Minimization of an Energy Defined via an Attractive-Repulsive Interaction Potential Related to Nonlocal Aggregation Models

Ihsan Topaloglu
McGill University and CRM Applied Math Lab

Recently, aggregation models given by the active transport equation of the form $\rho_t - \nabla \cdot (\rho(\nabla K \ast \rho)) = 0$, where $\rho$ denotes the density of aggregation and $K$ is an interaction potential, have received much attention. This PDE is, indeed, the gradient flow of the energy $E[\rho]=\iint_{\mathbb{R}^n\times\mathbb{R}^n} K(x-y)\rho(x)\rho(y)dxdy$ with respect to the Wasserstein metric. These models have a wide range of applications including biological swarms, granular media and self-assembly of nanoparticles.

In this talk I will concentrate on the energy $E$ defined via interaction potentials of the form $K(|x|)=|x|^q/q - |x|^p/p$ in the parameter regime $ -n < p < q $. After establishing the existence of minimizers of the constrained minimization problem, I will consider the ground states in certain parameter regimes of the powers $p$ and $q$. Moreover, I will comment on the minimizers of the energy $E$ defined over binary densities $\rho\in\{0,1\}$.

This is a joint work with Rustum Choksi and Razvan Fetecau.