Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics
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String Theory: Phenomenology; Superstring Theories;
Two-Dimensional Conformal Field Theory and Vertex
Operator Algebras; Viscous Incompressible Fluids:
Mathematical Theory.
Further Reading
Aharony O (2000) A brief review of ‘‘little string theories.’’
Classical and Quantum Gravity 17: 929–938.
Aspinwall PS (1996) K3 surfaces and string duality, 1996
preprint, arXiv:hep-th/9611137.
Bachas C et al. (eds.) (2002) Les Houches 2001: Unity from
Dual ity: Gravity, Gauge Theory and Strings.Berlin:
Springer.
de Boer J et al. (2002) Triples, fluxes, and strings. Advances in
Theoretical and Mathematical Physics 4: 995.
Connes A and Gawe¸dzki K (eds.) (1998) Les Houches 1995:
Quantum Symmetries. Amsterdam: North-Holland.
Deligne P et al. (eds.) (1999) Quantum Fields and Strings: A Course for
Mathematicians. Providence, RI: American Mathematical Society.
Douglas M et al. (eds.) (2004) Strings and Geometry: Proceedings
of the 2002 Clay School. Providence, RI: American Mathe-
matical Society.
Gauntlett J (2004) Branes, calibrations and supergravity. In:
Douglas M et al. (eds.) Strings and Geometry, pp. 79–126.
Providence, RI: American Mathematical Society.
Green MB, Schwarz JH, and Witten E (1987) Superstring Theory,
2 vols. Cambridge: Cambridge University Press.
Li J and Yau S-T (2004) The existence of supersymmetric string
theory with torsion, 2004 preprint, arXiv:hep-th/0411136.
Polchinski J (1998) String Theory, 2 vols. Cambridge: Cambridge
University Press.
Compressible Flows: Mathematical Theory
G-Q Chen, Northwestern University,
Evanston, IL, USA
ª 2006 Elsevier Ltd. All rights reserved.
Introduction
The Euler equations for compressible fluids consist of
the conservation laws of mass, momentum, and energy:
@
t
þr
x
m ¼ 0; x 2 R
d
½1
@
t
m þr
x
m m
þr
x
p ¼ 0 ½2
@
t
E þr
x
m
ðE þ pÞ
¼ 0 ½3
Equivalently, these correspond to the general form of
nonlinear hyperbolic systems of conservation laws:
@
t
u þr
x
f ð uÞ¼0; x 2 R
d
; u 2 R
n
½4
System [1]–[3] is closed by the following constitut ive
relations:
p ¼ pð; eÞ; E ¼
1
2
jmj
2
þ e ½5
In [1]–[3] and [5], = 1= is the deformation
gradient (specific volume for fluids, strain for
solids), v = (v
1
, ..., v
d
)
>
is the fluid velocity with
v = m the momentum vector, p is the scalar
pressure, and E is the total energy with e the
internal energy which is a given function of (, p)or
(, p) defined through thermodynamical relations.
The other two thermodynamic variables are tem-
perature and entropy S.If(, S) are chosen as
independent variables, then the constitutive relations
can be written as
ðe; p;Þ¼ðeð; SÞ ; pð; SÞ;ð; SÞÞ ½6
governed by dS = de þ pd = de pd=
2
. For
polytropic gases,
p ¼ pð; SÞ¼
e
S=c
v
e ¼
p
ð 1Þ
¼
p
R
½7
where R > 0 may be taken to be the universal gas
constant divided by the effective molecular weight of
the particular gas, c
v
> 0 is the specific heat at constant
volume, = 1 þ R=c
v
> 1 is the adiabatic exponent,
and can be any positive constant under scaling.
The most important criterion of applicability of
any mathematical model is its well -posedness:
existence, uniqueness, and stability. The well-posedness
theory for compressible fluid flows is far from being
complete, and many further issues are still unexplored.
In particular, the global existence and uniqueness of
solutions in R
d
, d 2, is still a major open problem, and
only partial results shed some lights on the amazing
complexity of the problem. Below, we will mainly focus
on the well-posedness issues with emphasis on the
Cauchy problem, the initial value problem:
uj
t¼0
¼ u
0
½8
first for inviscid compressible fluid flows and then
for viscous compressible fluid flows.
Throughout this article, where a cited reference is
not shown in the ‘‘Further reading’’ section, it may
usually be found by consulting Bressan (2000),
Compressible Flows: Mathematical Theory 595
Chen (2005), Dafermos (2005), Feireisl (2004),
Lions (1986, 1988) or Liv (2000).
Inviscid Compressible Fluid Flows:
Euler Equations
Solutions to the Euler equations [1]–[3] are generically
discontinuous functions obeying the Clausius–Duhem
inequality, the second law of thermodynamics:
@
t
ðSÞþr
x
ðmSÞ0 ½9
in the sense of distributions. Such discontinuous
solutions are called entropy solutions.
