Communications in Mathematical Sciences

Volume 15 (2017)

Number 8

Existence of weak solutions to a kinetic flocking model with cut-off interaction function

Pages: 2177 – 2193

DOI: http://dx.doi.org/10.4310/CMS.2017.v15.n8.a4

Author

Chunyin Jin (Beijing International Center for Mathematical Research, Peking University, Beijing, China)

Abstract

We prove the existence of weak solutions to a kinetic flocking model with cut-off interaction function by using the weak convergence method. Under the natural assumption that the $v$-support of the initial distribution function $f_0(x,v)$ is bounded, we show that the $v$-support of the distribution function $f(t,x,v)$ is uniformly bounded in time. Employing this property, we remove the constraint in the paper of Karper, Mellet, and Trivisa (SIAM. J. Math. Anal., 45, 215–243, 2013) that the initial distribution function should have better integrability for large $\lvert x \rvert$.

Keywords

Cucker–Smale model, kinetic flocking model, velocity averaging lemma, Schauder fixed point theorem

2010 Mathematics Subject Classification

35D30, 35Q83, 35Q92, 74H20

Full Text (PDF format)

Paper received on 29 April 2017.

Paper accepted on 23 July 2017.