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

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. This 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

[1]  Maria Weber.  The fundamental solution of a degenerate partial differential equation of parabolic type , 1951 .

[2]  A. J. Stam.  Some Inequalities Satisfied by the Quantities of Information of Fisher and Shannon , 1959, Inf. Control..

[3]  Amiel Feinstein,et al.  Information and information stability of random variables and processes , 1964 .

[4]  H. Grad.  On Boltzmann’s H-Theorem , 1965 .

[5]  S. Kullback,et al.  A lower bound for discrimination information in terms of variation (Corresp.) , 1967, IEEE Trans. Inf. Theory.

[6]  L. Hörmander.  Hypoelliptic second order differential equations , 1967 .

[7]  Clifford Ambrose Truesdell,et al.  Fundamentals of Maxwell's kinetic theory of a simple monatomic gas , 1980 .

[8]  G. Toscani.  H-theorem and asymptotic trend of the solution for a rarefied gas in the vacuum , 1987 .

[9]  Leif Arkeryd,et al.  Stability in L1 for the spatially homogenous Boltzmann equation , 1988 .

[10]  H. Risken,et al.  The Fokker-Planck Equation; Second Edition , 1989 .

[11]  L. Desvillettes.  Convergence to equilibrium in large time for Boltzmann and B.G.K. equations , 1990 .

[12]  Scaling limit for interacting Ornstein-Uhlenbeck processes , 1991 .

[13]  Eric A. Carlen,et al.  Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation , 1992 .

[14]  L. Arkeryd.  On the strong L1 trend to equilibrium for the Boltzmann equation , 1992 .

[15]  L. Gross.  Logarithmic Sobolev inequalities and contractivity properties of semigroups , 1993 .

[16]  M. Ledoux.  L'algèbre de Lie des gradients itérés d'un générateur markovien , 1993 .

[17]  Eric A. Carlen,et al.  Entropy production estimates for Boltzmann equations with physically realistic collision kernels , 1994 .

[18]  P. Lions,et al.  Compactness in Boltzmann’s equation via Fourier integral operators and applications. III , 1994 .

[19]  R. Illner,et al.  The mathematical theory of dilute gases , 1994 .

[20]  François Bouchut,et al.  On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with Coulombic and Newtonian potentials , 1995, Differential and Integral Equations.

[21]  B. Wennberg.  Stability and exponential convergence for the Boltzmann equation , 1995 .

[22]  Laurent Desvillettes,et al.  About the regularizing properties of the non-cut-off Kac equation , 1995 .

[23]  Cédric Villani,et al.  On the spatially homogeneous landau equation for hard potentials part i : existence, uniqueness and smoothness , 2000 .

[24]  Benoît Perthame,et al.  Boltzmann equation with infinite energy: renormalized solutions and distributional solutions for small initial data and initial data close to a Maxwellian , 1997 .

[25]  Giuseppe Toscani,et al.  Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation , 1999 .

[26]  Xuguang Lu,et al.  Spatial decay solution of the Boltzmann equation: converse properties of long time limiting behavior , 1999 .

[27]  Giuseppe Toscani,et al.  Sharp Entropy Dissipation Bounds and Explicit Rate of Trend to Equilibrium for the Spatially Homogeneous Boltzmann Equation , 1999 .

[28]  Giuseppe Toscani,et al.  On the Trend to Equilibrium for Some Dissipative Systems with Slowly Increasing a Priori Bounds , 2000 .

[29]  Claudia Garetto.  Pseudo-Differential Operators in Algebras of Generalized Functions and Global Hypoellipticity , 2005, math/0502283.