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DCDS

In this survey paper I describe the convoluted links between the
regularity theory of optimal transport and the geometry of cut locus.

KRM

We investigate the behavior in $N$ of the $N$--particle entropy functional for Kac's stochastic
model of Boltzmann dynamics, and its relation to the entropy function for
solutions of Kac's one dimensional nonlinear model Boltzmann equation. We prove
results that bring together the notion of propagation of chaos, which Kac introduced in the context of this model, with the problem of estimating the rate of equilibration in the model in entropic terms, showing that the entropic rate of convergence can be arbitrarily slow. Results proved here
show that one can in fact use entropy production bounds in Kac's stochastic model to obtain entropic convergence bounds for his non linear model Boltzmann equation, though the problem of obtaining optimal lower bounds of this sort for the original Kac model remains open
and the upper bounds obtained here show that this problem is somewhat subtle.

KRM

Cercignani's conjecture assumes a linear inequality between the
entropy and entropy production functionals for Boltzmann's nonlinear
integral operator in rarefied gas dynamics. Related to the field of
logarithmic Sobolev inequalities and spectral gap inequalities, this
issue has been at the core of the renewal of the mathematical theory
of convergence to thermodynamical equilibrium for rarefied gases
over the past decade. In this review paper, we survey the various
positive and negative results which were obtained since the
conjecture was proposed in the 1980s.

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