# Convergence to the Landau equation from the truncated BBGKY hierarchy in the weak-coupling limit

## TL;DR

This paper rigorously demonstrates the convergence of the truncated BBGKY hierarchy to the Landau equation in three dimensions under the weak-coupling limit, preserving the full singularity of the kernel.

## Contribution

It provides a rigorous proof of convergence to the Landau equation, including the full singularity, addressing previous issues of artificial removal of the singular region.

## Key findings

- Convergence to the Landau equation established
- Full singularity of the Landau kernel preserved
- Addresses previous gaps in understanding particle interactions

## Abstract

We consider the evolution of the one-particle function in the weak-coupling limit in three space dimensions, obtained by truncating the BBGKY hierarchy under a propagation of chaos approximation. For this dynamics, we rigorously show the convergence to a solution of the Landau equation, keeping the full singularity of the Landau kernel. This resolves the issue arising from [10], where the singular region has been removed artificially. Since the singularity appears in the Landau equation due to the geometry of particle interactions, it is an intrinsic physical property of the weak-coupling limit which is crucial to the understanding of the transition from particle systems to the Landau equation.