A new understanding of grazing limit

2023/08/16 by Tong Yang, Yang, Tong, Yulong Zhou +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Gas Dynamics and Kinetic Theory #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2308.08375

openalex publication_date 2023/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The grazing limit of the Boltzmann equation to Landau equation is well-known and has been justified by using cutoff near the grazing angle with some suitable scaling. In this paper, we will provide a new understanding by simply applying a natural scaling on the Boltzmann operator without angular cutoff. The proof is based on a new well-posedness theory on the Boltzmann equation without angular cutoff in the regime with optimal ranges of parameters so that the grazing limit can be justified directly for any γ>-5 that includes the Coulomb potential corresponding to γ=-3. With this new understanding, the scaled Boltzmann operator in fact can be decomposed into two components. The first one converges to the Landau operator when the singular parameter s of interaction angle tends to 1^- and the second one vanishes in this limit.

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