# Revista Matemática Iberoamericana

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**Volume 31, Issue 4, 2015, pp. 1375–1402**

**DOI: 10.4171/RMI/872**

Published online: 2015-12-23

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

Qionglei Chen^{[1]}, Changxing Miao

^{[2]}and Zhifei Zhang

^{[3]}(1) Institute of Applied Physics and Computational Mathematics, Beijing, China

(2) Institute of Applied Physics and Computational Mathematics, Beijing, China

(3) Peking University, Beijing, China

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 $p>3$.

*Keywords: *Compressible Navier–Stokes equations, ill-posedness, Besov space

Chen Qionglei, Miao Changxing, Zhang Zhifei: On the ill-posedness of the compressible Navier–Stokes equations in the critical Besov spaces. *Rev. Mat. Iberoam.* 31 (2015), 1375-1402. doi: 10.4171/RMI/872