2012/03/31 by Ricardo Alonso, José A. Cañizo, José Alfredo Cañizo +2 · 27 citations
Engineering · Mathematics · #Boltzmann equation #Collision #Exponential function #Gas Dynamics and Kinetic Theory #Homogeneous #Integrable system #Kernel (algebra) #Mathematical Biology Tumor Growth #Moment (physics) #Thermoelastic and Magnetoelastic Phenomena #math.AP
paper · pdf · doi:10.1080/03605302.2012.715707
published in Communications in Partial Differential Equations 38(1), 155-169 (Taylor & Francis) · 14 pages. Many typos corrected in this revised version
arxiv created 2012/07/23 · openalex publication_date 2012/08/28 · arxiv updated 2014/01/15 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
We study the creation and propagation of exponential moments of solutions to the spatially homogeneous d-dimensional Boltzmann equation. In particular, when the collision kernel is of the form |v − v *|β b(cos (θ)) for β ∈ (0, 2] with cos (θ) = |v − v *|−1(v − v *)·σ and σ ∈ 𝕊 d−1, and assuming the classical cut-off condition b(cos (θ)) integrable in 𝕊 d−1, we prove that there exists a > 0 such that moments with weight exp (amin t, 1|v|β) are finite for t > 0, where a only depends on the collision kernel and the initial mass and energy. We propose a novel method of proof based on a single differential inequality for the exponential moment with time-dependent coefficients.