Sharp well-posedness and ill-posedness for the 3-D micropolar fluid system in Fourier-Besov spaces

We study the Cauchy problem of the incompressible micropolar fluid system in ${\mathbb{R}}^3$. In a recent work of the first author and Jihong Zhao \cite{ZhuZ18}, it is proved that the Cauchy problem of the incompressible micropolar fluid system is locally well-posed in the Fourier–Besov spaces ${\text{F}}_{p,r}^{2-\frac{3}{p}}$ for $1<p\le \mathrm{\infty }$ and $1\le r<\mathrm{\infty }$, and globally well-posed in these spaces with small initial data. In this work we consider the critical case $p=1$. We show that this problem is locally well-posed in ${\text{F}}_{1,r}^{-1}$ for $1\le r\le 2$, and is globally well-posed in these spaces with small initial data. Furthermore, we prove that such problem is ill-posed in ${\text{F}}_{1,r}^{-1}$ for $2<r\le \mathrm{\infty }$, which implies that the function space ${\text{F}}_{1,2}^{-1}$ is sharp for well-posedness. In addition, using a similar argument we also prove that this problem is ill-posed in the Besov space ${\text{B}}_{\mathrm{\infty },r}^{-1}$ with $2<r\le \mathrm{\infty }$.