When a flow is isentropic, that is, entropy S is a
uniform constant S
0
in the flow, then the Euler
equations for the flow take the simpler form:
@
t
þr
x
m ¼ 0
@
t
m þr
x
m m=ðÞþr
x
p ¼ 0
½10
where the pressure is a function of the density,
p = p(, S
0
), with constant S
0
. For a polytropic gas,
pðÞ¼
;>1 ½11
where can be any positive constant by scaling. This
system can be derived from [1] to [3] as follows: for
smooth solutions of [1]–[3],entropyS(, m, E)is
conserved along fluid particle trajectories, that is,
@
t
ðSÞþr
x
ðmSÞ¼0
If the entropy is initially a uniform constant and
the solution remains smooth, then the energy
equation can be eliminated and entropy S keeps the
same constan t in later time. Thus, under constant
initial entropy, a smooth solution of [1]–[3] satisfies
the equations in [10]. Furthermore, solutions of
system [10] are also a good approximation to
solutions of system [1]–[3] even after shocks form,
since the entropy increases across a shock to the
third order in wave strength for solutions of [1]–[3],
while in [10] the entropy is constant. Moreover,
system [10] is an excellent model for the isothermal
fluid flow with = 1 and for the shallow-water flow
with = 2. For such barotropic flows (i.e., p = p()),
the energy equation [3] serves as an entropy
inequality (see Lax (1973)):
@
t
E þr
x
ðmðE þ pðÞÞ=Þ0
in the sense of distributions
In the one-dimensional case, system [1]–[3] in
Eulerian coordinates is
@
t
þ @
x
m ¼ 0;@
t
m þ @
x
m
2
= þ p
¼ 0
@
t
E þ @
x
mðE þ pÞ=ðÞ¼0
½12
The system above can be rewritten in Lagrangian
coordinates:
@
t
@
x
v ¼ 0;@
t
v þ @
x
p ¼ 0
@
t
ðe þ v
2
=2Þþ@
x
ðpvÞ¼0
½13
with v = m=, where the coordinates (t, x)are
the Lagrangian coordinates, which are different
from the Eulerian coordinates for [12];forsimp-
licity of notations, we do not distinguish them.
For the barotropic case, systems [12] and [13]
reduce to
@
t
þ @
x
m ¼ 0;@
t
m þ @
x
m
2
= þ p
¼ 0 ½14
and
@
t
@
x
v ¼ 0;@
t
v þ @
x
p ¼ 0 ½15
respectively, where press ure p = p() =
~
p(), = 1=.
The solutions of [12] and [13], as well as [14] and
[15], are equivalent even for entropy solutions with
vacuum where = 0.
The potential flow is well known in transonic
aerodynamics, beyond the isentropic approxi-
mation [10] from [1] to [3]. Denote D
t
= @
t
þ
P
d
k = 1
v
k
@
x
k
the convective derivative along fluid
particle trajectories. From [1] to [3],wehave
D
t
S ¼ 0 ½16
and, by taking the curl of the momentum equations,
D
t
!
¼
!
r
x
v þ
p
S
ð; SÞ
3
r
x
r
x
S ½17
The identities [16] and [17] imply that a smooth
solution of [1]–[3] which is both isentropic and
irrotational at time t = 0 remains isentropic and
irrotational for all later times, as long as this
solution stays smooth. Then, the conditions
S = S
0
= const. and ! = curl
x
v = 0 are reasona ble for
smooth solutions. For a smooth irrotational solu-
tion, we integrate the d-momentum equations in
[10] through Bernoulli’s law:
@
t
v þr
x
ðjvj
2
=2Þþr
x
hðÞ¼0
where h
0
() = p
(, S
0
)=. On a simply connected
space region, the condition curl
x
v = 0 implies that
there exists such that v = r
x
. Then,
@
t
þr
x
ðr
x
Þ¼0
@
t
þ
1
2
jr
x
j
2
þ hðÞ¼K
½18
for some constant K. From the second equation in
[18], we have
ðDÞ¼h
1
ðK ð@
t
þ
1
2
jr
x
j
2
ÞÞ
596 Compressible Flows: Mathematical Theory
Then, system [18] can be rewritten as the following
time-dependent potential flow equation of second
order:
@
t
ðDÞþr
x
ððDÞr
x
Þ¼0 ½19
For a steady solution =(x), that is, @
t
=0,
we obtain the celebrated steady potential flow
equation of aerodynamics:
r
x
ððr
x
Þr
x
Þ¼0 ½20
In applications in aerodynamics, [18] or [19] is
used for discontinuous solutions, and the empirical
evidence is that entropy solutions of [18] or [19] are
fairly good approximations to entropy solutions for
[1]–[3] provided that (1) the shock strengths are
small, (2) the curvature of shock fronts is not too
large, and (3) there is a small amount of vorticity in
the region of interest. Model [19] or [18] is an
excellent model to capture multidimensional shock
waves by ignoring vorticity waves, while the
incompressible Euler equations are an excellent
model to capture multidimensional vorticity waves
by ignoring shock waves.
Local Well-Posedness for Classical Solutions
Consider the Cauchy problem for the Euler equatio ns
[1]–[3] with Cauchy data [8]:
Assume that u
0
: R
d
!Dis in H
s
\ L
1
with s > d=2 þ 1.
Then, for the Cauchy problem [1]–[3] and [8],there
exists a finite time T = T(ku
0
k
s
, ku
0
k
L
1
) 2 (0, 1)such
that there is a unique, stable bounded classical solution
u 2 C
1
([0, T] R
d
)withu(t , x) 2Dfor (t, x) 2 [0, T]
R
d
and u 2 C([0, T]; H
s
) \ C
1
([0, T]; H
s1
). Moreover,
the interval [0, T)withT < 1 is the maximal interval
of the classical H
s
existence for [1]–[3] if and only if
either k(u
t
,r
x
u)k
L
1
!1 or u(t, x) escapes every
compact subset K ! D as t ! T.
This local existence can be established by relying
solely on the elementary linear existence theory for
symmetric hyperbolic systems with smoot h coeffi-
cients (cf. Majda (1984)), or by the abstract
semigroup theory (Kato 1975) .
Formation of Singularities
For the one-dimensional case, singularities include
the development of shock wave s and formation of
vacuum states. For the multidimensional case, the
situation is much more complicated: besides shock
waves and vacuum states, singularities can also be
generated from vortex sheets, focusing and breaking
of waves, among others.
