# Homogenization of the unsteady compressible Navier-Stokes equations for adiabatic exponent $\gamma>3$

## TL;DR

This paper proves that for unsteady compressible Navier-Stokes equations in perforated domains, the homogenized limit holds for a lower adiabatic exponent , expanding previous results which required , using novel pressure estimate techniques.

## Contribution

It establishes homogenization results for  in perforated domains, extending known limits and introducing new pressure estimate methods.

## Key findings

- Homogenization limit holds for  in perforated domains.
- New pressure estimates enable lower  bounds.
- Results extend to Navier-Stokes-Fourier system and arbitrary dimensions.

## Abstract

We consider the unsteady compressible Navier-Stokes equations in a perforated three-dimensional domain, and show that the limit system for the diameter of the holes going to zero is the same as in the perforated domain provided the perforations are small enough. The novelty of this result is the lower adiabatic exponent $\gamma>3$ instead of the known value $\gamma>6$. The proof is based on the use of two different restriction operators leading to two different types of pressure estimates. We also discuss the extension of this result for the unsteady Navier-Stokes-Fourier system as well as the optimality of the known results in arbitrary space dimension for both steady and unsteady problems.