A cross-diffusion system derived from a Fokker–Planck equation with partial averaging

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• 作者：Ansgar Jüngel ; Nicola Zamponi
• 关键词：Cross ; diffusion system ; Fokker–Planck equation ; Entropy methods ; Global existence of weak solutions ; Large ; time asymptotics ; Positivity of solutions
• 刊名：Zeitschrift für angewandte Mathematik und Physik
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：68
• 期：1
• 全文大小：568KB
• 刊物主题：Theoretical and Applied Mechanics; Mathematical Methods in Physics;
• 出版者：Springer International Publishing
• ISSN：1420-9039
• 卷排序：68

文摘

A cross-diffusion system for two components with a Laplacian structure is analyzed on the multi-dimensional torus. This system, which was recently suggested by P.-L. Lions, is formally derived from a Fokker–Planck equation for the probability density associated with a multi-dimensional Itō process, assuming that the diffusion coefficients depend on partial averages of the probability density with exponential weights. A main feature is that the diffusion matrix of the limiting cross-diffusion system is generally neither symmetric nor positive definite, but its structure allows for the use of entropy methods. The global-in-time existence of positive weak solutions is proved and, under a simplifying assumption, the large-time asymptotics is investigated.