<div dir="ltr"><p class="gmail-MsoTitle" style="margin:0cm 0cm 0.0001pt;text-align:center;font-size:24pt;font-family:"New York",serif;font-weight:bold;text-decoration-line:underline"><span style="font-size:18pt;font-family:"Times New Roman",serif">SEMINARIO
DI MATEMATICA</span></p>
<p class="gmail-MsoTitle" style="margin:0cm 0cm 0.0001pt;text-align:center;font-size:24pt;font-family:"New York",serif;font-weight:bold;text-decoration-line:underline"><br></p>
<p class="MsoNormal" align="center" style="margin:0cm 0cm 0.0001pt 9pt;text-align:center;line-height:18pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><b><span style="font-size:16pt;font-family:"Times New Roman",serif">lunedì
29 aprile 2019</span></b><b><span style="font-size:16pt;font-family:"Times New Roman",serif"></span></b></p>
<p class="MsoNormal" align="center" style="margin:0cm 0cm 0.0001pt 9pt;text-align:center;line-height:18pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><span style="font-size:16pt;font-family:"Times New Roman",serif">ore 11:00</span></p>
<p class="MsoNormal" align="center" style="margin:0cm 0cm 0.0001pt 9pt;text-align:center;line-height:18pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><span style="font-size:16pt;font-family:"Times New Roman",serif"> </span></p>
<p class="MsoNormal" align="center" style="margin:0cm 0cm 0.0001pt 9pt;text-align:center;line-height:18pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><u><span style="font-size:16pt;font-family:"Times New Roman",serif">Scuola Normale Superiore</span></u></p>
<p class="MsoNormal" align="center" style="margin:0cm 0cm 0.0001pt 9pt;text-align:center;line-height:18pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><span style="font-size:16pt;font-family:"Times New Roman",serif">Pisa</span></p>
<p class="MsoNormal" align="center" style="margin:0cm 0cm 0.0001pt 9pt;text-align:center;line-height:18pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><span style="font-size:16pt;font-family:"Times New Roman",serif">Aula Tonelli</span></p>
<p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><br></p>
<p class="MsoNormal" align="center" style="text-align:center;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><b><i><span style="font-size:26pt;font-family:"Times New Roman",serif">Michael Röckner</span></i></b></p>
<p class="MsoNormal" align="center" style="text-align:center;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><i><span style="font-family:"Times New Roman",serif">Bielefeld
University</span></i><i><span style="font-family:"Times New Roman",serif"></span></i></p>
<p class="MsoNormal" align="center" style="text-align:center;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><span style="font-family:"Times New Roman",serif"> </span></p>
<p class="MsoNormal" align="center" style="text-align:center;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><span style="font-size:16pt;font-family:"Times New Roman",serif">Terrà un seminario
dal titolo:</span></p>
<p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><span style="font-family:"Times New Roman",serif"> </span></p>
<p class="MsoNormal" align="center" style="text-align:center;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><b><i><span lang="EN-US" style="font-size:28pt;font-family:"Times New Roman",serif">“</span></i></b><b><span lang="EN-US" style="font-size:28pt;font-family:"Times New Roman",serif">The evolution to equilibrium of solutions to nonlinear
Fokker-Planck equations</span></b><b><i><span lang="EN-US" style="font-size:28pt;font-family:"Times New Roman",serif">”</span></i></b><b><span lang="EN-US" style="font-size:28pt;font-family:"Times New Roman",serif"></span></b></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><b><i><span lang="EN-US" style="font-size:10pt;font-family:"Times New Roman",serif"> </span></i></b></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><b><i><span lang="EN-US" style="font-size:10pt;font-family:"Times New Roman",serif"> </span></i></b></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><b><i><span style="font-size:10pt;font-family:"Times New Roman",serif">Abstract:</span></i></b></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><i><span style="font-size:10pt;font-family:"Times New Roman",serif">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</span></i><b><i><span style="font-size:10pt;font-family:"Times New Roman",serif"></span></i></b></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><span style="font-size:16pt;font-family:"Times New Roman",serif"> </span></p>
<p class="MsoNormal" style="text-align:justify;margin:0cm 0cm 0.0001pt;font-size:12pt;font-family:TradeGothicPl-CondEighteen"><span style="font-size:16pt;font-family:"Times New Roman",serif">Tutti gli interessati sono invitati a
partecipare.</span></p>
<p class="gmail-Corpodeltesto" style="margin:0cm 0cm 0.0001pt;text-align:justify;font-size:12pt;font-family:"Avant Garde""><span style="font-size:16pt;font-family:"Times New Roman",serif"> </span></p>
<p class="gmail-Corpodeltesto" style="margin:0cm 0cm 0.0001pt;text-align:justify;font-size:12pt;font-family:"Avant Garde""><span style="font-size:16pt;font-family:"Times New Roman",serif"> Classe
di Scienze</span></p><div><br></div>-- <br><div dir="ltr" class="gmail_signature" data-smartmail="gmail_signature"><div dir="ltr"><div><div dir="ltr"><div><div dir="ltr"><span style="color:rgb(84,141,212);font-family:Cambria,serif;font-size:9pt"><br></span></div><div dir="ltr"><span style="color:rgb(84,141,212);font-family:Cambria,serif;font-size:9pt">- - - - -</span><br></div><div dir="ltr"><p><span style="font-size:9.0pt;font-family:"Cambria","serif";color:#548dd4">
Michele VERDE<br>
Scuola Normale Superiore</span></p>
<p><span style="font-size:9.0pt;font-family:"Cambria","serif";color:#548dd4">Servizio alla didattica e allievi<br>
Piazza Dei Cavalieri, 7<br>
I - 56126 Pisa <br></span></p>
<p><span style="font-size:9.0pt;font-family:"Cambria","serif";color:#548dd4">Tel. 050-509048<br>
E-Mail: <a href="mailto:michele.verde@sns.it" target="_blank"><span style="color:#548dd4">michele.verde@sns.it</span></a><br>
E-Mail: <a href="mailto:classi@sns.it" target="_blank">classi@sns.it</a><br>
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