Consider the Cauchy problem of the Euler
equations [1]–[3] in R
3
for polytropic gases with
smooth initial data:
ð; v; SÞj
t¼0
¼ð
0
; v
0
; S
0
ÞðxÞ
0
ðxÞ > 0; x 2 R
3
½21
satisfying (
0
, v
0
, S
0
)(x) = (
,0,
S) for jxjL, where
>0,
S, and L are given constants. The equations
possess a unique local C
1
solution (, v, S)(t, x) with
(t, x) > 0 provided that the initial data [21] is
sufficiently regular. The support of the smooth
disturbance (
0
(x)
, v
0
(x), S
0
(x)
S) propagates
with speed at most =
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
(
,
S)
q
(the sound speed),
that is,
ð; v; SÞðt; xÞ¼ð
; 0 ;
SÞ if jxjL þ t ½22
Define
PðtÞ¼
Z
R
3
pððt ; xÞ; Sðt; xÞÞ
1=
pð
;
SÞ
1=
dx
FðtÞ¼
Z
R
3
ðvÞðt; xÞx dx
which, roughly speaking, measure the entropy and the
radial component of momentum. Then, if (, v, S)(t, x)
is a C
1
solution of [1]–[3] and [21] for 0 < t < T,and
Pð0Þ0; Fð0Þ >R
4
max
x
0
ðxÞ
with ¼ 16=3 ½23
then the lifespan T of the C
1
solution is finite
(Sideris 1985).
To illustrate a way in which the conditions in
[23] may be satisfied, consider the initial data:
0
=
, S
0
=
S. Then P(0) = 0, and [23] holds if
Z
jxj<R
v
0
ðxÞx dx >R
4
Comparing both sides, one finds that the initial
velocity must be supersonic in some region relative
to the sound speed at infinity. The formation of a
singularity (presumably a shock wave) is detected as
the disturbance overtakes the wave front forcing the
front to propagate with supersonic speed.
Singularities are formed even without the condi-
tion of largeness, such as [23], being satisfied. For
example, if S
0
(x)
S and, for some 0 < R
0
< R,
Z
jxj>r
jxj
1
ðjxjrÞ
2
ð
0
ðxÞ
Þdx > 0
Z
jxj>r
jxj
3
ðjxj
2
r
2
Þ
0
ðxÞv
0
ðxÞx dx 0
½24
for R
0
< r < R, then the lifespan T of the C
1
solution of [1]–[3] and [21] is finite. The
Compressible Flows: Mathematical Theory 597
assumptions in [24] mean that, in an average sense,
the gas must be slightly compresse d and outgoing
directly behind the wave front.
Local Well-Posedness for Shock-Front Solutions
For a general hyperbolic system of conservation laws
[4], shock-front solutions are discontinuous, piecewise
smooth entropy solutions with the following structure:
1. There exists a C
2
spacetime hypersurface S(t)
defined in (t, x) for 0 t T with spacetime
normal (
t
,
x
) = (
t
,
1
, ...,
d
) as well as two
C
1
vector-valued functions: u
þ
(t, x) and u
(t, x),
defined on respective domains D
þ
and D
on
either side of the hypersurface S( t) and satisfying
@
t
u
þr
x
f (u
) = 0inD
;
2. The jump across the hypersurface S(t) satisfies the
Rankine–Hugoniot condition:
{
t
ðu
þ
u
Þþ
x
ðf ðu
þ
Þf ðu
ÞÞ}
j
S
= 0
For [4], the surface S is not known in advance
and must be determined as part of the solution of
the problem; thus, the two equations in (1)–(2)
describe a multidimensional, highly nonlinear, free-
boundary-value problem. The initial data yielding
shock-front solutions is defined as follows. Let S
0
be
a smooth hypersurface parametrized by , and let
() = (
1
, ...,
d
)() be a unit normal to S
0
. Define
the piecewise smooth initial values for respective
domains D
þ
0
and D
0
on either side of the hypersur-
face S
0
as
u
0
ðxÞ¼
u
þ
0
ðxÞ; x 2D
þ
0
u
0
ðxÞ; x 2D
0
½25
It is assumed that the initial jump in [25] satisfies the
Rankine–Hugoniot condition, that is, there is a
smooth scalar function () so that
ðÞ u
þ
0
ðÞu
0
ðÞ
þ ðÞ fu
þ
0
ðÞ
fu
0
ðÞ
¼ 0 ½26
and that () does not define a characteristic
direction, that is,
ðÞ 6¼
i
u
0
;2
S
0
; 1 i n ½27
where
i
, i = 1, ..., n, are the eigenvalues of [4].Itis
natural to require that S(0) = S
0
.
Consider the Euler equations [1]–[3] in R
3
for
polytropic gases with piecewise smooth initial data:
ð; v; EÞj
t¼0
¼
þ
0
; v
þ
0
; E
þ
ðxÞ; x 2D
þ
0
0
; v
0
; E
ðxÞ; x 2D
0
½28
Assume that S
0
is a smooth compact surface in R
3
and that (
þ
0
, v
þ
0
, E
þ
0
)(x) belongs to the uniform local
Sobolev space H
s
ul
(D
þ
0
), while (
0
, v
0
, E
0
)(x) belongs
to the Sobolev space H
s
(D
0
), for some fixed s 10.
Assume also that there is a function () 2 H
s
(S
0
)
so that [26] and [27] hold , and the compatibility
conditions up to order s 1 are satisfied on S
0
by
the initial data, togeth er with the entropy condition:
v
þ
0
ðÞþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
ð
þ
0
; S
þ
0
Þ
q
<ðÞ
< v
0
ðÞþ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
ð
0
; S
0
Þ
q
½29
Then, there are a C
2
hypersurface S(t) and C
1
functions (
, v
, E
)(t, x) defined for t 2 [0, T],
with T sufficiently small, so that
ð;v;EÞðt;xÞ¼
ð
þ
;v
þ
;E
þ
Þðt; xÞ; ðt;xÞ2D
þ
ð
;v
;E
Þðt; xÞ; ðt;xÞ2D
½30
is the discontinuous shock-front solution of the
Cauchy problem [1]–[3] and [28]. Here a vector
function u is in H
s
ul
, provided that there exists
some r > 0 so that max
y2R
d
kw
r,y
uk
H
s
< 1 with
w
r, y
(x)=w((xy )=r), where w 2C
1
0
(R
d
)isa
function so that w(x) 0, w(x)=1 when jxj1=2,
and w(x)=0 when jxj> 1.
