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Global existence and uniqueness of measure valued solutions to a Vlasov-type equation with local alignment. (English) Zbl 1384.35128

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

Citations:

Zbl 1230.82037
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