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

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 $(\dot{B}^{3/p}_{p,1}+\bar{\rho},\,\dot{B}^{3/p-1}_{p,1})$ for $p>6$ in the sense that a "norm inflation" happens in finite time, here $\bar{\rho}$ is a positive constant. While, the compressible viscous heat-conductive flows is ill-posed for the initial density, velocity and temperature belonging to the critical Besov space $(\dot{B}^{3/p}_{p,1}+\bar{\rho},\,\dot{B}^{3/p-1}_{p,1},\,\dot{B}^{3/p-2}_{p,1})$ for $p>3$.