×

Regularity results for linear elliptic problems related to the primitive equations. (English) Zbl 1165.86003

The authors deal with the regularity of solutions of the GFD-Stokes problem, namely:
\[ \begin{aligned} -\Delta v-\frac{\partial^2v}{\partial x_3^2}+ \operatorname{grad}p= f_1 &\quad\text{in }M^\varepsilon,\\ \int_{-\varepsilon h}^0 \operatorname{div}v=0 &\quad\text{on }\Gamma_i,\\ \frac{\partial v}{\partial x_3}+ \alpha_vv=g_v &\quad\text{on }\Gamma_i,\\ v=0 &\quad\text{on }\Gamma_b\cup\Gamma_\ell, \end{aligned}\tag{1} \]
where \(v=v(x_1,x_2,x_3)\in\mathbb R^2\), and \(p=p(x_1,x_2)\) are the unknown functions, \(f_1\) is the external volume force, \(\alpha_v>0\), \(g_v\) are given. Here the domain occupied by the ocean is of the form
\[ M_\varepsilon= \{(x_1,x_2,x_3)\in\mathbb R^3\mid (x_1,x_2)\in\Gamma_i,\;-\varepsilon h(x_1,x_2)< x_3<0\}, \]
and its boundary \(\partial M_\varepsilon= \Gamma_i\cup\Gamma_b\cup\Gamma_\ell\), where \(\Gamma_i\) is the interface between the ocean and atmosphere, \(\Gamma_b\) is the bottom boundary of the ocean and \(\Gamma_\ell\) is its lateral boundary. Moreover the authors are also interested in the regularity of solutions of the following elliptic problem related to the equation of the temperature \(T\) or the equation of salinity:
\[ \begin{aligned} -\Delta T- \frac{\partial^2T}{\partial x_3^2}= f_2 &\quad\text{in }M_\varepsilon,\\ \frac{\partial T}{\partial x_3}+ \alpha_TT=g_T &\quad\text{on }\Gamma_i,\\ \frac{\partial T}{\partial n}=0 &\quad\text{on }\Gamma_b\cup\Gamma_l, \end{aligned}\tag{2} \]
where \(T=T(x_1,x_2,x_3)\in\mathbb R\) is the unknown function, \(f_2\) is the heating source inside the ocean, \(\alpha_T>0\), \(g_T\) is given, \(n\) is the unit outward normal to the boundary.

MSC:

86A05 Hydrology, hydrography, oceanography
35J65 Nonlinear boundary value problems for linear elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35Q35 PDEs in connection with fluid mechanics
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
86A10 Meteorology and atmospheric physics
76U05 General theory of rotating fluids
PDFBibTeX XMLCite
Full Text: DOI