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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 $[\dot{FB}^{2-\frac{3}{p}}_{p,r}(\mathbb{R}^3)]^4$ 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 $(u_0,θ_0)$ 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 the results stated above are sharp.