We study the large-data Cauchy problem for Boltzmann equations with general collision kernels. We prove that sequences of solutions which satisfy only the physically natural a priori bounds converge weakly in L' to a solution. From this stability result we deduce global existence of a solution to the Cauchy problem. Our method relies upon recent compactness results for velocity averages, a new formulation of the Boltzmann equation which involves nonlinear normalization and an analysis of subsolutions and supersolutions. It allows us to overcome the lack of strong a priori estimates and define a meaningful collision operator for general configurations.

https://www.jstor.org/stable/pdfplus/1971423.pdf

On the Cauchy problem for Boltzmann equations: global existence and weak stability

Chapter 2 – A Review of Mathematical Topics in Collisional Kinetic Theory

A recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal diffusion problems described by a linear integral equation on bounded domains in $\mbRn$. The nonlocal vector calculus also enables striking analogies to be drawn between the nonlocal model and classical models for diffusion, including a notion of nonlocal flux. The ubiquity of the nonlocal operator in applications is illustrated by a number of examples ranging from continuum mechanics to graph theory. In particular, it is shown that fractional Laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that we consider. The numerous applications elucidate different interpretations of the operator and the associated governing equations. For example, a probabilistic perspective explains that the nonlocal spatial operator appearing in our model corresponds to the infinitesimal generator for a symmetric jump process. Sufficie...

Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints

/pdf/a-kinetic-formulation-of-multidimensional-scalar-mfczat3uc0.pdf

A kinetic formulation of multidimensional scalar conservation laws and related equations

In order to study weak continuity of quadratic forms on spaces of L2 solutions of systems of partial differential equations, we define defect measures on the space of positions and frequencies.A sy...

Microlocal defect measures

Regularity of the moments of the solution of a Transport Equation

This paper is devoted to the mathematical definition of the extrapolation length which appears in the diffusion approximation. To obtain this result, we describe the spectral properties of the transport equation and we show how the diffusion approximation is related to the computation of the critical size. The paper also contains some simple numerical examples and some new results for the Milne problem. Introduction. The computation of the critical size and the diffusion approximation for the transport equation have been closely related and this is due to the following facts. First, the computation of the critical size is much easier for the diffusion approximation than for the original transport equation. Second, for the critical size one can consider a host media X = nXQ with X0 given and 17 a positive number. The problem of the critical size is then reduced to the computation of the parameter zj. It turns out that when the transport operator is almost conservative, the critical value of the parameter 17 is large and it is exactly for this range of value that the diffusion approximation is accurate. On the other hand the "physical" boundary condition for the diffusion approxi- mation is of the form

/pdf/diffusion-approximation-and-computation-of-the-critical-size-2g57u5jgcy.pdf

Diffusion approximation and computation of the critical size

http://www.cmls.polytechnique.fr/perso/golse.francois/Articles/BardosFGPerthameSentis1988.pdf

The nonaccretive radiative transfer equations: Existence of solutions and Rosseland approximation

The Maxwell-Boltzmann approximation for ion kinetic modeling

The present paper discusses the diffusion approximation of the linear Boltzmann equation in cases where the collision frequency is not uniformly large in the spatial domain. Our results apply for instance to the case of radiative transfer in a composite medium with optically thin inclusions in an optically thick background medium. The equation governing the evolution of the approximate particle density coincides with the limit of the diffusion equation with infinite diffusion coefficient in the optically thin inclusions.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cms/2015/0013/0003/CMS-2015-0013-0003-a003.pdf

Rémi Sentis

Papers

Regularity of the moments of the solution of a Transport Equation

Diffusion approximation and computation of the critical size

The nonaccretive radiative transfer equations: Existence of solutions and Rosseland approximation

The Maxwell-Boltzmann approximation for ion kinetic modeling

The diffusion approximation for the linear Boltzmann equation with vanishing scattering coefficient