We focus on a family of nonlinear continuity equations for the evolution of a non-negative density $\rho$ with a continuous and compactly supported nonlinear mobility $\mathrm{m}(\rho)$ not necessarily concave. The velocity field is the negative gradient of the variation of a free energy including internal and confinement energy terms. Problems with compactly supported mobility are often called saturation problems since the values of the density are constrained below a maximal value. Taking advantage of a family of approximating problems, we show the existence of $C_0$-semigroups of $L^1$ contractions. We study the $\omega$-limit of the problem, its most relevant properties, and the appearance of free boundaries in the long-time behaviour. This problem has a formal gradient-flow structure, and we discuss the local/global minimisers of the corresponding free energy in the natural topology related to the set of initial data for the $L^\infty$-constrained gradient flow of probability densities. Furthermore, we analyse a structure preserving implicit finite-volume scheme and discuss its convergence and long-time behaviour.

翻译：本文研究一类非线性连续性方程，用于描述具有连续紧支撑非线性迁移率$\mathrm{m}(\rho)$（未必为凹函数）的非负密度$\rho$的演化过程。速度场为自由能变分的负梯度，该自由能包含内能与约束能项。具有紧支撑迁移率的问题常被称为饱和问题，因为密度值被限制在最大值以下。通过利用一族逼近问题，我们证明了$L^1$压缩的$C_0$-半群的存在性。我们研究了该问题的$\omega$-极限及其最重要性质，并分析了长时间行为中自由边界的出现。该问题具有形式上的梯度流结构，我们在与概率密度$L^\infty$约束梯度流初始数据集相关的自然拓扑中，讨论了相应自由能的局部/全局极小元。此外，我们分析了一种保结构的隐式有限体积格式，并讨论了其收敛性与长时间行为。