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 $1<p\le \infty$ and $1\le r<\infty$, 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 $1\le r\le 2$, and is globally well-posed in these spaces with small initial data. Furthermore, we prove that such problem is ill-posed in $\F^{-1}_{1,r}$ for $2<r\le \infty$, 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 $2<r\le \infty$.

20 pages

References in corpus (1)

• Global well-posedness for the micropolar fluid system in the critical Besov spaces