Research Article
Received 14 January 2010 Published online 17 January 2011 in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/mma.1405
MOS subject classification: 35 B 25; 35 L 45; 76 N 10
A note on incompressible limit
for compressible Euler equations
Jiang Xu
a ∗ †
and Wen-An Yong
b
Communicated by X. Wang
This paper presents a simple justification of the classical low Mach number limit in critical Besov spaces for compressible
Euler equations with prepared initial data. As the first step of this justification, we formulate a continuation principle
for general hyperbolic singular limit problems in the framework of critical Besov spaces. With this principle, it is also
shown that, for the Mach number sufficiently small, the smooth compressible flows exist on the (finite) time interval
where the incompressible Euler equations have smooth solutions, and the definite convergence orders are obtained.
Copyright © 2011 John Wiley & Sons, Ltd.
Keywords: compressible Euler equations; continuation principle; Besov spaces
1. Introduction
In a suitable nondimensional form (see, e.g. [1]), the compressible Euler equations for a polytropic fluid read as
t
+div(v) = 0,
(v)
t
+div(v ⊗v) +
-2
∇P = 0.
(1)
Here = (t,x) is the fluid density function of (t,x) ∈ [0, +∞) ×R
d
with d1, v = (v
1
,v
2
,... ,v
d
)
⊤
(⊤ represents the transpose) denotes
the fluid velocity, is the Mach number and the pressure P = P() is given as the -law
P() =
with 1. The notations div, ∇ and ⊗ are the respective divergence operator, gradient operator and tensor products. The system (1)
is supplemented with initial data
(, v)(0,x) = ( ¯ (x, ), ¯ v(x, )). (2)
Our interest is to investigate the limit when →0 goes to zero (zero Mach number limit). This limit problem has attracted much
attention since the pioneering works of Ebin [2] and Klainerman–Majda [3, 4]. Many significant contributions have been made by
Schochet, Ukai, Isozaki, Beira ¯ o da Veiga, Secchi, Métivier, Alazard, just few to mention. The interested reader is referred to [5] for
a comprehensive survey of the literature, also on Navier–Stokes equations. Subsequently, the second author [6] presented a short
and elementary approach to the above limit problem, with a definite rate of convergence from (1) to the incompressible Euler
equations. The proof relies mainly on a continuation principle formulated in [7, 8] for general hyperbolic singular limit problems.
However, all of these results are established in Sobolev spaces H
ℓ
(R
d
) with the regularity index ℓ>1 +d/ 2.
In this paper, we shall follow from the strategy in [6] to investigate the limit case = 1 +d/ 2 and verify the above zero Mach
number limit in Besov spaces. For this purpose, we first develop the continuation principle from [7, 8] in the Besov space B
2,1
(R
d
).
This principle is based on Iftimie’s local existence theory of smooth solutions to generally symmetric hyperbolic systems [9]. Actually,
the local existence for the Euler equations and Euler–Poisson equations in the same critical spaces was independently obtained
in [10, 11]. These results dealt with some concrete contents, e.g. from Poisson equation and can be regarded as a supplement of
Iftimie’s result.
With the continuation principle above, we exploit the theory of Besov spaces and quickly obtain the following result for well-
prepared initial data.
a
Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, People’s Republic of China
b
Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing 100084, People’s Republic of China
∗
Correspondence to: Jiang Xu, Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 211106, People’s Republic of China.
†
E-mail: jiangxu_79@yahoo.com.cn
Copyright © 2011 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2011, 34 831–838
831