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

Abstract

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 \((u_0,\theta_0)\) 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 the results stated above are sharp.