Variational characterizations of the effective multiplication factor of a nuclear reactor core

We derive some new inf-sup and sup-inf formulae for the so-called effective multiplication factor arising in the study of reactor analysis. We treat in a same formalism the transport equation and the energy-dependent diffusion equation

Semigroup generation properties of streaming operators with non--contractive boundary conditions

We present $c_0$--semigroup generation results for the free streaming operator with abstract boundary conditions. We recall some known results on the matter and establish a general theorem. We motivate our study with a lot of examples and show that our result applies to the physical cases of Maxwell boundary conditions in the kinetic theory of gases as well as to the non--local boundary conditions involved in transport--like equations from population dynamics.Comment: 28 page

Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation

Information Geometry generalizes to infinite dimension by modeling the tangent space of the relevant manifold of probability densities with exponential Orlicz spaces. We review here several properties of the exponential manifold on a suitable set $\mathcal E$ of mutually absolutely continuous densities. We study in particular the fine properties of the Kullback-Liebler divergence in this context. We also show that this setting is well-suited for the study of the spatially homogeneous Boltzmann equation if $\mathcal E$ is a set of positive densities with finite relative entropy with respect to the Maxwell density. More precisely, we analyse the Boltzmann operator in the geometric setting from the point of its Maxwell's weak form as a composition of elementary operations in the exponential manifold, namely tensor product, conditioning, marginalization and we prove in a geometric way the basic facts i.e., the H-theorem. We also illustrate the robustness of our method by discussing, besides the Kullback-Leibler divergence, also the property of Hyv\"arinen divergence. This requires to generalise our approach to Orlicz-Sobolev spaces to include derivatives.%Comment: 39 pages, 1 figure. Expanded version of a paper presente at the conference SigmaPhi 2014 Rhodes GR. Under revision for Entrop

Existence of self-similar profile for a kinetic annihilation model

We show the existence of a self-similar solution for a modified Boltzmann equation describing probabilistic ballistic annihilation. Such a model describes a system of hard-spheres such that, whenever two particles meet, they either annihilate with probability $\alpha \in (0,1)$ or they undergo an elastic collision with probability $1 - \alpha$. For such a model, the number of particles, the linear momentum and the kinetic energy are not conserved. We show that, for $\alpha$ smaller than some explicit threshold value $\alpha_*$, a self-similar solution exists.Comment: This new version supersedes and replaces the previous one. We found a mistake in the previous (and published) version of the manuscript and explained how to fix it in "Erratum to "Existence of self-similar profile for a kinetic annihilation model" [J. Differential Equations 254 (7) (2013) 3023-3080]. J. Differential Equations 257 (2014), no. 8, 3071-3074." This version provides a complete and corrected version of the previous manuscrip

Long time behavior of non-autonomous Fokker-Planck equations and the cooling of granular gases

We analyze the asymptotic behavior of linear Fokker-Planck equations with time-dependent coefficients. Relaxation towards a Maxwellian distribution with time-dependent temperature is shown under explicitly computable conditions. We apply this result to the study of Brownian motion in granular gases as introduced J. J. Brey, J. Dufty and A. Santos (1999), by showing that the Homogenous Cooling State attracts any solution at an algebraic rate.Comment: 15 pages, submitted to Ukrainian Math.

Uniqueness of the self-similar profile for a kinetic annihilation model

We show the existence of a self-similar solution for a We prove the uniqueness of the self-similar profile solution for a modified Boltzmann equation describing probabilistic ballistic annihilation. Such a model describes a system of hard spheres such that, whenever two particles meet, they either annihilate with probability $\alpha \in (0,1)$ or they undergo an elastic collision with probability $1-\alpha$. The existence of a self-similar profile for $\alpha$ smaller than an explicit threshold value $\underline{\alpha}_1$ has been obtained in our previous contribution (J. Differential Equations, 254, 3023--3080, 2013). . We complement here our analysis of such a model by showing that, for some $\alpha^{\sharp}$ explicit, the self-similar profile is unique for $\alpha \in (0,\alpha^{\sharp})$

Exponential convergence to equilibrium for subcritical solutions of the Becker-D\"oring equations

We prove that any subcritical solution to the Becker-D\"{o}ring equations converges exponentially fast to the unique steady state with same mass. Our convergence result is quantitative and we show that the rate of exponential decay is governed by the spectral gap for the linearized equation, for which several bounds are provided. This improves the known convergence result by Jabin & Niethammer (see ref. [14]). Our approach is based on a careful spectral analysis of the linearized Becker-D\"oring equation (which is new to our knowledge) in both a Hilbert setting and in certain weighted $\ell^1$ spaces. This spectral analysis is then combined with uniform exponential moment bounds of solutions in order to obtain a convergence result for the nonlinear equation