勷勤数学•专家报告

题      目：On the Interactions of Flocking Particles with the Navier-Stokes Flow in an Infinite Channel

报  告  人： 周韬 博士  (邀请人：罗天文)

                                       香港中文大学

时      间： 10月15日  11:30-12:30

地     点：数学科学学院西楼114

报告人简介:

        周韬博士毕业于香港中文大学，研究方向为偏微分方程。

摘      要：

        In this talk, we study the global existence of weak solutions for the Cucker-Smale-Navier-Stokes system in an infinitely long cylindrical domain with specular boundary condition. This system models the collective behavior of flocking particles in an incompressible viscous fluid, whereas the specular boundary condition allows particles to collide with the cylinder wall elastically.

      This generalize the previous results for Stokes type equations, and is the first result to study the flocking particles interacting with Navier-Stokes flows in an infinitely long channel. This involves substantial analytical challenges, especially the treatments of nonlinear convection term. In this work, we overcome these difficulties by employing Ladyzhenskaya's inequality and Gronwall's lemma. Moreover, we establish the flocking estimates and demonstrate convergence toward flocking states asymptotically.

      Additionally, we perform several numerical simulations to compare the effects of the Navier-Stokes system and its Stokes counterpart and examine the long-time asymptotic behaviors of the corresponding solutions. We also compare the interactions with or without flocking terms, from these simulations, we furthermore conclude that such interactions exhibit shear thinning non-Newtonian flow behaviors.

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