Aimed at graduate students, researchers and academics in mathematics, engineering, oceanography, meteorology and mechanics, this text provides a detailed introduction to the physical theory of rotating fluids, a significant part of geophysical fluid dynamics. The text is divided into four parts, with the first part providing the physical background of the geophysical models to be analysed. Part II is devoted to a self contained proof of the existence of weak (or strong) solutions to the incompressible Navier-Stokes equations. Part III deals with the rapidly rotating Navier-Stokes equations, first in the whole space, where dispersion effects are considered. The case where the domain has periodic boundary conditions is then analysed, and finally rotating Navier-Stokes equations between two plates are studied, both in the case of periodic horizontal coordinates and those in R². In Part IV the stability of Ekman boundary layers, and boundary layer effects in magnetohydrodynamics and quasigeostrophic equations are discussed. The boundary layers which appear near vertical walls are presented and formally linked with the classical Prandlt equations. Finally spherical layers are introduced, whose study is completely open.

Preface; General Introduction; On the Navier-Stokes equations; 1. Some elements of functional analysis; 2. Weak solutions of the Navier-Stokes equations; 3. Stability of the Navier-Stokes equations; 4. References and remarks on the Navier-Stokes equations; Rotating Fluids; 5. Dispersive cases; 6. The periodic case; 7. Ekman boundary layers for rotating fluids; 8. References and remarks on rotating fluids; Perspectives; 9. Stability of horizontal boundary layers; 10. Other systems; 11. Vertical layers; 12. Other layers; References

Jean-Yves CheminLaboratoire J.-L. Lions, University of Paris 6, France, Benoit DesjardinsCentre of Atomic Studies, France, Isabelle GallagherInstitut de Mathématiques de Jussieu, University of Paris 7, France, Emmanuel GrenierÉcole Normale Superiore de Lyon, France