The compatibility conditions are needed in order
to avoid the formation of discontinuities in higher
derivatives along other characteristic surfaces ema-
nating from S
0
: Once the main condition [26] is
satisfied, the compatibility conditions are automati-
cally guaranteed for a wide class of initial data. The
idea of the proof is to use the existence of a strictly
convex entropy and the symmetrization of [4]; the
shock-front solutions are defined as the limit of a
convergent classical iteration scheme based on
a linearization by using the theory of linearized
stability for shock fronts (Majda 1984). The uni-
form existence time of shock-front solutions in
shock strength can be achieved (Me´tivier 1990).
Global Theory in L
1
for the Isentropic Euler
Equations for x 2 R
Consider the Cauchy problem for [14] with initial
data:
ð; mÞj
t¼0
¼ð
0
; m
0
ÞðxÞ½31
where
0
and m
0
are in the physical region
{(, m): 0, jmjC
0
} for some C
0
> 0. System
[14] is strictly hyperbolic at the states with >0,
and strict hyperbolicity fails at the vacuum states
V := {(, m=): = 0, jm=j < 1 }. Then, we have:
1. There exists a global solution (, m)(t, x) of the
Cauchy problem [14] and [31] satisfying
0 ðt; xÞC; jmðt; xÞj Cðt; xÞ½32
598 Compressible Flows: Mathematical Theory
for some C > 0 depending only on C
0
and , and
the entropy inequality
@
t
ð ; mÞþ@
x
qð; mÞ0 ½33
in the sense of distributions for any convex weak
entropy–entropy flux pair (, q), that is,
rqð; mÞ¼rð; mÞrf ð; mÞ
with
r
2
ð ; mÞ0 and j
V
¼ 0
2. The solution operator (, m)(t, ) = S
t
(
0
, m
0
)( ),
determined by (1), is compact in L
1
loc
(R) for t > 0;
3. Furthermore, if (
0
, m
0
)(x) is periodic with period
P, then there exists a global periodic solution
(, m)(t, x) with [32] such that (, m)(t, x) asymp-
totically decays to
1
jPj
Z
P
ð
0
; m
0
ÞðxÞdx
in L
1
.
The con vergence of the Lax–Friedrichs scheme,
the Godunov scheme, and the vanishing viscosity
method for system [14] have also been established.
The results are based on a compensated compact-
ness framework to replace the BV compactness
framework. For a gas obeying the -law, the case
= (N þ 2)=N, N 5 odd, was first studied by
DiPerna (1983), and the case 1 < 5=3 for
usual gases was first solved by Chen (1986) and
Ding-Chen-Luo (1985). The cases 3 and 5=3 <
<3 were treated by Lions–Perthame–Tadmor
(1994) and Lions–Perthame–Souganidis (1996),
respectively. The case of general pressure laws was
solved by Chen–LeFloch (2000, 2003). All the
results for entropy solutions to [14] in Eulerian
coordinates can equivalently be prese nted as the
corresponding results for entropy solutions to [15]
in Lagrangian coordinates. The isothermal case
= 1 was treated by Huang–Wang (2002).
Global Theory in BV for the Adiabatic Euler
Equations for x 2R
Consider the Euler equations [13] for polytropic
gases with the Cauchy data:
ð;v; SÞj
t¼0
¼ð
0
; v
0
; S
0
ÞðxÞ½34
Then we have (Liu 1977, Temple 1981, Chen and
Wagner 2003):
Let K {(, v, S):>0} be a compact set in R
þ
R
2
,
and let N 1 be any constant. Then there exists a
constant C
0
= C
0
(K, N), independent of 2 (1, 5= 3],
such that, for every initial data (
0
, v
0
, S
0
) 2 K with
TV
R
(
0
, v
0
, S
0
) N,when
ð 1ÞTV
R
ð
0
; v
0
; S
0
ÞC
0
for any 2ð1; 5=3
the Cauchy problem [13] and [34] has a global
entropy solution (, v, S)(t, x) which is bounded and
satisfies
TV
R
ð;v; SÞðt; Þ CTV
R
ð
0
; v
0
; S
0
Þ
for some constant C > 0 independent of .
This result specially includes that for the baro-
tropic case (Nishida 1968, Nishida–Smoller 1973,
DiPerna 1973). Some efforts in the direction of
relaxing the requirement of small total variation
have been made. Some extensions to the initial-
boundary value problems have also been made. In
addition, an entropy solution in BV with periodic
data or compact support decays when t ! 0.
Furthermore, even for a general hyperbolic system
[4] for x 2 R, we have:
If the initial data functions u
0
(x) and v
0
(x) have
sufficiently small total variation and u
0
v
0
2 L
1
(R),
then, for the corresponding exact Glimm, or wave-
front tracking, or vanishing viscosity solutions u(t, x)
and v(t, x) of the Cauchy problem [4] and [8], there
exists a constant C > 0 such that
kuðt; Þ vðt; Þk
L
1
ðRÞ
Cku
0
v
0
k
L
1
ðRÞ
for all t > 0 ½35
An immediate consequence is that the whole
sequence of the approximate solutions constructed
by the Glimm (1965) scheme, as well as the wave-
front tracking method and the vanishing viscosity
method, converges to a unique entropy solution of
[4] and [8] when the mesh size or the viscosity
coefficient tends to zero. More detailed discussions
and extensive references about the L
1
-stability of BV
entropy solutions and related topics can be found in
Bressan (2000) and Daferm os (2000); also see Chen
and Wang (2002). Furthermore, the Riemann solu-
tion is unique and asymptotically stable in the class
of entropy solutions to [13] with large variation
satisfying only one physical entropy inequality
(Chen-Frid-Li 2002).
