On the ill-posedness for the full system of compressible Navier-Stokes equations

Abstract

We consider the Cauchy problem for compressible Navier--Stokes equations of the ideal gas in the three-dimensional spaces. It is known that the Cauchy problem in the scaling critical spaces of the homogeneous Besov spaces ${\dot{B}}_{p,1}^{\frac{3}{p}}\times {\dot{B}}_{p,1}^{-1+\frac{3}{p}}\times {\dot{B}}_{p,1}^{-2+\frac{3}{p}}$ is uniquely solvable for all $1<p<3$ and is ill-posed for all $p>3$. However, it is an open problem whether or not it is well-posed in the case when $p=3$. In this paper, we prove that for the case $p=3$ the Cauchy problem is ill-posed by constructing a sequence of initial data, which shows that the solution map is discontinuous.