Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics

TL;DR

This paper analyzes equilibrium solutions of an anisotropic nonlocal aggregation equation with nonlinear diffusion, deriving conditions, studying dimension reduction, and validating results through convergence proofs and numerical simulations.

Contribution

It introduces a novel analysis of anisotropic aggregation equilibria without a gradient flow structure, including energy functional convergence and numerical scheme validation.

Findings

Equilibrium conditions for stationary line patterns are derived.

Gamma-convergence of energy functionals is established as diffusion vanishes.

Numerical scheme convergence is proven and demonstrated with results.

Abstract

In this paper, we study the equilibria of an anisotropic, nonlocal aggregation equation with nonlinear diffusion which does not possess a gradient flow structure. Here, the anisotropy is induced by an underlying tensor field. Anisotropic forces cannot be associated with a potential in general and stationary solutions of anisotropic aggregation equations generally cannot be regarded as minimizers of an energy functional. We derive equilibrium conditions for stationary line patterns in the setting of spatially homogeneous tensor fields. The stationary solutions can be regarded as the minimizers of a regularised energy functional depending on a scalar potential. A dimension reduction from the two- to the one-dimensional setting allows us to study the associated one-dimensional problem instead of the two-dimensional setting. We establish $\Gamma$-convergence of the regularised energy functionals as the diffusion coefficient vanishes, and prove the convergence of minimisers of the regularised energy functional to minimisers of the non-regularised energy functional. Further, we investigate properties of stationary solutions on the torus, based on known results in one spatial dimension. Finally, we prove weak convergence of a numerical scheme for the numerical solution of the anisotropic, nonlocal aggregation equation with nonlinear diffusion and any underlying tensor field, and show numerical results.