# Non-Existence of Classical Solutions with Finite Energy to the Cauchy Problem of the Compressible Navier-Stokes Equations

## TL;DR

This paper proves that classical solutions with finite energy do not exist for the compressible Navier-Stokes equations near vacuum, highlighting the importance of the homogeneous Sobolev space in well-posedness analysis.

## Contribution

It establishes the non-existence of finite energy classical solutions in inhomogeneous Sobolev spaces for the Cauchy problem near vacuum, emphasizing the role of homogeneous spaces.

## Key findings

- Finite energy classical solutions do not exist near vacuum.
- Homogeneous Sobolev space is essential for well-posedness.
- Results hold even for short time under natural initial data assumptions.

## Abstract

In this paper, we investigate the well-posedess of classical solutions to the Cauchy problem of Navier-Stokes equations,and prove that the classical solution with finite energy does not exist even in the inhomogeneous Sobolev space for any short time under some natural assumptions on initial data near the vacuum. This implies in particular that the homogeneous Sobolev space is crucial as studying the well-posedness for the Cauchy problem of compressible Navier-Stokes equations in the presence of vacuum at far fields even locally in time.