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

作者：Jungel Ansgar; Zamponi Nicola

来源：Zeitschrift fur Angewandte Mathematik und Physik, 2017, 68(1): 28.

DOI：10.1007/s00033-017-0772-1

• Cross-diffusion system
• Fokker-Planck equation
• Entropy methods
• Global existence of weak solutions
• Large-time asymptotics
• Positivity of solutions

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 It (o) over bar 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.

• 出版日期2017-2