Moments and Regularity for a Boltzmann Equation via Wigner Transform

TL;DR

This paper investigates the properties of solutions to the Boltzmann equation using Wigner transform techniques, establishing regularity, moment propagation, and fundamental physical properties like non-negativity and energy conservation.

Contribution

It introduces new methods to propagate moments and regularity for Boltzmann solutions via Wigner transform, extending previous well-posedness results and ensuring physical properties without extra assumptions.

Findings

Propagation of moments in space and derivatives in velocity for regular solutions

Persistence of Sobolev regularity and continuity of the solution map

Non-negativity, energy conservation, and H-theorem for regular solutions

Abstract

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 $\mathbb{R}^d$, $d\geq 2$, which we use as a model in this paper. Local well-posedness for this equation has been proven using the Wigner transform when $\left< v \right>^\beta f_0 \in L^2_v H^\alpha_x$ for $\min (\alpha,\beta) > \frac{d-1}{2}$. We prove that if $\alpha,\beta$ are large enough, then it is possible to propagate moments in $x$ and derivatives in $v$ (for instance, $\left< x \right>^k \left< \nabla_v \right>^\ell f \in L^\infty_T L^2_{x,v}$ if $f_0$ is nice enough). 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 $H^{\alpha,\beta}$; and, continuity of the solution map in $H^{\alpha,\beta}$. 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 $H^{\alpha,\beta}$ for any $\alpha,\beta > \frac{d-1}{2}$, without any additional regularity or decay assumptions.