Local Well-Posedness for Boltzmann’s Equation and the Boltzmann Hierarchy via Wigner Transform

Abstract

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.

Moments and regularity for a Boltzmann equation via Wigner transform

“…As an application of our results, we are able to prove that for constant collision kernels, the solutions constructed in [11] propagate non-negativity assuming only that the data itself is non-negative. This is true under minimal assumptions for which the theory of [11] applies; in particular, we do not require higher moment or regularity estimates to prove non-negativity.…”

“…As an application of our results, we are able to prove that for constant collision kernels, the solutions constructed in [11] propagate non-negativity assuming only that the data itself is non-negative. This is true under minimal assumptions for which the theory of [11] applies; in particular, we do not require higher moment or regularity estimates to prove non-negativity. The reason is that our persistence of regularity results allow us to approximate low-regularity solutions by higher regularity solutions for a short time interval; even though the higher Sobolev norms may be very large, they will at least be finite on a time interval bounded from below uniformly with respect to the mollification parameter.…”

“…In particular, we prove persistence of regularity results for the scale of Sobolev spaces which arise naturally from the analysis of [11]. Effectively, as soon as the solution has enough regularity and decay to apply the methods of [11], we can prove that more regular data leads to more regular solutions for as long as the solution exists in the less regular 2 There is nothing particularly special about L 1 moments; for example, a theorem of Lanford on the derivation of Boltzmann's equation [23] relies upon exponential L ∞ moments. 3 Technically one must show that the solution of the new fixed point problem is related to the solution the original fixed point problem in the expected way; this is formally trivial but inconvenient to prove rigorously.…”

“…For this reason, in the present work we cannot hope to prove that the solution is smoother than the data; we can only hope to show that the solution is as regular as the data and that is what we achieve. More precisely, we obtain a fairly complete description of the persistence of regularity for local-in-time solutions using the functional framework of [11], at least when the collision kernel is constant.…”

“…We also remark that we address L 2 moments, whereas the space homogeneous theory is primarily concerned with L 1 moments. 2 As is typically the case when one proves a propagation result, the proofs in this paper are based on the following idea: we write an equation for a desired moment or derivative, and then solve that equation by a fixed point argument as in [11]. 3 In all cases we must pay a cost in terms of regularity/moments in order to propagate regularity/moments in some other variable.…”