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

### Qionglei Chen

Institute of Applied Physics and Computational Mathematics, Beijing, China### Changxing Miao

Institute of Applied Physics and Computational Mathematics, Beijing, China### Zhifei Zhang

Peking University, Beijing, China

## 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 $p>3$.

## Cite this article

Qionglei Chen, Changxing Miao, Zhifei Zhang, On the ill-posedness of the compressible Navier–Stokes equations in the critical Besov spaces. Rev. Mat. Iberoam. 31 (2015), no. 4, pp. 1375–1402

DOI 10.4171/RMI/872