About \( L^p \) estimates for the spatially homogeneous Boltzmann equation

2004/09/24 by Laurent Desvillettes, Clément Mouhot · 4 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Applied mathematics #Boltzmann constant #Boltzmann equation #Gas Dynamics and Kinetic Theory #Homogeneous #Mathematics #Numerical methods in inverse problems #Physics #Statistical physics #Thermodynamics #math.AP #msc:76P05 #msc:82B40 #msc:82C40

paper · pdf · doi:10.1016/j.anihpc.2004.03.002

published as Annales de l'Institut Henri Poincaré Analyse non linéaire 22 (2005) 127-142 · 19 pages

openalex publication_date 2004/09/24 · arxiv created 2006/07/21 · arxiv updated 2016/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

For the homogeneous Boltzmann equation with (cutoff or noncutoff) hard potentials, we prove estimates of propagation of L^p norms with a weight (1 + |x|²)^{q/ 2} ( 1 < p < + ∞ , q ∈ ℝ ₊ large enough), as well as appearance of such weights. The proof is based on some new functional inequalities for the collision operator, proven by elementary means. Résumé On prouve la propagation de normes L^p avec poids (1 + |x|²)^{q/ 2} et l'apparition de tels poids pour l'équation de Boltzmann homogène dans le cas des potentiels durs (avec ou sans troncature angulaire). La démonstration est basée sur de nouvelles inégalités fonctionnelles pour l'opérateur de collision, que l'on prouve par des moyens élémentaires.

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