Gao, Yu; Xue, Xiaoping Global existence and uniqueness of measure valued solutions to a Vlasov-type equation with local alignment. (English) Zbl 1384.35128 Math. Methods Appl. Sci. 40, No. 18, 7640-7662 (2017). Summary: We use a particle method to study a Vlasov-type equation with local alignment, which was proposed by S. Motsch and E. Tadmor [J. Stat. Phys. 144, No. 5, 923–947 (2011; Zbl 1230.82037)]. For \(N\)-particle system, we study the unconditional flocking behavior for a weighted Motsch-Tadmor model and a model with a “tail”. When \(N\) goes to infinity, global existence and stability (hence uniqueness) of measure valued solutions to the kinetic equation of this model are obtained. We also prove that measure valued solutions converge to a flock. The main tool we use in this paper is Monge-Kantorovich-Rubinstein distance. MSC: 35Q83 Vlasov equations 35B40 Asymptotic behavior of solutions to PDEs 92D50 Animal behavior 35R06 PDEs with measure 92C15 Developmental biology, pattern formation 35B35 Stability in context of PDEs 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness 35Q92 PDEs in connection with biology, chemistry and other natural sciences Keywords:flocking; Monge-Kantorovich-Rubinstein distance; particle method Citations:Zbl 1230.82037 PDFBibTeX XMLCite \textit{Y. Gao} and \textit{X. Xue}, Math. Methods Appl. Sci. 40, No. 18, 7640--7662 (2017; Zbl 1384.35128) Full Text: DOI