Title: Convergence of a finite-volume scheme for aggregation-diffusion equations with saturation Show Chinese title

Title: 一种有限体积格式在具有饱和的聚集-扩散方程中的收敛性

Abstract: In [Bailo, Carrillo, Hu. SIAM J. Appl. Math. 2023] the authors introduce a finite-volume method for aggregation-diffusion equations with non-linear mobility. In this paper we prove convergence of this method using an Aubin--Simons compactness theorem due to Gallou\"et and Latch\'e. We use suitable discrete $H^1$ and $W^{-1,1}$ discrete norms. We provide two convergence results. A first result shows convergence with general entropies ($U$) (including singular and degenerate) if the initial datum does not have free boundaries, the mobility is Lipschitz, and the confinement ($V$) and aggregation ($K$) potentials are $W^{2,\infty}_0$. A second result shows convergence when the initial datum has free boundaries, mobility is just continuous, and $V$ and $K$ are $W^{1,\infty}$, but under the assumption that the entropy $U$ is $C^1$ and strictly convex. Show Chinese abstract

Abstract: 在[Bailo, Carrillo, Hu. SIAM J. Appl. Math. 2023]中，作者引入了一种用于具有非线性迁移率的聚集-扩散方程的有限体积方法。在本文中，我们使用Gallouët和Latché的Aubin--Simons紧性定理证明了该方法的收敛性。我们使用合适的离散$H^1$和$W^{-1,1}$离散范数。我们提供了两个收敛结果。第一个结果表明，如果初始数据没有自由边界，迁移率是Lipschitz连续的，并且约束（$V$）和聚集（$K$）势是$W^{2,\infty}_0$，则使用一般的熵（$U$）（包括奇异和退化的情况）可以得到收敛。 一个第二个结果表明，当初始数据具有自由边界时，迁移率只是连续的，且$V$和$K$是$W^{1,\infty}$，但假设熵$U$是$C^1$且严格凸。