Abstract EN:

In this thesis, we study the Navier-Stokes-Coriolis equations with free surface in the Sobolev-Slobodetski spaces which describe the parabolic regularity of their solutions, The methods based on these spaces were used by J. T. Beale [5], [4], V. A Solonnikov [50] and A. Tani [52] to study the initial value problem for the Navier-Stokes equations with free surface. We introduce a mathematical model of geophysical fluids and derive the Navier- Stokes-Coriolis equations. We first study the global well-posedness of the incompressible Navier-Stokes equations on the tridimensionnal torus without rotation in the case of small initial data in Sobolev spaces with high regularity. This illustrates the parabolic regularity methods. The main chapter deals with a long time existence and uniqueness result for the Navier-Stokes-Coriolis System with free surface when the initial data is close to the equilibrium. This work extends the results of J. T. Beale [4] and D. G. Sylvester [51] to the case of rotating fluids. The Chapter 4 then gathers the essential properties of Sobolev- Slobodetski in arbitrary domains and the particular case of reference domain introduced in the Chapter 4. We finally formulate in the Chapter 5 some perspectives on highly rotating fluids with free surface.