Friday, November 22, 2019 | 1:00pm | Yost 107

PhD Candidate: Robert Volkin

Program: Applied Mathematics

Advisor: Dr. Alethea Barbaro

Title: Spherical Solutions for the Radially Symmetric Aggregation Equation: Analysis and a Novel Numerical Method

Abstract

Biological models of collective motion have become a rich subject of study in kinetic theory. These models treat biological agents as particles with positions and velocities that evolve according to interaction rules with the other agents. Such models can be studied at different levels of abstraction ranging from discrete space and time individual particle systems to continuous particle density models.

We focus on certain nonlocal equations with a radially symmetric, repulsive-attractive interaction potential called the aggregation equation. A density of particles evolves according to a continuity equation whose dynamics are driven by the potential, which we take to follow a power law.

By analyzing the associated interaction energy functional, we deduce the existence of special distributional solutions in which particles concentrate on spherical shells for all time. For spherical shell initial data, we determine exact solutions for certain attractive and repulsive powers. For more general parameter choices, we show exponential convergence of solutions to steady-state. When the initial data is formed from convex combinations of spherical shells, we examine the connection between their steady-state solutions and those for continuous initial data.

Furthermore, we use the existence of these spherical shell solutions to construct a novel numerical method that lies between a particle and continuous density model. We use this method to investigate continuous steady-states and properties of the energy functional for solutions.

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A reception will be held following the defense in Yost 207.