Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics

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.

References in corpus (13)

• A Finite-Volume Method for Nonlinear Nonlocal Equations with a Gradient Flow Structure
• The derivation of Swarming models: Mean-Field Limit and Wasserstein distances
• On the Dynamics of Large Particle Systems in the Mean Field Limit
• Dimensionality of Local Minimizers of the Interaction Energy
• Existence of Compactly Supported Global Minimisers for the Interaction Energy
• Existence of Ground States of Nonlocal-Interaction Energies
• The Filippov characteristic flow for the aggregation equation with mildly singular potentials
• On global minimizers of repulsive-attractive power-law interaction energies
• Existence of ground states for aggregation-diffusion equations
• Pattern formation of a nonlocal, anisotropic interaction model
• An Anisotropic Interaction Model for Simulating Fingerprints
• Stability analysis of line patterns of an anisotropic interaction model
• Uniqueness and non-uniqueness of steady states of aggregation-diffusion equations