# Graduate Student Seminar-Analysis and Modeling of Self-organized Systems with Long Range Interaction

Title: Analysis and Modeling of Self-organized Systems with Long Range Interaction

Speaker: Chong Wang

Date and Time: Monday, April 16, 2:30-3:30pm

Place: Rome 204

Place: Rome 204

Abstract: Energy-driven pattern formation induced by competing short and long range interaction is common in many biological and physical systems. We report on our work through two models. The sharp interface model is a nonlocal and non-convex geometric variational problem. The admissible class of the energy functional is a collection of sets where each set is of finite perimeter. The original problem is recast as a variational problem on a Hilbert space through introducing internal variables. We prove the existence of the core-shell assemblies and the existence of the disc assemblies as the stationary points of the energy functional in ternary systems. We also prove the existence of a triple bubble in a quaternary system. The other model is the diffuse interface model concerning minimizers of the Ginzburg-Landau free energy supplemented with long range interaction in inhibitory systems. As model parameters vary, a large number of morphological phases appear as stable stationary states. One open question related to the polarity direction of double-bubble assemblies is answered numerically. More importantly, it is shown that the average size of bubbles in a single-bubble assembly does not depend on the ratio of volume fractions but rather is determined by the long range interaction coefficients and the sum of the minority constituent volumes. In double-bubble assemblies, a two-thirds power law between the number of double bubbles and the long range interaction coefficients in the strong segregation regime is justified both numerically and theoretically. A range of parameters is identified that yields double-bubble assemblies. These two models can be connected through gamma convergence.