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 Γ-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.

Mean-field optimal control for biological pattern formation

“…The unusual form of the attraction force F A is motivated by [5,10,11,14,24] where the direction of the interaction force depends on a spatially homogeneous or inhomogeneous tensor field T = T (x) with…”

“…Note that these conditions are satisfied for the exponentially decaying force coefficients in [5,10,11,14,24]. The rather unusual assumption (2.5) is considered to guarantee physically relevant forces through periodic extension on the torus which will be introduced in the following.…”

“…An example of this class of models is the Kücken-Champod model [5,24] describing the formation of fingerprint patterns based on the interaction of N cells. The interacting particle model (1.1) has been studied in [5,10,11,14]. In particular, the particles align in line patterns.…”

Fokker-Planck modeling of many-agent systems in swarm manufacturing: asymptotic analysis and numerical results

“…Particular attention has been paid to self-organizing systems and their emerging patterns in biology, social sciences, traffic dynamics and robotics. Without intending to review the huge literature on these topics, we point the reads to [4,8,11,12,15,19,36]. In these works Langevin-type equations are designed to produce several patterns stemming from interaction forces.…”