We use the dispersive properties of the linear Schrödinger equation to prove local well-posedness results for the Boltzmann equation and the related Boltzmann hierarchy, set in the spatial domain R d for d ≥ 2. The proofs are based on the use of the (inverse) Wigner transform along with the spacetime Fourier transform. The norms for the initial data f0 are weighted versions of the Sobolev spaces L 2 v H α x with α ∈ d−1 2 , ∞ . Our main results are local well-posedness for the Boltzmann equation for cutoff Maxwell molecules and hard spheres, as well as local well-posedness for the Boltzmann hierarchy for cutoff Maxwell molecules (but not hard spheres); the latter result holds without any factorization assumption for the initial data.

We discuss old and new results on the mathematical justification of Boltzmann's equation. The classical result along these lines is a theorem which was proven by Lanford in the 1970s. This paper is naturally divided into three parts. I. Classical. We give new proofs of both the uniform bounds required for Lanford's theorem, as well as the related bounds due to Illner & Pulvirenti for a perturbation of vacuum. The proofs use a duality argument and differential inequalities, instead of a fixed point iteration. II. Strong chaos. We introduce a new notion of propagation of chaos. Our notion of chaos provides for uniform error estimates on a very precise set of points; this set is closely related to the notion of strong (one-sided) chaos and the emergence of irreversibility. III. Supplemental. We announce and provide a proof (in Appendix A) of propagation of partial factorization at some phase-points where complete factorization is impossible. 1 L ∞ (without at least an assumption like factorization or exchangeability) but it is wellposed for continuous data. 8 The strong chaos result in [5] requires |xi − xj | ε log 1 ε . 9By contrast, the authors of [28] have provided a very precise but averaged (not pointwise) description of correlations.

Abstract. We consider a gas of N identical hard spheres in the whole space, and we enforce the Boltzmann-Grad scaling. We may suppose that the particles are essentially independent of each other at some initial time; even so, correlations will be created by the dynamics. We will prove a structure theorem for the correlations which develop at positive time. Our result generalizes a previous result which states that there are phase points where the three-particle marginal density factorizes into two-particle and one-particle parts, while further factorization is impossible. The result depends on uniform bounds which are known to hold on a small time interval, or globally in time when the mean free path is large.

In this paper, we continue our study of the Boltzmann equation by use of tools originating from the analysis of dispersive equations in quantum dynamics. Specifically, we focus on properties of solutions to the Boltzmann equation with collision kernel equal to a constant in the spatial domain R d , d ≥ 2, which we use as a model in this paper. Local well-posedness for this equation has been proven using the Wigner transform whenWe prove that if α, β are large enough, then it is possible to propagate moments in x and derivatives in v (for instance,The mechanism is an exchange of regularity in return for moments of the (inverse) Wigner transform of f . We also prove a persistence of regularity result for the scale of Sobolev spaces H α,β ; and, continuity of the solution map in H α,β . Altogether, these results allow us to conclude non-negativity of solutions, conservation of energy, and the H-theorem for sufficiently regular solutions constructed via the Wigner transform. Non-negativity in particular is proven to hold in H α,β for any α, β > d−1 2 , without any additional regularity or decay assumptions.

We present a fast summation method for lattice sums of the type which arise when solving wave scattering problems with periodic boundary conditions. While there are a variety of effective algorithms in the literature for such calculations, the approach presented here is new and leads to a rigorous analysis of Wood's anomalies. These arise when illuminating a grating at specific combinations of the angle of incidence and the frequency of the wave, for which the lattice sums diverge. They were discovered by Wood in 1902 as singularities in the spectral response. The primary tools in our approach are the Euler-Maclaurin formula and a steepest descent argument. The resulting algorithm has super-algebraic convergence and requires only milliseconds of CPU time.