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We prove the ill-posedness of the 3-D baratropic Navier-Stokes equation forthe 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 thesense that a ``norm inflation" happens in finite time, here $\bar{\rho}$ is apositive constant. Our argument also shows that the compressible viscousheat-conductive flows is ill-posed for the initial density, velocity andtemperature belonging to the critical Besov space $(\dot{B}^{\f3p}_{p,1}+\bar{\rho},\,\dot{B}^{\f 3p-1}_{p,1},\,\dot{B}^{\f 3p-2}_{p,1})$ for$p>3$. These results shows that the compressible Navier-Stokes equations areill-posed in the smaller critical spaces compared with the incompressibleNavier-Stokes equations.

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