Anton Arnold, Jan Erb

"Entropy method for hypocoercive & non-symmetric Fokker-Planck equations with linear
drift"


In the last 15 years the entropy method has become a powerful tool for analyzing the
large-time behavior of the Cauchy problem for linear and non-linear Fokker-Planck
type equations (advection-diffusion equations). These problems appear, e.g., as the
kinetic Fokker-Planck equation of plasma physics, porous media equations, or
electron transport models for
semiconductor devices. In particular, this entropy method can be used to analyze
the rate of convergence to the equilibrium, and in parallel to prove the validity of
logarithmic Sobolev
inequalities and a variety of Poincare-type inequalities.
The essence of the method is to first derive a differential inequality between
the first and second time derivative of the relative entropy, and then between the
entropy dissipation and the entropy.  

For degenerate parabolic equations, the entropy dissipation may vanish for states
other than the equilibrium. Hence, the standard entropy method does not carry over. 
For hypocoercive Fokker-Planck equations we introduce an auxiliary functional (of
entropy dissipation type) to prove exponential decay of the solution towards the
steady state in relative entropy.  We show that the obtained rate is indeed sharp
(both for the logarithmic and quadratic entropy). Finally, we extend the method to
the kinetic Fokker-Planck equation (with non-quadratic potential) and
non-degenerate, non-symmetric Fokker-Planck equations.

References:

1) A. Arnold, J. Erb. Sharp entropy decay for hypocoercive and non-symmetric
Fokker-Planck equations with linear drift, arXiv 2014.

2) A. Arnold, P. Markowich, G. Toscani, A. Unterreiter. On logarithmic Sobolev
inequalities
and the rate of convergence to equilibrium for Fokker-Planck type equations.
   Comm. PDE 26/1-2 (2001) 43-100.