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

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 $(\dot{B}^{\f 3p}_{p,1}+\barρ,\,\dot{B}^{\f 3p-1}_{p,1})$ for $p>6$ in the sense that a ``norm inflation" happens in finite time, here $\barρ$ 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 $(\dot{B}^{\f 3p}_{p,1}+\barρ,\,\dot{B}^{\f 3p-1}_{p,1},\,\dot{B}^{\f 3p-2}_{p,1})$ 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.