Local well-posedness for Boltzmann's equation and the Boltzmann hierarchy via Wigner transform

2017/03/02 by Thomas Chen, Chen, Thomas, Ryan Denlinger +3 · 2 citations
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.1703.00751

openalex publication_date 2017/03/02 · openalex created_date 2017/03/16 · openalex updated_date 2026/07/28

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 L²_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.

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• Moments and Regularity for a Boltzmann Equation via Wigner Transform
• From Kac particles to the Landau equation with hard potentials: BBGKY hierarchy method

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