Variational approach to coarse-graining of generalized gradient flows
File(s)
Published version
Author(s)
Duong, Manh Hong
Lamacz, Agnes
Peletier, Mark A
Sharma, Upanshu
Type
Journal Article
Abstract
In this paper we present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (a) a natural interaction between the duality structure and the coarse-graining, (b) application to systems with non-dissipative effects, and (c) application to coarse-graining of approximate solutions which solve the equation only to some error. As examples, we use this technique to solve three limit problems, the overdamped limit of the Vlasov–Fokker–Planck equation and the small-noise limit of randomly perturbed Hamiltonian systems with one and with many degrees of freedom.
Date Issued
2017-06-28
Date Acceptance
2017-05-04
Citation
Calculus of Variations and Partial Differential Equations, 2017, 56 (4)
ISSN
0944-2669
Publisher
Springer
Journal / Book Title
Calculus of Variations and Partial Differential Equations
Volume
56
Issue
4
Copyright Statement
© 2017 The Author(s). This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Identifier
https://doi.org/10.1007/s00526-017-1186-9
Subjects
Science & Technology
Physical Sciences
Mathematics, Applied
Mathematics
SMOLUCHOWSKI-KRAMERS APPROXIMATION
FOKKER-PLANCK EQUATION
NON-LINEAR DIFFUSION
AVERAGING PRINCIPLE
DIFFERENTIAL-EQUATIONS
RANDOM PERTURBATIONS
PARABOLIC EQUATIONS
GAMMA-CONVERGENCE
LIMIT
HOMOGENIZATION
0101 Pure Mathematics
0102 Applied Mathematics
General Mathematics
Notes
mrclass: 35K67 (35B25 35K10 35K20 49J45 49S99 60F10 70F40) mrnumber: 3666792 mrreviewer: Joseph L. Shomberg
Publication Status
Published
Article Number
100
Date Publish Online
2017-06-28