# [CvGmt News] [Notiziario] [Settimanale] Avviso seminario di Matematica Prof. M. Röckner (29 apr. 19)

Michele Verde michele.verde at sns.it
Wed Apr 24 09:03:52 CEST 2019

```SEMINARIO DI MATEMATICA

*lunedì 29 aprile 2019*

ore 11:00

*Scuola Normale Superiore*

Pisa

Aula Tonelli

*Michael Röckner*

*Bielefeld University*

Terrà un seminario dal titolo:

*“**The evolution to equilibrium of solutions to nonlinear Fokker-Planck
equations**”*

*Abstract:*

*The talk is about the so-called H -Theorem for a class of nonlinear
Fokker-Planck equations which are of porous media type on the whole
Euclidean space perturbed by a transport term. We first construct a
solution in the sense of mild solutions on L^1 through a nonlinear
semigroup of contractions. Then we study the asymptotic behavior of the
solutions when time tends to infinity. For a large class M of initial
conditions we show their relative compactness with respect to local L^1
convergence, while all limit points belong to L^1. Under an additional
assumption we obtain that we in fact have  convergence in L^1, if the
initial condition is a probability density. The limit is then identified as
the unique stationary solution in M to the nonlinear Fokker-Planck
equation. This solution is thus an invariant measure of the solution to the
corresponding distribution dependent SDE whose time marginals converge to
it in L^1. It turns out that under our conditions the underlying nonlinear
Kolmogorov operator is a (both in the second and first order part)
nonlinear analog of the generator of a distorted Brownian motion. The
solution of the above mentioned distribution dependent SDE can thus be
interpreted as a “nonlinear distorted Brownian motion“. Our main technique
for the proofs is to construct a suitable Lyapunov function acting
nonlinearly on the path in L^1, which is given by the nonlinear contraction
semigroup  applied to the initial condition, and then adapt a classical
technique of Pazy to our situation. This Lyapunov function is given by a
generalized entropy function (which in the linear case specializes to the
usual Boltzmann-Gibbs entropy) plus a mean energy part*

Tutti gli interessati sono invitati a partecipare.

Classe di Scienze

--

- - - - -

Michele VERDE
Scuola Normale Superiore

Servizio alla didattica e allievi
Piazza Dei Cavalieri, 7
I - 56126 Pisa

Tel. 050-509048
E-Mail: michele.verde at sns.it
E-Mail: classi at sns.it
- - - - -

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Grazie

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