Emergent dynamics of Cucker--Smale particles under the effects of random communication and incompressible fluids

Seminar | April 27, 2017 | 2:30-3:30 p.m. | No. 224 room in Nanhai Building

Speaker: XIAO Qing-hua
Sponsor: JNU Information and Technological Institute

ABOUT XIAO QINGHUA
Dr. Xiao Qinghua graduated from Wuhan University, now is the associate researcher and graduate tutor in Wuhan Institute of Physics and Mathematics of Chinese Academy of Sciences. Mainly engaged in the study of nonlinear partial differential equations, Dr. Xiao has made a series of work on the Boltzmann equation and the stability of the dynamical solutions associated with the Cucker Smale equation and the stability of the hyperbolic partial differential equations. He also published nearly 20 papers in JFA, SIAM J .Math.Anal, J. Stat. Phys., J. Differential Equations and other international famous journals.
ABOUT THE LECTURE
We study the dynamics of infinitely many Cucker-Smale(C-S) flocking particles under the interplay of a random communication and incompressible fluid. For the dynamics of ensemble of flocking particles, we use the kinetic Cucker-Smale-Fokker-Planck (CS-FP) equation with a degenerate diffusion coefficient, whereas for the fluid part, we use the incompressible Navier-Stokes(N-S) equations. These two subsystems are coupled via the drag force. For this coupled model, we present global existence of weak and strong solutions in $bbr^d $ $(d=2 or 3)$. Under extra regularity assumptions of initial data, the unique solvability of strong solutions is also established in $bbr^2$. In a large coupling regime and a periodic spatial domain, we show that the velocities of C-S particles and fluids are asymptotically aligned to constant velocities in a two-dimensional periodic spatial domain $bbt^2:=bbr^2/bbz^2$.