Some mathematical aspects of geophysical fluid dynamic equations.

*(English)*Zbl 1048.86005The author presents a survey of mathematical models describing the behavior of an ocean and of the surrounding atmosphere. Notice that the coupling between the ocean and the atmosphere is not considered in this paper.

In the first part of his paper, the author presents a hierarchy of the models which are either available or not yet described. He then focuses on the hydrostatic ones with viscosity.

Starting from the classical conservation laws for fluids, the author first describes the so-called primitive equations for the ocean. These equations are deduced from the conservation laws introducing some classical simplifications. Considering prognostic variables, these equations can be reduced to evolution equations of the kind \(u^{\prime }+Au+R\left( u\right) =0\) where \(u^{\prime }\) denotes the time derivative of \(u\) and \(A\) is a linear operator which looks like Stokes operator. Appropriate boundary conditions (in terms of this Stokes operator) are imposed and the solution \( u \) starts from an initial value \(u_{0}\).

Doing the same construction for the atmosphere, but with other simplifications, the author gets the primitive equations for the atmosphere, leading to a similar evolution equation.

The last part of the paper gathers existence results from the literature concerning weak solutions either in the 3D or in the 2D cases. The main tools of the proofs are highlighted.

In the first part of his paper, the author presents a hierarchy of the models which are either available or not yet described. He then focuses on the hydrostatic ones with viscosity.

Starting from the classical conservation laws for fluids, the author first describes the so-called primitive equations for the ocean. These equations are deduced from the conservation laws introducing some classical simplifications. Considering prognostic variables, these equations can be reduced to evolution equations of the kind \(u^{\prime }+Au+R\left( u\right) =0\) where \(u^{\prime }\) denotes the time derivative of \(u\) and \(A\) is a linear operator which looks like Stokes operator. Appropriate boundary conditions (in terms of this Stokes operator) are imposed and the solution \( u \) starts from an initial value \(u_{0}\).

Doing the same construction for the atmosphere, but with other simplifications, the author gets the primitive equations for the atmosphere, leading to a similar evolution equation.

The last part of the paper gathers existence results from the literature concerning weak solutions either in the 3D or in the 2D cases. The main tools of the proofs are highlighted.

Reviewer: Alain Brillard (Mulhouse)

##### MSC:

86A05 | Hydrology, hydrography, oceanography |

86A10 | Meteorology and atmospheric physics |

76D05 | Navier-Stokes equations for incompressible viscous fluids |

35K20 | Initial-boundary value problems for second-order parabolic equations |

35Q35 | PDEs in connection with fluid mechanics |