Multidimensional Steady Theory
The mathematical study of two-dimensional steady
supersonic flows past wedges, whose vertex angles
are less than the critical angle, can date back to the
1940s, since the stability of such flows is fundamental
in applications (cf. Courant–Friedrichs (1948)). Local
solutions around the wedge vertex were first
constructed (Gu 1962, Schaeffer 1976, Li 1980).
Compressible Flows: Mathematical Theory 599
Such global potential solutions were constructed
when the wedge has some convexity, or is a small
perturbation of the straight wedge with fast decay in
the flow direction (Chen 2001, Chen-Xin-Yin 2002),
or is piecewise smooth which is a small perturba-
tion of straight wedge (Zhang 2003). For the
two-dimensional steady s upersonic flows gov-
erned by the full Euler equations past Lipschitz
wedges, it indicates (Chen-Zhang-Zhu 2005a)
that, when the wedge vertex angle is less than
the c ritical angle, the strong shock front
emanating f rom the wedge vertex is nonlinearly
stable in structure g lobally, although there may be
many weak shocks and vortex sheets between the
wedge boundary and the strong shock front, under
the BV p erturbation of the wedge so that the total
variation of the tangent function along the wedge
boundary is suitably small. This asserts that any
supersonic shock for the wedge problem is non-
linearly stable.
A self-similar gas flow past an infinite cone in R
3
with small vertex angle is also nonlinearly stable
upon the BV perturbation of the obstacle (Lien-Liu
1999). It is still open for the nonlinear stability when
the infinite cone in R
3
has arbitrary vertex angle.
The stability issues of supersonic vertex sheets have
been studied by classical linearized stability analysis,
large-scale numerical simulations, and asymptotic
analysis. In particular, the nonlinear development of
instabilities of supersonic vortex sheets at high
Mach number was predicted as time evolves
(Woodward 1985, Artola-Majda 1989). In contrast
with the prediction of evolution instability, steady
supersonic vortex sheets, as time-asymptotics, are
stable globally in structure, even under the BV
perturbation of the Lipschitz walls, although there
may be many weak shocks and supersonic vortex
sheets away from the strong vortex sheet (Chen-
Zhang-Zhu 2005b).
Transonic shock problems for steady fluid flows
are important in applications (cf. Courant and
Friedrichs (1948)). A program on the existence and
stability of multidimensional transonic shocks has
been initiated and three new analytical approaches
have been developed (Chen-Feldman 2003, 2004).
The transonic problems include the existence and
stability of transonic shocks in the whole R
d
, the
existence and stability of transonic flows past finite
or infinite nozzles, the stability of transonic flows
past infinite nonsmooth wedges, and the existence of
regular shock reflection solutions. The first
approach is an iteration scheme based on the
nondegeneracy of the free boundary condition: the
jump of the normal derivative of a solution across
the free bounda ry has a strictly positive lower bound
(Chen-Feldman 2003, 2004), which works for the
nonlinear equations whose coeffic ients may depend
on not only the solution itself but also the gradients
of the solution. The second approach is a partial
hodograph procedure, with which the existence and
stability of multidimensional transonic shocks that
are not nearly orthogonal to the flow direction can
be handled (Chen-Feldman 2004): one of the main
ingredients in this approach is to employ a partial
hodograph transform to reduce the free boundary
problem into a conormal boundary value problem
for the corresponding nonlinear equations of diver-
gence form and then develop techniques to solve the
conormal boundary value problem. When the reg-
ularity of the steady perturbation is C
3,
or higher,
the third approach is to employ the implicit function
theorem to deal with the existence and stability
problem. Another iteration approach, which works
well for the two-dimensional equations whose coeffi-
cients depend only on the solution itself, has also
been developed (Canic-Keyfitz-Lieberman 2000).
Further longstanding open problems include the
existence of global transonic flows past an airfoil or
a smooth obstacle (Morawetz 1956–58, 1985).
Multidimensional Unsteady Problems
Now we present some multidimensional time-
dependent problems with a simplifying feature that
the data (domain and/or the initial data) coupled
with the structure of the underlying equations
obey certain geometric structure so that the multi-
dimensional problems can be reduced to lower-
dimensional problems with more complicated
couplings. Different types of geometric structure
call for different techniques.
The Euler equations for compressible fluids
with geometric structure describe many important
fluid flows, including spherically symmetric flows
and self-similar flows. Such geometric flows
are motivated by many physical problems such as
shock diffractions, supernovas formation in stellar
dynamics, inertial confinement fusion, and under-
water explosions. For the initial data with large
amplitude having geometric structure, the requi-
red physical insight is: (1) whether the solution
has the same geometric structure globally and
(2) whether the solution blows up to infinity in a
finite time. These questions are not easily under-
stood in physical experiments and numer ical simula-
tions, especially for the blow-up, because of the
limited capacity of available instruments and
computers.
600 Compressible Flows: Mathematical Theory
The first type of geometric structure is spherical
symmetry. A criterion for L
1
Cauchy data functions
of arbitrarily large amplitude was observed to
guarantee the existence of spherically symmetric
solutions in L
1
in the large for the isentropic flows,
which model outgoing blast waves and large-time
asymptotic solutions (Chen 1997). On the other hand,
it is evident that the density blows up as jxj!0in
general, especially for the focusing case; the singular-
ity at the origin makes the problem truly multi-
dimensional due to the reflection of waves from
infinity and their strengthening as they move radially
inwards. One of the important open questions is to
understand the order of singularity, (t, jxj) jxj
,
at the origin for bounded Cauchy data.
