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

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### 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.

### Author and article information

###### Journal
24 October 2015
###### Article
1510.07134
fe3c8654-bedc-4dbc-8213-8339aea96870