A new understanding of grazing limit

TL;DR

This paper offers a new perspective on the grazing limit of the Boltzmann equation to the Landau equation by applying a natural scaling without angular cutoff, extending the validity to a broader parameter range.

Contribution

It introduces a simplified approach using natural scaling and a new well-posedness theory to justify the grazing limit for a wider class of potentials, including Coulomb.

Findings

The scaled Boltzmann operator decomposes into two parts, with one converging to the Landau operator.

The new method applies to the regime with $ ext{γ} > -5$, including Coulomb potential.

The proof avoids angular cutoff assumptions, simplifying the analysis.

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 $\gamma>-5$ that includes the Coulomb potential corresponding to $\gamma=-3$. 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.