The second type of geometric structure is self-
similarity, that is, the solutions with initial data
functions that give rise to self-similar solutions,
especially including Riemann solutions. Compressi-
ble flow equations in R
d
, d 2, with one or more
linearly degenerate modes of wave propagation have
additional difficulties. In that case, the global flow is
governed by a reduced (self-similar) system which is
of composite (hyperbolic–elliptic) type in the sub-
sonic region. The linearly degenerate waves give rise
to one or more families of degenerate characteristics
which remain real in the subsonic region. In some
cases, the reduced equations couple an elliptic
(degenerate elliptic) problem for the density with a
hyperbolic (transport) equation for the vorticity.
An important prototype for both practical
applications and the theory of multidimensional
complex wave patterns is the problem of diffraction
of a shock wave which is incident along an inclined
ramp (see Glimm and Majda (1991)). When a
plane shock hits a wedge head-on, a self-similar
reflected shock moves outw ard as the original
shock moves forward. The computational and
asymptotic analysis shows that various patterns of
reflected shocks may occur, including regular
reflection and (simple, double, and complex)
Mach reflections. The main part or whole reflected
shock is a transonic shock in the self-similar
coordinates, for which the corresponding equation
changes the type from hyperbolic to elliptic across
the shock. There are few rigorous mathematical
results on the global existence and stability of
shock reflection solutions and the transition among
regular, simple Mach, double Mach, and complex
Mach reflections for the potential flow equa-
tion [19] and the full Euler equations [1]–[3].
Some resul ts were recently obtained for simplified
models including the transonic small-disturbance
equation near the reflection point and the pressure
gradient equation when the wedge is close to a flat
wall.
For the potential flow equation [19], a self-
similar solution is a solution of the form:
=t(y), y = x=t. Letting ’(y) = y
2
=2 þ (y),
then the system can be rewritten in the form of a
second-order equation of mixed hyperbolic–elliptic
type in y 2 R
d
by scaling:
r
y
ððjr
y
’j
2
;’Þr
y
’Þþdðjr
y
’j
2
;’Þ¼0 ½36
with (q
2
, z) = (1 (q
2
þ 2z)=2)
1=(1)
. Equation [36]
at jr
y
’j= q is hyperbolic (pseudosupersonic) if
(q
2
, z) þ q
q
(q
2
, z) < 0 and elliptic (pseudosubsonic)
if (q
2
, z) þ q
q
(q
2
, z) > 0. Under this framework,
the nature of the shock reflection pattern has been
explored for weak incident shocks (strength b)and
small wedge angles 2
w
by a number of different
scalings, a study of mixed equations, and matching
asymptotics for the different scalings, where the
parameter = c
1
2
w
=b( þ 1) ranges from 0 to 1
and c
1
is the speed of sound behind the incident
shock (Morawetz 1994). For >2, a reg ular
reflection of both strong and weak kinds is
possible as well as a Mach reflection; for <
1=2, a Mach reflection occurs and the flow behind
the reflection is subsonic and can be c onstructed in
principle (with an elliptic problem) and matched;
and for 1=2 <<2, the flow behind a Mach
reflection may be t ransonic which is a solution of
a nonlinear boundary-value problem of mixed
type. The basic pattern of reflection has been
shown to be an almost semicircular shock issuing,
for a regular reflection, from the reflection point
on the wedge and, for a Mach reflection, matched
with a local interaction flow. Some related
observations were also made (Keller-Blank 1951,
Hunter-Keller 1984, Hunter 1988). It is important
to establish rigorous proofs. Recently, a rigorous
ex istence proof was established for global solutions
to shock reflection by large-angle wedges in Chen
and Feldman (2005).
Analytical Frameworks for Entropy Solutions
The recen t great progress for entropy solutions for
one-dimensional time-dependent Euler equations
and two-dimensional steady Euler equations, based
on BV, L
1
, or even L
1
estimates, naturally arises the
expectation that a similar approach may also be
effective for the multidimensio nal Euler equations,
or more generally, hyperbolic systems of conserva-
tion laws, especially,
kuðt ; Þk
BV
Cku
0
k
BV
½37
Compressible Flows: Mathematical Theory 601
Unfortunately, this is not the case. The necessary
condition for [37] to be held for p 6¼ 2 (Rauch
1986) is
rf
k
ðuÞrf
l
ðuÞ¼rf
l
ðuÞrf
k
ðuÞ
for all k; l ¼ 1; 2; ...; d ½38
The analysis suggests that only systems in which the
commutativity relation [38] holds offer any hope for
treatment in the framework of BV. This special case
includes the scalar case n = 1 and the case of one
space dimension d = 1. Beyond that, it contains very
few systems of physical interest.
In this regard, it is important to identify effective
analytical frameworks for studying entropy solu-
tions of the multidimensional Euler equations [1]–
[3], which are not in BV. Naturally, we want to
approach the questions of existence, stability,
uniqueness, and long-time behavior of entropy
solutions with as much generality as possible. For
this purpose, a theory of divergence-measure fields
to construct such a global framework has been
developed for studying entropy solutions (Chen-Frid
1999, 2000, Chen-Torres 2005, Chen-Torres-Ziemer
2005). For more details, see Chen (2005).
Viscous Compressible Fluid Flows:
Navier–Stokes Equations
Compressible fluid flows that are viscous and
conduct heat are governed by the following
Navier–Stokes equations:
@
t
þr
x
m ¼ 0; x 2 R
d
½39
@
t
m þr
x
m m
þr
x
p ¼r
x
S ½40
@
t
E þr
x
m
ðE þpÞ
¼r
x
m
S
r
x
q ½41
Here, S =S( r
x
v, , ) is the viscous stress tensor
which is symmetric from the conservation of angular
momentum and q is the heat flux. If the fluid is
isotropic and the viscous tensor S is a linear function
of r
x
v and invariant under a change of reference
frame (translation and rotation), then we deduce
from elementary algebraic manipulations that
necessarily
S ¼ ð; Þr
x
v þ 2ð; ÞD ½42
which corresponds to the Newtonian fluids, where
D = (r
x
v þ (r
x
v)
>
)=2 is the deformation tensor and
and are the Lame´ viscosity coefficients.
