On the ill-posedness of the compressible Navier-Stokes equations in the critical Besov spaces

TL;DR

This paper demonstrates that the 3-D compressible Navier-Stokes equations are ill-posed in certain critical Besov spaces, with solutions experiencing norm inflation in finite time, highlighting fundamental differences from incompressible cases.

Contribution

It establishes ill-posedness results for the compressible Navier-Stokes equations in specific critical Besov spaces, extending understanding of their mathematical behavior.

Findings

Norm inflation occurs in finite time for certain initial data.

Ill-posedness holds for initial density and velocity in specified Besov spaces.

Results show greater ill-posedness in smaller critical spaces compared to incompressible cases.

Abstract

We prove the ill-posedness of the 3-D baratropic Navier-Stokes equation for the initial density and velocity belonging to the critical Besov space $(\dot{B}^{\f 3p}_{p,1}+\bar{\rho},\,\dot{B}^{\f 3p-1}_{p,1})$ for $p>6$ in the sense that a ``norm inflation" happens in finite time, here $\bar{\rho}$ is a positive constant. Our argument also shows that the compressible viscous heat-conductive flows is ill-posed for the initial density, velocity and temperature belonging to the critical Besov space $(\dot{B}^{\f 3p}_{p,1}+\bar{\rho},\,\dot{B}^{\f 3p-1}_{p,1},\,\dot{B}^{\f 3p-2}_{p,1})$ for $p>3$. These results shows that the compressible Navier-Stokes equations are ill-posed in the smaller critical spaces compared with the incompressible Navier-Stokes equations.