# Hydrodynamic Limit of a Kinetic Gas Flow Past an Obstacle

## TL;DR

This paper proves the convergence of steady Boltzmann solutions to Navier-Stokes solutions for gas flow past an obstacle, providing error estimates and new analytical techniques in unbounded domains.

## Contribution

It constructs unique steady Boltzmann solutions around an obstacle with small velocity at infinity and establishes their approximation by Navier-Stokes solutions as the mean-free path diminishes.

## Key findings

- Existence of unique steady Boltzmann solutions around an obstacle.
- Quantitative error estimates between Boltzmann and Navier-Stokes solutions.
- Development of new $L^6$ and $L^3$ estimates in unbounded exterior domains.

## Abstract

Given an obstacle in $\mathbb{R}^3$ and a non-zero velocity with small amplitude at the infinity, we construct the unique steady Boltzmann solution flowing around such an obstacle with the prescribed velocity as $|x|\to \infty$, which approaches the corresponding Navier-Stokes steady flow, as the mean-free path goes to zero. Furthermore, we establish the error estimate between the Boltzmann solution and its Navier-Stokes approximation. Our method consists of new $L^6$ and $L^3$ estimates in the unbounded exterior domain, as well as an iterative scheme preserving the positivity of the distribution function.