First-order aggregation models with alignment

TL;DR

This paper extends a first-order aggregation model by incorporating alignment interactions, resulting in an implicit velocity equation with potential non-uniqueness, and analyzes its well-posedness and numerical solutions.

Contribution

It introduces an implicit velocity formulation with alignment in a macroscopic aggregation model and rigorously studies its well-posedness and numerical implementation.

Findings

The model can be derived as a macroscopic limit of a second-order kinetic equation.

The implicit velocity equation may have multiple solutions.

Numerical schemes successfully approximate the model in 1D and 2D.

Abstract

We include alignment interactions in a well-studied first-order attractive-repulsive macroscopic model for aggregation. The distinctive feature of the extended model is that the equation that specifies the velocity in terms of the population density, becomes {\em implicit}, and can have non-unique solutions. We investigate the well-posedness of the model and show rigorously how it can be obtained as a macroscopic limit of a second-order kinetic equation. We work within the space of probability measures with compact support and use mass transportation ideas and the characteristic method as essential tools in the analysis. A discretization procedure that parallels the analysis is formulated and implemented numerically in one and two dimensions.