Article Contents
Article Contents

Equilibria of an anisotropic nonlocal interaction equation: Analysis and numerics

• 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.

Mathematics Subject Classification: Primary: 35B33, 65M08; Secondary: 35B36, 35Q92, 65M12.

 Citation:

• Figure 1.  Stationary solution to the anisotropic interaction equation (10), obtained with the numerical scheme (33) on a grid of size 50 in each spatial direction and different diffusion coefficients $\delta$ for the spatially homogeneous tensor field with $s = (0,1)$ and $l = (1,0)$ and uniformly distributed initial data on a disc on the computational domain $[-0.5,0.5]^2$

Figure 2.  Stationary solution to the anisotropic interaction equation (10), obtained with the numerical scheme (33) on grids of sizes 50,100 and 200 in each spatial direction for the diffusion coefficient $\delta = 10^{-10}$ for the spatially homogeneous tensor field with $s = (0,1)$ and $l = (1,0)$ and uniformly distributed initial data on a disc on the computational domain $[-0.5,0.5]^2$

Figure 3.  Stationary solution to the anisotropic interaction equation (10), obtained with the numerical scheme (33) on grids of sizes 100 and 200 in each spatial direction for the diffusion coefficient $\delta = 10^{-10}$ for the spatially homogeneous tensor field with $s = (0,1)$ and $l = (1,0)$ and uniformly distributed initial data on the computational domain $[-0.5,0.5]^2$

Figure 4.  Stationary solution to the anisotropic interaction equation (10), obtained with the numerical scheme (33) on a grid of size 200 in each spatial direction and different diffusion coefficients $\delta$ for the spatially homogeneous tensor field with $s = (0,1)$ and $l = (1,0)$ and uniformly distributed initial data on the computational domain $[-0.5,0.5]^2$

Figure 5.  Cross-section of stationary solution to the anisotropic interaction equation (10), obtained with the numerical scheme (33) on a grid of size 200 in each spatial direction and diffusion coefficient $\delta = 10^{-9}$ for the spatially homogeneous tensor field with $s = (0,1)$ and $l = (1,0)$ and uniformly distributed initial data on the computational domain $[-0.5,0.5]^2$

Figure 6.  Stationary solution to the anisotropic interaction equation (10), obtained with the numerical scheme (33) on a grid of size 50 in each spatial direction and diffusion coefficient $\delta = 10^{-10}$ for different spatially inhomogeneous tensor fields from real fingerprint images and uniformly distributed initial data on the computational domain $[-0.5,0.5]^2$

Figure 7.  Numerical solution to the anisotropic interaction equation (10) after $n$ iterations for different $n$, obtained with the numerical scheme (33) on a grid of size 50 in each spatial direction with diffusion coefficient $\delta = 10^{-10}$ for the spatially inhomogeneous tensor field of part of a fingerprint and uniformly distributed initial data on the computational domain $[-0.5,0.5]^2$

Figure 8.  Stationary solution to the anisotropic interaction equation (10), obtained with the numerical scheme (33) on a grid of size 50 in each spatial direction for different values of the diffusion coefficient $\delta$ for a given spatially inhomogeneous tensor field and uniformly distributed initial data on the computational domain $[-0.5,0.5]^2$

Figure 9.  Stationary solution to the anisotropic interaction equation (10), obtained with the numerical scheme (33) on a grid of size 50 in each spatial direction, diffusion coefficient $\delta = 10^{-10}$ and different force rescalings $\eta$ for a given spatially inhomogeneous tensor field and uniformly distributed initial data on the computational domain $[-0.5,0.5]^2$

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