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ACMS Abstracts: Fall 2017

Jinzi Mac Huang (Courant)

Sculpting of a dissolving body

In geology, dissolution in fluids leads to natural pattern formations. For example the Karst topography occurs when water dissolves limestone, and travertine terraces form as a balance of dissolution and precipitation. In this talk, we consider the shape dynamics of a soluble object immersed in water, with either external flow imposed or convective flow under gravity. We find that different flow configurations lead to different shape dynamics, for example a terminal self-similar shape emerges from dissolving in external flow, while fine scale patterns form when no external flow is imposed. We also find that under gravity, a dissolving body with initially smooth surface evolves into an increasingly sharp needle shape. A mathematical model predicts that a geometric shock forms at the tip of dissolved body, with the tip curvature becoming infinite in finite time.

Dongnam Ko (Seoul National Univ.)

On the emergence of local flocking phenomena in Cucker-Smale ensembles

Emergence of flocking groups are often observed in many complex network systems. The Cucker-Smale model is one of the flocking model, which describes the dynamics of attracting particles. This talk concerns time-asymptotic behaviors of Cucker-Smale particle ensembles, especially for mono-cluster and bi-cluster flockings. The emergence of flocking phenomena is determined by sufficient initial conditions, coupling strength, and communication weight decay. Our asymptotic analysis uses the Lyapunov functional approach and a Lagrangian formulation of the coupled system. We derive a system of differential inequalities for the functionals that measure the local fluctuations and group separations along particle trajectories. The bootstrapping argument is the key idea to prove the gathering and separating behaviors of Cucker-Smale particles simultaneously.