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

Abstract: 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 $\F^{2-\frac{3}{p}}_{p,r}$ for $1<p\leq\infty$ and $1\leq r<\infty$, 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 $\F^{-1}_{1,r}$ for $1\leq r\leq 2$, and is globally well-posed in these spaces with small initial data. Furthermore, we prove that such problem is ill-posed in $\F^{-1}_{1,r}$ for $2< r\leq \infty$, which implies that the function space $\F^{-1}_{1,2}$ is sharp for well-posedness. In addition, using a similar argument we also prove that this problem is ill-posed in the Besov space $\B^{-1}_{\infty,r}$ with $2<r\leq\infty$.