2018/04/11 by Thomas Chen, Chen, Thomas, Ryan Denlinger +3
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1804.04019
openalex publication_date 2018/04/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ℝd, d≥ 2, which we use as a model in this paper. Local well-posedness for this equation has been proven using the Wigner transform when < v >βf0 ∈ L2v Hαx for min (α,β) > (d-1)/(2). We prove that if α,β are large enough, then it is possible to propagate moments in x and derivatives in v (for instance, < x >k < ∇v >^ℓ f ∈ L^∞T L2x,v if f0 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 Hα,β; and, continuity of the solution map in Hα,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 Hα,β for any α,β> (d-1)/(2), without any additional regularity or decay assumptions.