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A minicourse on the low Mach number limit Thomas Alazard

Summary: A minicourse on the low Mach number limit
Thomas Alazard
CNRS & Univ. Paris-Sud 11, France
1. Introduction
These lectures are devoted to the study of the so-called low Mach number limit for
classical solutions of the compressible Navier-Stokes or Euler equations for non-isentropic
fluids. The Mach number, hereafter denoted by , is a fundamental dimensionless number.
By definition, it is the ratio of a characteristic velocity in the flow to the sound speed in
the fluid. Hence, the target of the mathematical analysis of the low Mach number limit
0 is to justify some usual simplifications that are made when discussing the fluid
dynamics of highly subsonic flows (which are very common).
For highly subsonic flows, a basic assumption that is usually made is that the com-
pression due to pressure variations can be neglected. In particular, provided the sound
propagation is adiabatic, it is the same as saying that the flow is incompressible. We can
simplify the description of the governing equations by assuming that the fluid velocity is
divergence-free. (The fact that the incompressible limit is a special case of the low Mach
number limit explains why the limit 0 is a fundamental asymptotic limit in fluid
mechanics.) On the other hand, if we include heat transfert in the problem, we cannot
ignore the entropy variations. In particular we have to take into account the compression
due to the combined effects of large temperature variations and thermal conduction. In


Source: Alazard, Thomas - Département de Mathématiques, Université de Paris-Sud 11


Collections: Mathematics