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SUMMARY:Factorisation for non-symmetric operators and exponential H-theore
 ms - Mischler\, S (Paris-Dauphine)
DTSTART:20101102T150000Z
DTEND:20101102T154500Z
UID:TALK27787@talks.cam.ac.uk
CONTACT:Mustapha Amrani
DESCRIPTION:We present a factorization method for estimating resolvents of
  non-symmetric operators in Banach or Hilbert spaces in terms of estimates
  in another (typically smaller) ``reference'' space. This applies to a cla
 ss of operators writing as a ``regularizing'' part (in a broad sense) plus
  a dissipative part. Then in the Hilbert case we combine this factorizatio
 n approach with an abstract Plancherel identity on the resolvent into a me
 thod for enlarging the functional space of decay estimates on semigroups. 
 In the Banach case\, we prove the same result however with some loss on th
 e norm. We then apply these functional analysis approach to several PDEs: 
 the Fokker-Planck and kinetic Fokker-Planck equations\, the linear scatter
 ing Boltzmann equation in the torus\, and\, most importantly the linearize
 d Boltzmann equation in the torus (at the price of extra specific work in 
 the latter case). In addition to the abstract method in itself\, the main 
 outcome of the paper is indeed the first proof of exponential decay toward
 s global equilibrium (e.g. in terms of the relative entropy) for the full 
 Boltzmann equation for hard spheres\, conditionnally to some smoothness an
 d (polynomial) moment estimates. This improves on the result in [Desvillet
 tes-Villani\, Invent. Math.\, 2005] where the rate was ``almost exponentia
 l''\, that is polynomial with exponent as high as wanted\, and solves a lo
 ng-standing conjecture about the rate of decay in the H-theorem for the no
 nlinear Boltzmann equation\, see for instance [Cercignani\, Arch. Mech\, 1
 982] and [Rezakhanlou-Villani\, Lecture Notes Springer\, 2001].
LOCATION:Seminar Room 1\, Newton Institute
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