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We study well-posedness and ill-posedness for Cauchy problem of thethree-dimensional viscous primitive equations describing the large scale oceanand atmosphere dynamics. By using the Littlewood-Paley analysis technique, inparticular Chemin-Lerner's localization method, we prove that the Cauchyproblem with Prandtl number $P=1$ is locally well-posed in the Fourier-Besovspaces $[\dot{FB}^{2-\frac{3}{p}}_{p,r}(\mathbb{R}^3)]^4$ for$1<p\leq\infty,1\leq r<\infty$ and $[\dot{FB}^{-1}_{1,r}(\mathbb{R}^3)]^4$ for$1\leq r\leq 2$, 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$[\dot{FB}^{-1}_{1,r}(\mathbb{R}^3)]^4$ for $2<r\leq\infty$, showing that theresults stated above are sharp.

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