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.

中文翻译：

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各向异性非局部相互作用方程的平衡：分析和数值

在本文中，我们研究了具有非线性扩散且不具有梯度流结构的各向异性非局部聚集方程的平衡。在这里，各向异性是由下面的张量场引起的。各向异性力通常不能与势相关，各向异性聚集方程的固定解通常不能视为能量函数的极小值。我们在空间齐次张量场的设置中得出静止线模式的平衡条件。取决于标量势，固定解可以看作是正则化能量函数的极小值。从二维设置到一维设置，我们可以研究相关的一维问题，而不是二维设置。随着扩散系数的消失，我们建立了正则化能量泛函的γ收敛，并证明了正则化能量泛函的极小值向非正则化能量泛函的极小值的收敛性。此外，我们基于一个空间维度上的已知结果，研究圆环上固定解的性质。最后，我们证明了具有非线性扩散和任何潜在张量场的各向异性非局部聚集方程数值解的数值格式的弱收敛性，并给出了数值结果。我们基于一个空间维度上的已知结果，研究圆环上固定解的性质。最后，我们证明了具有非线性扩散和任何潜在张量场的各向异性非局部聚集方程数值解的数值格式的弱收敛性，并给出了数值结果。我们基于一个空间维度上的已知结果，研究圆环上固定解的性质。最后，我们证明了具有非线性扩散和任何基础张量场的各向异性非局部聚集方程数值解的数值格式的弱收敛性，并给出了数值结果。