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 $({\dot{B}}_{p,1}^{3/p}+\bar{\rho },{\dot{B}}_{p,1}^{3/p-1})$ 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 $({\dot{B}}_{p,1}^{3/p}+\bar{\rho },{\dot{B}}_{p,1}^{3/p-1},{\dot{B}}_{p,1}^{3/p-2})$ for $p>3$.