On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation
type: Item
instance of: scholarly article
zbMATH Open document ID: 1029.82032
author: Cédric Villani
author: Laurent Desvillettes
publication date: 2001-04-26T00:00:00Z
Mathematics Subject Classification ID: 82C40
Mathematics Subject Classification ID: 35Q99
Mathematics Subject Classification ID: 46E99
Mathematics Subject Classification ID: 47N55
Mathematics Subject Classification ID: 60H99
Mathematics Subject Classification ID: 60J99
Mathematics Subject Classification ID: 82C70
zbMATH DE Number: 1592005
zbMATH Keywords: Fokker-Planck equation
zbMATH Keywords: logarithmic Sobolev inequality
zbMATH Keywords: entropy
zbMATH Keywords: Boltzmann equation
zbMATH Keywords: degenerate collision operator
zbMATH Keywords: kinetic equations
MaRDI profile type: Publication
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title: On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation
published in: Communications on Pure and Applied Mathematics
review text: In this interesting article the authors study the long-time behaviour of kinetic equations with a degenerate collision operator action only on the velocity variable \(v\). The main result proves a decay to equilibrium. NEWLINENEWLINENEWLINEThis is the first in a series of two papers exposing a general method to overcome the problem of infinitely many equilibria. It is based on log-Sobolev inequalities and entropy. This article explains clearly the method and applies it for simplicity of presentation to a linear Fokker-Planck equation of the type NEWLINE\[NEWLINE \partial_t f + v\nabla_x f-\nabla V (x) \cdot \nabla_v f = \nabla_v(\nabla_v f + f v) NEWLINE\]NEWLINE where \(f(t,x,v)\geq 0\) is for fixed \(t\) a probability density, and \(V\) is a smooth potential strictly convex at \(\infty\). NEWLINENEWLINENEWLINEThe main result proves a decay to equilibrium, which is faster that \({\mathcal O}(t^{-1/\varepsilon})\) for any \(\varepsilon>0\). This is not optimal for their example as the authors point out themselves. In a second article [Preprint (2003)] the method is applied to nonlinear problems of Boltzmann-type. NEWLINENEWLINENEWLINEThe article also contains a well written general overview of the problem and an extensive reference list.
DOI: 10.1002/1097-0312(200101)54:1%3C1::AID-CPA1%3E3.0.CO;2-Q
description: scientific article