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Many transport equations, such as the neutron transport, radiative transfer, and transport equations for waves in random media, have a diiusive scaling that leads to the diiusion equations. In many physical applications, the scaling parameter (mean free path) may diier in several orders of magnitude from the rareeed regimes to the hydrodynamic (diiusive)… (More)

We show that Shannon's entropy–power inequality admits a strengthened version in the case in which the densities are log-concave. In such a case, in fact, one can extend the Blachman–Stam argument [25, 5] to obtain a sharp inequality for the second derivative of Shannon's entropy functional with respect to the heat semigroup.

The date of receipt and acceptance will be inserted by the editor Summary We are concerned with efficient numerical simulation of the radiative transfer equations. To this end, we follow the Well-Balanced approach's canvas and reformulate the relaxation term as a nonconservative product regularized by steady-state curves while keeping the velocity variable… (More)

—We associate to the p-th Rényi entropy a definition of entropy power, which is the natural extension of Shannon's entropy power and exhibits a nice behaviour along solutions to the p-nonlinear heat equation in R n. We show that the Rényi entropy power of general probability densities solving such equations is always a concave function of time, whereas it… (More)

We study certain nonlinear continuous models of opinion formation derived from a kinetic description involving exchange of opinion between individual agents. These models imply that the only possible final opinions are the extremal ones, and are similar to models of pure drift in magnetization. Both analytical and numerical methods allow to recover the… (More)

We show that an information-theoretic property of Shannon's entropy power, known as concavity of entropy power [7], can be fruitfully employed to prove inequalities in sharp form. In particular, the concavity of entropy power implies the logarithmic Sobolev inequality, and the Nash's inequality with the sharp constant. 1 Introduction In information theory,… (More)

The concern of this paper is the derivation and the analysis of a simple explicit numerical scheme for general one-dimensional filtration equations. It is based on an alternative formulation of the problem using the pseudo-inverse of the density's repartition function. In particular, the numerical approximations can be proven to satisfy a contraction… (More)

In this paper we study the numerical passage from the spatially homogeneous Boltzmann equation without cutoff to the Fokker-Planck-Landau equation in the so-called grazing collision limit. To this aim we derive a Fourier spectral method for the non cutoff Boltzmann equation in the spirit of [21, 23]. We show that the kernel modes that define the spectral… (More)