We show that degenerate nonlinear diffusion equations can be asymptotically obtained as a limit from a class of nonlocal partial differential equations. The nonlocal equations are obtained as gradient flows of interaction-like energies approximating the internal energy. We construct weak solutions as the limit of a (sub)sequence of weak measure solutions by using the Jordan-Kinderlehrer-Otto scheme from the context of 2-Wasserstein gradient flows. Our strategy allows to cover the porous medium equation, for the general slow diffusion case, extending previous results in the literature. As a byproduct of our analysis, we provide a qualitative particle approximation.

我们证明，退化非线性扩散方程可以作为一类非局部偏微分方程的极限渐近获得。非局部方程是作为近似内能的类相互作用能量的梯度流而获得的。我们通过使用 2-Wasserstein 梯度流上下文中的 Jordan-Kinderlehrer-Otto 方案，将弱解构造为弱测度解（子）序列的极限。我们的策略允许覆盖多孔介质方程，对于一般的慢扩散情况，扩展了文献中的先前结果。作为我们分析的副产品，我们提供了定性粒子近似。