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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:Factorisation for non-symmetric operators and expo
nential H-theorems - Mischler\, S (Paris-Dauphine)
DTSTART;TZID=Europe/London:20101102T150000
DTEND;TZID=Europe/London:20101102T154500
UID:TALK27787AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/27787
DESCRIPTION:We present a factorization method for estimating r
esolvents of non-symmetric operators in Banach or
Hilbert spaces in terms of estimates in another (t
ypically smaller) ``reference'' space. This applie
s to a class of operators writing as a ``regulariz
ing'' part (in a broad sense) plus a dissipative p
art. Then in the Hilbert case we combine this fact
orization approach with an abstract Plancherel ide
ntity on the resolvent into a method for enlarging
the functional space of decay estimates on semigr
oups. In the Banach case\, we prove the same resul
t however with some loss on the norm. We then appl
y these functional analysis approach to several PD
Es: the Fokker-Planck and kinetic Fokker-Planck eq
uations\, the linear scattering Boltzmann equation
in the torus\, and\, most importantly the lineari
zed Boltzmann equation in the torus (at the price
of extra specific work in the latter case). In add
ition to the abstract method in itself\, the main
outcome of the paper is indeed the first proof of
exponential decay towards global equilibrium (e.g.
in terms of the relative entropy) for the full Bo
ltzmann equation for hard spheres\, conditionnally
to some smoothness and (polynomial) moment estima
tes. This improves on the result in [Desvillettes-
Villani\, Invent. Math.\, 2005] where the rate was
``almost exponential''\, that is polynomial with
exponent as high as wanted\, and solves a long-sta
nding conjecture about the rate of decay in the H-
theorem for the nonlinear Boltzmann equation\, see
for instance [Cercignani\, Arch. Mech\, 1982] and
[Rezakhanlou-Villani\, Lecture Notes Springer\, 2
001].
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:Mustapha Amrani
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