On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear Fokker-Planck equation (Q2710672)

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.