[Zurück]

F. Achleitner:
"Short- and long-time behavior in (hypo)coercive ODE-systems and kinetic equations";
Vortrag: Online Workshop for Nonlinear Partial Differential Equations, Kobe University (Japan) (eingeladen); 22.04.2021.

see english abstract

We will discuss hypocoercivity on the level of ODEs and devise a new way to construct strict Lyapunov functionals:
Systems of ODEs dx/dt = Ax with semi-dissipative matrix A (i.e. the Hermitian part of matrix A is negative semi-definite) are Lyapunov stable but not necessarily asymptotically stable.
There exist many equivalent conditions, to decide if the ODE system is asymptotically stable or not.
Some conditions allow to construct a strict Lyapunov functional in a natural way.
We will review these classical conditions/approaches and identify a "hypocoercivity index" which e.g. characterizes the short-time asymptotics of the propagator norm for semi-dissipative ODEs.

Finally, we apply these results to study the long-time behavior of (hypocoercive) nonlinear BGK-type model with constant collision frequency, and (kinetic) Fokker-Planck equations.
In particular, we will compare our strict Lyapunov functionals for the linear(ized) kinetic equations with other classical approaches.

Kinetic equations, BGK models, hypocoercivity, Lyapunov functionals, perturbation methods for matrix equations.