论文标题
多孔培养基方程和交叉扩散系统作为非局部相互作用的极限
Porous medium equation and cross-diffusion systems as limit of nonlocal interaction
论文作者
论文摘要
本文研究了非本地相互作用的二次多孔培养基方程和一类交叉扩散系统的推导。我们证明了非局部相互作用方程的溶液的收敛。系统，用于二次多孔介质方程的解决方案。交叉扩散系统，处于局部相互作用内核的极限。分析是在（非本地）部分微分方程的水平上进行的，我们使用梯度流技术来得出能量，二阶矩和对数熵的界限。后者的耗散产生了足够的规律性以获得紧凑的结果并将其传递到局部卷积的极限。我们提出的策略依赖于离散化方案，该方案可以稍作修改，以将我们的结果扩展到没有梯度流结构的PDE。特别是，它不需要相关能量的凸度。我们的分析允许在相关低规律性下以限制非粘性多孔培养基方程的弱解，假设初始值具有有限的能量和熵。
This paper studies the derivation of the quadratic porous medium equation and a class of cross-diffusion systems from nonlocal interactions. We prove convergence of solutions of a nonlocal interaction equation, resp. system, to solutions of the quadratic porous medium equation, resp. cross-diffusion system, in the limit of a localising interaction kernel. The analysis is carried out at the level of the (nonlocal) partial differential equations and we use gradient flow techniques to derive bounds on energy, second order moments, and logarithmic entropy. The dissipation of the latter yields sufficient regularity to obtain compactness results and pass to the limit in the localised convolutions. The strategy we propose relies on a discretisation scheme, which can be slightly modified in order to extend our result to PDEs without gradient flow structure. In particular, it does not require convexity of the associated energies. Our analysis allows to treat the case of limiting weak solutions of the non-viscous porous medium equation at relevant low regularity, assuming the initial value to have finite energy and entropy.