Furthermore, since the fluid is isotropic, we are led
to the Fourier law:
q ¼kð; ; jr
x
jÞr
x
for scalar function k which, in most cases, is taken
to be simply a function of and , or even a
constant called the thermal conduction coefficient.
Again, system [39]–[41] is closed by the constitutive
relations in [5]. The equation for entropy S is
@
t
ðSÞþr
x
mS þ
q
¼
Sðr
x
vÞ : r
x
v
q r
x
2
½43
The second law of thermodynamics indicates that
the right-hand side of [43] should be non-negative
which yields the restriction:
kð; ; jr
x
jÞ 0; 0;þ2=d 0
The case >0 and þ >0 is the viscous case
with heat conductivity k > 0. In particular, the
kinetic theory indicates that the Stokes relationship
should hold, namely = 2=d and the adiabat ic
component = 5=3 for monatomic gases.
In mathematic al viscous fluid dynamics, an
important model is the barotropic model for
viscous fluids, that is, p = p(). Then, the specific
energy E can be taken in the form of
E = (1=2)jvj
2
þ e() with e
0
() = p()=
2
. For clas-
sical solutions, the energy of a barotropic flow
satisfies the equality:
@
t
E þr
x
ððE þ pÞvÞ¼r
x
ðSvÞS : r
x
v
which is now a direct consequence of [39] and [40].
The question of local existence of classical
solutions to [39]–[41] for regular initial data was
addressed by Nash (1962), where there is no
indication whether or not these solutions exist for
all times.
In the case of one space dimension, the well-
posedness is largely settled. The basic result for the
existence of classical solutions is that of Kazhikhov
(1976); see Lions (1998) and Feireisl (20 04) for
extensive references. The discontinuous solutions
have been constructed (Shelukhin 1979, Serre 1986,
Hoff 1987, Chen-Hoff-Trivisa 2000).
For the Navier–Stokes equations in R
3
with
general equation of state, the global classical
solutions for the Cauchy problem and various
initial-boundary value problems whose initial data
is small around a constant state have been
602 Compressible Flows: Mathematical Theory
constructed (Matsumura-Nishida 1980, 1983). The
approach is to obtain a priori estimates via energy
methods for extending the local solution or for a
difference method globally. These results have been
extended to the Cauchy problem or the initial-
boundary value problems with small discontinuous
initial data (Hoff 1997).
For the Navier–Stokes equations in R
d
for
barotropic flows with [11] and large initial data,
the global existence of solutions containing vacuum
for the Cauchy problem or various initial-boundary
value problems was first established by Lions
(1998) for 3=2ifd = 2, 9= 5ifd = 3, and
>d=2ifd 4. The gap was closed by Feireisl–
Novotny´–Petzeltova´ (2001) for the full range
>d=2. These results have been extended to the
full Navier–Stokes equations describing the moti on
of a genera l compressible, viscous, and heat con-
ducting fluid (see Feireisl (2004)). The physically
relevant isothermal case, = 1, is completely open
even if d = 2. The only large data existence result is
that for radially symmetric data (Hoff 1992). The
general case 1 and d = 3 for radially symmetric
data was solved only recently (Jiang-Zhang 2001).
The lower-bound estimate on the density is a
delicate issue. Weak solutions containing vacuum
for the isentropic viscous flows with constant
viscosity are unstable in general (Hoff-Serre
1991). Hence, it is important to see whether
vacuum will never develop if the initial data is
away from vacuum; this has been shown for the
one-dimensional case for large initial data and
for the multidimensional case with small data. O n
the other hand, from the kinetic theory, if
solutions contain vacuum, then the viscosity
coefficients in the N avier–Stokes equations should
depend on the density near vacuum; this indeed
stabilizes t he solutions for the one-dimensional
case.
The stability of viscous shock waves has been
studied for the one-dimensional case (see Liu (2000)
and the references therein). The compressible–
incompressible limits from the isentropic compres-
sible to incompressible Navier–Stokes equations
when the Mach number tends to zero have been
established for arbitrarily weak solutions (Lions-
Masmoudi 1998) and for smoot h solutions and a
class of initial data functions (Hoff 1998) .
The inviscid limits from the Navier–Stokes equa-
tions to the Euler equations have been established as
long as the solutions of the Euler equations are
smooth, when the viscosity and heat conductivity
coefficients tend to zero (Klainerman-Majda 1982).
It is completely open for general entropy solutions,
even in the one-dimensional case.
See also: Breaking Water Waves; Capillary Surfaces;
Fluid Mechanics: Numerical Methods; Geophysical
Dynamics; Incompressible Euler Equations:
Mathematical Theory; Inviscid Flows;
Magnetohydrodynamics; Newtonian Fluids and
Thermohydraulics; Non-Newtonian Fluids; Partial
Differential Equations: Some Examples; Stability of
Flows; Viscous Incompressible Fluids: Mathematical
Theory.
Further Reading
Bressan A (2000) Hyperbolic Systems of Conservation Laws: The
One-Dimensional Cauchy Problem. Oxford: Oxford Univer-
sity Press.
Chen G-Q (2005) Euler equations and related hyperbolic
conservation laws. In: Dafermos CM and Feireisl E (eds.)
Handbook of Differential Equations II: Evolutionary Differ-
ential Equations, Chapter 1, pp. 1–104. Amsterdam: Elsevier.
