Existence of Weak Solutions to Kinetic Flocking Models
Trygve K. Karper, Antoine Mellet, Konstantina Trivisa · SIAM Journal on Mathematical Analysis · 2013
We establish the global existence of weak solutions to a class of kinetic flocking equations. The models under consideration include the kinetic Cucker--Smale equation [Cucker and Smale, IEEE Trans. Automat. Control, 52 (2007), pp. 852--862, Cucker and Smale, Japan. J. Math., 2 (2007), pp. 197--227] with possibly nonsymmetric flocking potential, the Cucker--Smale equation with additional strong local alignment, and a newly proposed model by Motsch and Tadmor [J. Statist. Phys., 141 (2011), pp. 923--947]. The main tools employed in the analysis are the velocity-averaging lemma and the Schauder fixed-point theorem along with various integral bounds.