Emergence of bi-cluster flocking for the Cucker–Smale model

We present the mathematical analysis of bi-cluster flocking phenomenon for the short-ranged Cucker–Smale model with some well-prepared initial data. For this, we derive a system of differential inequalities for the functionals measuring the local spatial and velocity fluctuations and differences of local velocity averages, and then estimate the upper bound of spatial fluctuations and the lower bound of the difference between local velocity averages. We explicitly present an admissible class of initial configurations leading to the asymptotic emergence of bi-cluster flocking phenomenon. Unlike global flocking (a mono-cluster flocking configuration in velocity), where the convergence rate is always exponential, the asymptotic convergence to bi-cluster flocking configurations is affected by the far-field decay rate of communication weights so that it can be algebraic.

HOMOGENIZATION OF BOUNDARY VALUE PROBLEMS AND SPECTRAL PROBLEMS FOR NEUTRON TRANSPORT IN LOCALLY PERIODIC MEDIA

This paper deals with boundary value problems and spectral problems for neutron transport equations involving locally periodic (in space) collision frequencies and collision operators. We show strong convergence results to the solution of homogenized problems when the period goes to zero. The mathematical analysis relies mainly on smoothing effects of velocity averages.