Sharp well-posedness and ill-posedness in Fourier-Besov spaces for the viscous primitive equations of geophysics

We study well-posedness and ill-posedness for Cauchy problem of the three-dimensional viscous primitive equations describing the large scale ocean and atmosphere dynamics. By using the Littlewood-Paley analysis technique, in particular Chemin-Lerner’s localization method, we prove that the Cauchy problem with Prandtl number $P=1$ is locally well-posed in the Fourier-Besov spaces $[{\dot{FB}}_{p,r}^{2-\frac{3}{p}}({\mathbb{R}}^3){]}^4$ for $1<p\le \mathrm{\infty },1\le r<\mathrm{\infty }$ and $[{\dot{FB}}_{1,r}^{-1}({\mathbb{R}}^3){]}^4$ for $1\le r\le 2$, and globally well-posed in these spaces when the initial data $(u_0,{\theta }_0)$ are small. We also prove that such problem is ill-posed in $[{\dot{FB}}_{1,r}^{-1}({\mathbb{R}}^3){]}^4$ for $2<r\le \mathrm{\infty }$, showing that the results stated above are sharp.