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Zbl 1151.35074
Cao, Chongsheng; Titi, Edriss S.
Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics. (English)
[J] Ann. Math. (2) 166, No. 1, 245-267 (2007). ISSN 0003-486X; ISSN 1939-0980/e

The mathematical model of large scale ocean and atmosphere dynamics is studied in the paper. The initial boundary value problem describes this model in a cylindrical domain $V=\Omega\times(-h,0)$, where $\Omega$ is a smooth bounded domain in $\Bbb R^2$ $$\aligned &\frac{\partial v}{\partial t}+(v\cdot\nabla)v+w\frac{\partial v}{\partial z} +\nabla p+f\vec{k}\times v+L_1v=0 \quad x\in V,\quad t>0,\\ &\frac{\partial p}{\partial z}+T=0, \quad \nabla\cdot v+\frac{\partial w}{\partial z}=0\quad x\in V,\quad t>0,\\ &\frac{\partial T}{\partial t}+v\cdot\nabla T+w\frac{\partial T}{\partial z}+L_2T=Q\quad x\in V,\quad t>0. \endaligned\tag1$$ Here $v=(v_1,v_2)$ is the horizontal velocity, $(v_1,v_2,w)$ is the three-dimensional velocity, $p$ is the pressure, $T$ is the temperature, $f$ is the Coriolis parameter, $Q$ is a given heat source, $L_1$ and $L_2$ are elliptic operators $$L_1=-\frac{1}{Re_1}\Delta-\frac{1}{Re_2}\frac{\partial^2 }{\partial z^2},$$ $$L_2=-\frac{1}{Rt_1}\Delta-\frac{1}{Rt_2}\frac{\partial^2 }{\partial z^2},$$ where $Re_1, Re_2,Rt_1,Rt_2$ are positive constants, $\Delta$ is the horizontal Laplacian. The system (1) is complemented with boundary and initial conditions $$\aligned &\frac{\partial v}{\partial z}=h\tau,\quad w=0,\quad \frac{\partial T}{\partial z}=-\alpha(T-T^*)\quad\text{on}\ z=0,\\ &\frac{\partial v}{\partial z}=h\tau,\quad w=0,\quad \frac{\partial T}{\partial z}=0\quad\text{on}\ z=-h,\\ &v\cdot n=0,\quad \frac{\partial v}{\partial n}\times n=0,\quad \frac{\partial T}{\partial n}=0\quad\text{on}\ \partial\Omega, \endaligned\tag2$$ $$\aligned &v(x,y,z,0)=v_0(x,y,z),\quad T(x,y,z,0)=T_0(x,y,z) \endaligned\tag3$$ where $\tau(x,y)$ is the wind stress on the ocean surface, $n$ is the normal vector to $\partial\Omega$, $T^*$ is the typical temperature distribution on the top surface of the ocean. It is shown that the problem (1), (2), (3) has unique strong solution which continuously depends on initial data for a general cylinder $V$ and for any initial data.
[Il'ya Sh. Mogilevskij (Tver')]

MSC 2000:: *35Q35 Other equations arising in fluid mechanics
76B15 Wave motions (fluid mechanics)
76B60 Atmospheric waves
86A05 Hydrology, hydrography, oceanography
76B03 Existence, uniqueness, and regularity theory
86A10 Meteorology

Keywords: primitive equations; ocean and atmosphere dynamics; global existence; uniqueness; strong solution

Cited in: Zbl 1136.35069

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