Global well-posedness of the three-dimensional viscous primitive equations of large scale ocean and atmosphere dynamics. (English) Zbl 1151.35074

Cao, Chongsheng; Titi, Edriss S.

Ann. Math. (2) 166, No. 1, 245-267 (2007).

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 \(\mathbb R^2\)
\[ \begin{aligned} &\frac{\partial v}{\partial t}+(v\cdot\nabla)v+w\frac{\partial v}{\partial z} +\nabla p+f\vec{k}\times v+L_1v=0 \quad x\in V,\quad t>0,\\ &\frac{\partial p}{\partial z}+T=0, \quad \nabla\cdot v+\frac{\partial w}{\partial z}=0\quad x\in V,\quad t>0,\\ &\frac{\partial T}{\partial t}+v\cdot\nabla T+w\frac{\partial T}{\partial z}+L_2T=Q\quad x\in V,\quad t>0. \end{aligned}\tag{1} \]
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
\[ \begin{aligned} &\frac{\partial v}{\partial z}=h\tau,\quad w=0,\quad \frac{\partial T}{\partial z}=-\alpha(T-T^*)\quad\text{on}\;z=0,\\ &\frac{\partial v}{\partial z}=h\tau,\quad w=0,\quad \frac{\partial T}{\partial z}=0\quad\text{on}\;z=-h,\\ &v\cdot n=0,\quad \frac{\partial v}{\partial n}\times n=0,\quad \frac{\partial T}{\partial n}=0\quad\text{on}\;\partial\Omega, \end{aligned}\tag{2} \]
\[ \begin{aligned} &v(x,y,z,0)=v_0(x,y,z),\quad T(x,y,z,0)=T_0(x,y,z) \end{aligned}\tag{3} \]
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.

Reviewer: Il’ya Sh. Mogilevskij (Tver’)

MSC:

35Q35	PDEs in connection with fluid mechanics
76B15	Water waves, gravity waves; dispersion and scattering, nonlinear interaction
76B60	Atmospheric waves (MSC2010)
86A05	Hydrology, hydrography, oceanography
76B03	Existence, uniqueness, and regularity theory for incompressible inviscid fluids
86A10	Meteorology and atmospheric physics

Keywords:

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