Mean Field Limit of Interacting Filaments and Vector Valued Non-linear PDEs

详细信息    查看全文

• 作者：Hakima Bessaih ; Michele Coghi ; Franco Flandoli
• 关键词：Currents ; Mean field theory ; Propagation of chaos ; (random) vortex filaments ; Vector ; valued PDEs
• 刊名：Journal of Statistical Physics
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：166
• 期：5
• 页码：1276-1309
• 全文大小：
• 刊物类别：Physics and Astronomy
• 刊物主题：Statistical Physics and Dynamical Systems; Theoretical, Mathematical and Computational Physics; Physical Chemistry; Quantum Physics;
• 出版者：Springer US
• ISSN：1572-9613
• 卷排序：166

文摘

Families of N interacting curves are considered, with long range, mean field type, interaction. They generalize models based on classical interacting point particles to models based on curves. In this new set-up, a mean field result is proven, as \(N\rightarrow \infty \). The limit PDE is vector valued and, in the limit, each curve interacts with a mean field solution of the PDE. This target is reached by a careful formulation of curves and weak solutions of the PDE which makes use of 1-currents and their topologies. The main results are based on the analysis of a nonlinear Lagrangian-type flow equation. Most of the results are deterministic; as a by-product, when the initial conditions are given by families of independent random curves, we prove a propagation of chaos result. The results are local in time for general interaction kernel, global in time under some additional restriction. Our main motivation is the approximation of 3D-inviscid flow dynamics by the interacting dynamics of a large number of vortex filaments, as observed in certain turbulent fluids; in this respect, the present paper is restricted to smoothed interaction kernels, instead of the true Biot–Savart kernel.