 
Summary: Some recent asymptotic results in fluid mechanics
Thomas Alazard
The general equations of fluid mechanics are the law of mass conser
vation, the NavierStokes equation, the law of energy conservation and the
laws of thermodynamics. These equations are merely written in this general
ity. Instead, one often prefers simplified forms. To obtain reduced systems,
the easiest route is to introduce dimensionless numbers which quantify the
importance of various physical processes. Many recent works are devoted to
the study of the classical solutions when such a dimensionless number goes
to zero. A few results in this field are here reviewed.
Contents
1. Low Mach number limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1
2. Rotating fluid equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3. Planetary geostrophic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8
4. Relaxation limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1 Low Mach number limit
The target of the low Mach number limit is to justify some simplifications
that are made when discussing the fluid dynamics of highly subsonic flows.
For a fluid with density , velocity v, pressure P, internal energy e and
temperature T, the equations, written in a nondimensional way, are
