27/06/2013, 11:00 — 12:00 — Room P3.10, Mathematics Building Emanuele Dolera, Università di Modena e Reggio Emilia, Italy

Reaching the best possible rate of convergence to equilibrium of
Boltzmann-equation solutions

This talk concerns a definitive answer to the problem of
quantifying the relaxation to equilibrium of the solutions to the
spatially homogeneous Boltzmann equation for Maxwellian molecules.
Under really mild conditions on the initial datum - closed to
necessity - and a weak, physically consistent, angular cutoff
hypothesis, the main result states that the total variation
distance (i.e. the ${L}^{1}$-norm in the absolutely continuous case)
between the solution and the limiting Maxwellian distribution
admits an upper bound of the form $C\mathrm{exp}(-{\Lambda}_{b}^{*}t)$,
${\Lambda}_{b}^{*}$ being the spectral gap of the linearized collision
operator and $C$ a constant depending only on the initial datum.
Hilbert hinted at the validity of this quantification in 1912,
which was explicitly formulated as a conjecture by McKean in 1966.
The main line of the new proof is based on an analogy between the
problem of convergence to equilibrium and the central limit theorem
of probability theory, as suggested by McKean.