Chen G-Q and Wang D (2002) The Cauchy problem for the
Euler equations for compressible fluids. In: Friedlander S
and Serre D (e ds.) Handbook of Mathematical Fluid
Dynamics, vol. 1, ch. 5, pp. 421–543. Amsterdam: Elsevier
Science B.V.
Courant R and Friedrichs KO (1948) Supersonic Flow and Shock
Waves. New York: Springer.
Dafermos CM (2005) Hyperbolic Conservation Laws in Con-
tinuum Physics (2nd edn). Berlin: Springer.
Feireisl E (2004) Dynamics of Viscous Compressible Fluids.
Oxford: Oxford University Press.
Glimm J (1965) Solutions in the large for nonlinear hyperbolic
system of equations. Communications on Pure and Applied
Mathematics 18: 95–105.
Glimm J and Majda A (1991) Multidimensional Hyperbolic
Problems and Computations. New York: Springer.
Lax PD (1973) Hyperbolic Systems of Conservation Laws and
the Mathematical Theory of Shock Waves. Philadelphia:
SIAM.
Lions PL (1996, 1998) Mathematical Topics in Fluid Mechanics,
vols. 1–2. New York: Oxford University Press.
Liu T-P (2000) Hyperbolic and Viscous Conservation Laws,
CBMS-NSF RCSAM, vol. 72. Philadelphia: SIAM.
Majda A (1984) Compressible Fluid Flow and Systems of
Conservation Laws in Several Space Variables. New York:
Springer.
Compressible Flows: Mathematical Theory 603
Computational Methods in General Relativity: The Theory
M W Choptuik, University of British Columbia,
Vancouver, Canada
ª 2006 Elsevier Ltd. All rights reserved.
Conventions and Units
This article adopts many of the conventions and
notations of Misner, Thorne, and Wheeler (1973) –
hereafter denoted MTW – including metric signature
( þþþ); definitions of Christoffel symbols and
curvature tensors (up to index permutations per-
mitted by standard symmetries of the tensors in a
coordinate basis); the use of Greek indices
, , , ..., ranging over the spacetime coordinate
values (0, 1, 2, 3) !(t, x
1
, x
2
, x
3
), to denote the com-
ponents of spacetime tensor s such as g
; the similar
use of Latin indices i, j, k, ..., ranging over the
spatial coordinate values (1, 2, 3) !(x
1
, x
2
, x
3
), for
spatial tensors such as
ij
; the use of the Einstein
summation convention for both types of indices; the
use of standard Kronecker delta symbols (tensors),
and
i
j
; the choice of geometric units, G = c = 1;
and, finally, the normalization of the matter fields
implicit in the choice of the constant 8 in [1].
The majority of the equations that appear in this
article are tensor equations, or specific compone nts
of tensor equations, written in traditional index (not
abstract index) form. Thus, these equations are
generally valid in any coordinate system, (t, x
i
),
but, of course do require the introduction of a
coordinate basis and its dual. This approach is also
largely a matter of convention, since all of what
follows can be derived in a variety of fashions, some
of them purely geometrical, and there are also
approaches to numerical relativity based, for exam-
ple, on frames rather than coordinate bases.
This article departs from MTW in its use of ,
i
,
and
ij
to denote the lapse, shift, an d spatial metric,
respectively, rather than MTW’s N, N
i
,and
(3)
g
ij
.
Finally, the operations of partial differentiation
with respect to coordinates x
, t, and x
i
are denoted
@
, @
t
, and @
i
, respectively.
Introduction
The numerical analysis of general relativity, or
numerical relativity, is concerned with the use of
computational methods to derive approximate solu-
tions to the Einstein field equations
G
¼ 8T
½1
Here, G
is the Einstein tensor – that contracted
piece of the Riemann curvature tensor that has
vanishing divergence – and T
is the stress tensor of
the matter content of the spacetime. T
likewise has
vanishing divergence, an expression of the princ iple
of local conservation of stress–energy that general
relativity embodies.
The elegant tensor formulation [1] belies the fact
that, ultimately, the field equations are generically a
complicated and nonlinear set of partial differential
equations (PDEs) for the components of the space-
time metric tensor, g
(x
), in some coordinate
system x
. Moreover, implicit in a numerical
solution of [1] is the numerical solution of the
equations of motion for any matter fields that
couple to the gravitational field – that is, that
contribute to T
. The reader is reminded that it is a
hallmark of general relativity that, in principle, all
matter fields – including massless one s such as the
electromagnetic field – contribute to T
.
Now, in the 3 þ 1 approach to general relativity
that is described below, the task of solving the field
equations [1] is formulated as an initial-value or
Cauchy problem. Specifically, the spacetime metric,
g
(x
) = g
(t, x
k
), which encodes all geometric
information concerning the spacetime, M,is
viewed as the time history, or dynamical evolution,
of the spatial metric,
ij
(0, x
k
), of an initial space-
like hypersurface, (0). In any practical calculation,
the degree to which the matter fields ‘‘back-react’’
on the gravitational field, that is, contribute to T
substantially enough to cause perturbations in g
at or above the desired accuracy threshold, will
thus dep end on the specifics of the initial
configuration.
In astr ophysics, there are relatively few well-
identified environments in which it is generally
thought to be crucial to the faithful emulation of
the physics that the matter fields be fully coupled to
the gravitational field. However, both observation-
ally and theoretically, the existence of gravitation-
ally compact objects is quite clear. Gravitationally
compact means that a star with mass, M,hasa
radius, R, comparable to its Schwarzschil d radius,
R
M
, which is defined by
R
M
¼
2G
c
2
M 10
27
kg m
1
½2
Here, and only here, G and c – Newton’s gravita-
tional constant and the speed of light, respectively –
have been explicitly reintroduced. The fact that
R
M
=R is about 10
6
and 10
9
at the surfaces of the
sun and earth, respectively, is a reminder of just how
604 Computational Methods in General Relativity: The Theory