Moments and regularity for a Boltzmann equation via Wigner transform

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 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.

Well/ill-posedness bifurcation for the Boltzmann equation with constant collision kernel

“…The novelty of Theorem 1.3 is certainly the ill-posedness / part (b). Part (a) is essentially included in [11,12] already and is stated and proved here with different estimates, namely (1.13) and (1.15).…”

“…Of course, the Boltzmann equation is certainly very different from the nonlinear dispersive PDEs and the nuts and bolts designed for one, so far, do not fit the other. Interestingly, the recent series of papers [11][12][13] by T. Chen, Denlinger, and Pavlović suggests that a systematic study of the Boltzmann equation using tools built for the dispersive PDEs might indeed be possible.…”

“…Technical problems remain (if they do not get worse) though, since the form of the collision operators does not improve and derivatives are complicated under the Wigner transform while the analysis of (1.1) was not very developed for a long time. In [11][12][13], incorporating the new developments [14-37, 44, 46, 47, 52, 55, 63-65] on the quantum many-body hierarchy dynamics that rely on the analysis of (1.1), with some highly nontrivial technical improvisions, T. Chen, Denlinger, and Pavlović provided an alternate dispersive PDE based route for proving the local well-posedness of the Boltzmann equation. The solutions provided in [11][12][13] differ from previous work in the sense that they solve the Boltzmann equation a.e.…”

“…In [11][12][13], incorporating the new developments [14-37, 44, 46, 47, 52, 55, 63-65] on the quantum many-body hierarchy dynamics that rely on the analysis of (1.1), with some highly nontrivial technical improvisions, T. Chen, Denlinger, and Pavlović provided an alternate dispersive PDE based route for proving the local well-posedness of the Boltzmann equation. The solutions provided in [11][12][13] differ from previous work in the sense that they solve the Boltzmann equation a.e. instead of everywhere in time.…”

“…We follow the lead of [11][12][13], but we study ill-posedness instead of well-posedness in this paper (and we are not using the Wigner transform). It is of mathematical and physical interest to prove well-posedness at the "optimal" regularity as it would mean all the nonlinear interactions of the complicated underlying nonlinear equation have been analyzed and accounted for.…”