy

D. Balagué, J. A. Carrillo, T. Laurent, G. Raoul

In this work we consider local minimizers (in the topology of transport distances) of the interaction energy associated to a repulsive-attractive potential. We show how the imensionality of the support of local minimizers is related to the repulsive strength of the potential at the origin.

Journal reference: Archive for Rational Mechanics and Analysis, 2013.
arXiv Thu, 25 Oct 2012

Complementary materials

Confinement for Repulsive-Attractive Kernels

D. Balagué, J. A. Carrillo, Y. Yao

We investigate the confinement properties of solutions of the aggregation equation with repulsive-attractive potentials. We show that solutions remain compactly supported in a large fixed ball depending on the initial data and the potential. The arguments apply to the functional setting of probability measures with mildly singular repulsive-attractive potentials and to the functional setting of smooth solutions with a potential being the sum of the Newtonian repulsion at the origin and a smooth suitably growing at infinity attractive potential.

Discrete and Continuous Dynamical Systems-B 19, 1227–1248, 2014.

arXiv Mon, 1 Oct 2012

Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates

Daniel Balagué, José Cañizo, Pierre Gabriel

We are concerned with the long-time behavior of the growth-fragmentation equation. We prove fine estimates on the principal eigenfunctions of the growth-fragmentation operator, giving their first-order behavior close to 0 and $+\infty$. Using these estimates we prove a spectral gap result by following the technique in [Caceres, Canizo, Mischler 2011, JMPA], which implies that solutions decay to the equilibrium exponentially fast. The growth and fragmentation coefficients we consider are quite general, essentially only assumed to behave asymptotically like power laws.

Journal reference: Kinetic and Related Models, Vol. 6, No. 2, pp. 219-243 (2013)
arXiv Submitted on 28 Mar 2012 (v1), last revised 19 Feb 2013 (v2)

Nonlocal interactions by repulsive-attractive potentials: radial ins/stability

D. Balagué, J. A. Carrillo, T. Laurent, G. Raoul

In this paper, we investigate nonlocal interaction equations with repulsive-attractive radial potentials. Such equations describe the evolution of a continuum density of particles in which they repulse each other in the short range and attract each other in the long range. We prove that under some conditions on the potential, radially symmetric solutions converge exponentially fast in some transport distance toward a spherical shell stationary state. Otherwise we prove that it is not possible for a radially symmetric solution to converge weakly toward the spherical shell stationary state. We also investigate under which condition it is possible for a non-radially symmetric solution to converge toward a singular stationary state supported on a general hypersurface. Finally we provide a detailed analysis of the specific case of the repulsive-attractive power law potential as well as numerical results. We point out the the conditions of radial ins/stability are sharp.

arXiv Sat, 24 Sep 2011
Physica D: Nonlinear Phenomena, Volume 260, 1 October 2013, Pages 5–25