A note on the zero Mach number limit of compressible Euler equations

Author:
Wen-An Yong

Journal:
Proc. Amer. Math. Soc. **133** (2005), 3079-3085

MSC (2000):
Primary 35B25, 35L45, 76N10

DOI:
https://doi.org/10.1090/S0002-9939-05-08077-9

Published electronically:
April 20, 2005

MathSciNet review:
2159788

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This note presents a short and elementary justification of the classical zero Mach number limit for isentropic compressible Euler equations with prepared initial data. We also show the existence of smooth compressible flows, with the Mach number sufficiently small, on the (finite) time interval where the incompressible Euler equations have smooth solutions.

**1.**T. Hagstrom and J. Lorenz,*On the stability of approximate solutions of hyperbolic-parabolic systems and the all-time existence of smooth, slightly compressible flows*, Indiana Univ. Math. J.**51**(2002), pp. 1339-1387. MR**1948453 (2004h:35188)****2.**T. Kato,*Nonstationary flows of viscous and ideal fluids in*, J. Funct. Anal.**9**(1972), pp. 296-305. MR**0481652 (58:1753)****3.**S. Klainerman and A. Majda,*Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids*, Commun. Pure Appl. Math.**34**(1981), pp. 481-524. MR**0615627 (84d:35089)****4.**S. Klainerman and A. Majda,*Compressible and incompressible fluids*, Commun. Pure Appl. Math.**35**(1982), pp. 629-651. MR**0668409 (84a:35264)****5.**A. Majda,*Compressible fluid flow and systems of conservation laws in several space variables*, Springer, New York, 1984. MR**0748308 (85e:35077)****6.**G. Métivier and S. Schochet,*The incompressible limit of the non-isentropic Euler equations*, Arch. Rational Mech. Anal.**158**(2001), pp. 61-90. MR**1834114 (2002d:76095)****7.**R. Teman,*Local existence of solutions of the Euler equations of incompressible perfect fluids*, in Turbulence and the Navier-Stokes equations, Springer, New York, 1976, pp. 184-194. MR**0467033 (57:6902)****8.**W.-A. Yong,*Singular perturbations of first-order hyperbolic systems with stiff source terms*, J. Differ. Equations**155**(1999), pp. 89-132. MR**1693210 (2000c:35011)****9.**W.-A. Yong,*Basic aspects of hyperbolic relaxation systems*, in Advances in the Theory of Shock Waves, Birkhäuser, Boston, 2001, pp. 259-305. MR**1842777 (2002g:35136)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
35B25,
35L45,
76N10

Retrieve articles in all journals with MSC (2000): 35B25, 35L45, 76N10

Additional Information

**Wen-An Yong**

Affiliation:
IWR, Universität Heidelberg, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany

Email:
yong.wen-an@iwr.uni-heidelberg.de

DOI:
https://doi.org/10.1090/S0002-9939-05-08077-9

Keywords:
Compressible Euler equations,
incompressible limit,
symmetrizable hyperbolic systems,
continuation principle,
energy estimates

Received by editor(s):
June 4, 2004

Published electronically:
April 20, 2005

Communicated by:
M. Gregory Forest

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.