Explore an advanced lecture on entropy methods for hypocoercive BGK and Fokker-Planck equations. Delve into introdegenerate Fokker-Planck equations with linear drift, focusing on a degenerate prototype with w = 1 coefficients. Review the entropy method and examine admissible relative entropies for probability densities. Analyze the exponential decay of entropy dissipation for constant D and the exponential decay of relative entropy for non-constant D. Investigate the problem of entropy decay compared to the standard entropy method, and study the exponential decay of auxiliary functional S. Learn about the modified entropy method and (parabolic) regularization of semigroups. Compare local vs. global decay rates in non-symmetric Fokker-Planck equations and explore the decay of logarithmic entropy. Conclude by understanding why the hypocoercive method is effective in this context.
Entropy Method for Hypocoercive BGK and Fokker-Planck Equations
Overview
Syllabus
Intro
degenerate Fokker-Planck equations with linear drift
degenerate prototype with w = 1
coefficients C. Din Fokker-Planck equation
review of entropy method
admissible relative entropies (for entropy method) for probability densities
Step 1: exponent decay of entropy dissipation for D=const
Step 2. exponential decay of relative entropy for Deconst Theorem 4
problem of entropy decay (cp. standard entropy method)
Step 1: exponential decay of auxiliary functional S.
Proof of Proposition 2 - modified entropy method
(parabolic) regularization of semigroup e
local vs. global decay rate in non-symmetric FP equations decay of logarithmic entropy for the non-symmetric FP equation
Why does the hypocoercive method work?
Taught by
ICTP Mathematics
