# Continuity equation and vacuum regions in compressible flows

## TL;DR

This paper studies the formation and characteristics of vacuum regions in weak solutions of the continuity equation and compressible Navier--Stokes equations, focusing on the analysis of renormalized solutions and their relationships.

## Contribution

It provides new insights into vacuum formation in compressible flows through analysis of renormalized solutions and their interrelations.

## Key findings

- Vacuum regions can form in weak solutions of the continuity equation.
- Analysis of renormalized solutions reveals properties of vacuum creation.
- Results are relevant for understanding compressible fluid dynamics.

## Abstract

We investigate the creation and properties of eventual vacuum regions in the weak solutions of the continuity equation, in general, and in the weak solutions of compressible Navier--Stokes equations, in particular. The main results are based on the analysis of renormalized solutions to the continuity and pure transport equations and their inter-relations which are of independent interest.