| Contents |
| We consider a second-order model of self-propelled interacting agents, which has been frequently used to model complex behavior of swarms such as fish schools or birds flocks. Particular solutions such as aligned flocks and rotating mills can be found on the particle level and the associated kinetic PDEs. We present some recent advances in the analysis of such solutions: First, we consider the particle system and show that nonlinear stability of flock solutions is inherited from the first-order aggregation equation to the second-order model. Flocks are shown to be stable as a family of particular solutions \footnote{
J.A. Carrillo, Y. Huang, S. Martin:
\emph{Nonlinear stability of flock solutions in second-order swarming models}, {Nonlinear Analysis: Real World Applications}, vol. 17 (2014), pp. 332 - 343
}.
Second, we turn to the mean-field equations and present existence and uniqueness results for the so-called Quasi-Morse potential, which is a special interaction potential allowing for an explicit expression of density profiles of particular solutions in terms of special functions (in 2D and 3D)\footnote{
J.A. Carrillo, Y. Huang, S. Martin:
\emph{Explicit Flock Solutions for Quasi-Morse potentials}, preprint: \href{http://arxiv.org/abs/1308.2883}{arxiv/1308.2883}
}
\footnote{
J. A. Carrillo, S. Martin, V. Panferov: \href{http://dx.doi.org/10.1016/j.physd.2013.02.004}{\emph{A new interaction potential for swarming models}}, Physica D - Nonlinear Phenomena, vol. 260 (2013), pp. 112-126
}. |
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