Sharp well-posedness and ill-posedness of the three-dimensional primitive equations of geophysics in Fourier-Besov spaces. (English) Zbl 1453.35152
Summary: We study the well-posedness and ill-posedness for the Cauchy problem of the three-dimensional primitive equations describing the large-scale oceanic and atmospheric circulations. By using the Littlewood-Paley analysis technique, we prove that the Cauchy problem of the three-dimensional primitive equations with the Prandtl number \(P = 1\) is locally well-posed in the Fourier-Besov spaces \(\dot{F B}_{p, r}^{2 - \frac{3}{p}}(\mathbb{R}^3)\) for \(1 < p \leq \infty, 1 \leq r < \infty\) and \(\dot{F B}_{1, r}^{- 1}(\mathbb{R}^3)\) for \(1 \leq r \leq 2\), and is globally well-posed in these spaces when the initial data are small. We also verify that such problem is ill-posed in \(\dot{F B}_{1, r}^{- 1}(\mathbb{R}^3)\) for \(2 < r \leq \infty\), which implies that our work completes a dichotomy of well-posedness and ill-posedness for the three-dimensional primitive equations in the Fourier-Besov space framework.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q86 | PDEs in connection with geophysics |
| 35B30 | Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs |
| 35R25 | Ill-posed problems for PDEs |
| 76U05 | General theory of rotating fluids |
| 86A05 | Hydrology, hydrography, oceanography |
| 86A10 | Meteorology and atmospheric physics |
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