We study an implicit finite-volume scheme for nonlinear, non-local aggregation-diffusion equations which exhibit a gradient-flow structure, recently introduced in [R. Bailo, J. A. Carrillo and J. Hu, Fully discrete positivity-preserving and energy-dissipating schemes for aggregation-diffusion equations with a gradient flow structure, arXiv:1811.11502 ]. Crucially, this scheme keeps the dissipation property of an associated fully discrete energy, and does so unconditionally with respect to the time step. Our main contribution in this work is to show the convergence of the method under suitable assumptions on the diffusion functions and potentials involved.

中文翻译：

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聚合扩散方程的完全离散和能量耗散有限体积方案的收敛

我们研究了非线性、非局部聚集扩散方程的隐式有限体积方案，该方程呈现出梯度流动结构，最近在 [R. Bailo, JA Carrillo 和 J. Hu，具有梯度流结构的聚合扩散方程的完全离散正性保持和能量耗散方案，arXiv:1811.11502]。至关重要的是，该方案保持了相关的完全离散能量的耗散特性，并且相对于时间步无条件地这样做。我们在这项工作中的主要贡献是在对所涉及的扩散函数和势能的适当假设下展示该方法的收敛性。