Taylor M.E. Partial Differential Equations III: Nonlinear Equations
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448 16. Nonlinear Hyperbolic Equations
5. Compressible fluid motion
We begin with a brief derivation of the equations of ideal compressible fluid flow
on a region . Suppose a fluid “particle” has position F.t;x/ at time t, with
F.0;x/ D x. Thus the velocity field of the fluid is
(5.1) v.t; y/ D F
t
.t; x/ 2 T
y
; y D F.t;x/;
where F
t
.t; x/ D .@=@t/F .t; x/.Ify 2 @, we assume that v.t; y/ is tangent
to @. We want to write down a Lagrangian for the motion. At any time t,the
kinetic energy of the fluid is
(5.2)
K.t/ D
1
2
Z
jv.t; y/j
2
.t; y/ dy
D
1
2
Z
jF
t
.t; x/j
2
0
.x/ dx;
where .t; y/ is the density of the fluid, and
0
.x/ D .0; x/. Thus
(5.3)
0
.x/ D .t; y/ det D
x
F.t;x/; y D F.t;x/:
In the simplest models, the potential energy density is a function of fluid den-
sity alone:
(5.4)
V.t/ D
Z
W
.t; y/
.t; y/ dy
D
Z
W
.t;F.t;x//
0
.x/ dx:
Set W./ D Q.
1
/;
0
.x/ D
0
.x/
1
. In such a case, the Lagrangian action
integral is
(5.5) L.F / D
Z
I
Z
h
1
2
jF
t
.t; x/j
2
Q
0
.x/ det D
x
F.t;x/
i
0
.x/ dx dt
defined on the space of maps F W I ! ,whereI is an arbitrary time
interval Œt
0
;t
1
Œ0; 1/. We seek to produce a PDE describing the critical points
of L.
Split L.F / into L.F / D L
K
.F / L
V
.F /, with obvious notation. Then
(5.6)
DL
K
.F /w D
“
D
F
t
.t; x/;
d
dt
w
t;F.t;x/
E
0
.x/ dx dt
D
“
D
@v
@t
C v r
y
v; Qw.t; y/
E
.t; y/ dy dt;
5. Compressible fluid motion 449
upon an integration by parts, since .d=dt/v
t;F.t;x/
D @v=@t C v r
y
v.We
have set Qw.t; y/ D w.t; x/; y D F.t;x/.Next,
(5.7)
DL
V
.F /w D
“
Q
0
0
.x/ det D
x
F.t;x/
det D
x
F.t;x/
Tr
D
x
F.t;x/
1
D
x
w.t; x/
dx dt:
Now D
x
w.t; x/ D D
x
Qw
t;F.t;x/
D D
y
Qw
t;F.t;x/
D
x
F.t;x/,so
(5.8)
Tr
D
x
F.t;x/
1
D
x
w.t; x/
D Tr D
y
Qw
t;F.t;x/
D div Qw
t;F.t;x/
:
Hence
(5.9)
DL
V
.F /w D
“
Q
0
.t; y/
1
div Qw.t; y/ dy dt
D
“
Q
00
.
1
/
2
˝
r
y
.t; y/; Qw.t; y/
˛
dy dt:
Since W./ D Q.
1
/,wehaveQ
00
.
1
/
2
D
2
W
00
./2W
0
./ D X
00
./
if we set
(5.10) X./ D W ./;
so we can write
(5.11) DL
V
.F /w D
“
˝
X
00
./r
y
; Qw.t; y/
˛
.t; y/ dy dt:
Thus we have the Euler equations:
@v
@t
Cr
v
v C X
00
./r D 0;(5.12)
@
@t
C div.v/ D 0:(5.13)
Equation (5.12) expresses the stationary condition, DL.F /w D 0 for all smooth
vector fields w, tangent to @, while (5.13) simply expresses conservation of
matter. Replacing v r
x
v by r
v
v as we have done above makes these equations
valid when is a Riemannian manifold with boundary. The boundary condition,
as we have said, is
(5.14) v.t; x/ k @; x 2 @:
450 16. Nonlinear Hyperbolic Equations
Recognizing @v=@t Cr
v
v D .d=dt/v
t;F.t;x/
as the acceleration of a fluid
particle, we rewrite (5.12) in a form reflecting Newton’s law F D ma:
(5.15)
@v
@t
Cr
v
v
Dr
x
p:
The real-valued function p is called the pressure of the fluid. Comparison with
(5.12)givesp D p./ and
(5.16)
p
0
./
D X
00
./:
The relation p D p./ is called an equation of state; the function p./ depends
upon physical properties of the fluid.
Making use of the identity
(5.17) div.u ˝v/ D .div v/u Cr
u
v;
we can rewrite the system (5.12)–(5.13), with X
00
./r replaced by rp=,inthe
form
(5.18)
.v/
t
C div.v ˝ v/ Crp D 0;
t
C div.v/ D 0;
which is convenient for consideration of nonsmooth solutions.
It is natural to assume that W./ is an increasing function of . One common
model takes
(5.19) W./ D ˛
1
; ˛>0; 1<<2:
In such a case, we have an equation of state of the form
(5.20) p./ D A
;AD . 1/˛ > 0:
Experiments indicate that for air, under normal conditions, this provides a good
approximation to the equation of state if we take D 1:4. Obviously, these formu-
las lose validity when becomes so large that air becomes as dense as a liquid, but
in that situation other physical phenomena come into play, and the entire problem
has to be reformulated.
We will rewrite Euler’s equation, letting Qv denote the 1-form corresponding to
the vector field v via the Riemannian metric on .Then(5.12) is equivalent to
(5.21)
@ Qv
@t
Cr
v
Qv Ddx
0
./; X
0
./ D
Z
p
0
./
d:
5. Compressible fluid motion 451
In turn, we will rewrite this, using the Lie derivative. Recall that, for any vector
field Z; r
v
Z D L
v
Z Cr
Z
v, by the zero-torsion condition on r. Using this, we
deduce that
hL
v
Qv r
v
Qv; ZiDhQv; r
Z
viD
1
2
hd jvj
2
;Zi;
so (5.21) is equivalent to
(5.22)
@ Qv
@t
C L
v
Qv D d
1
2
jvj
2
X
0
./
:
A physically important object derived from the velocity field is the vorticity,
which we define to be
(5.23)
Q
D d Qv;
for each t a 2-form on . Applying the exterior derivative to (5.22)givesthe
Vorticity equation
(5.24)
@
Q
@t
C L
v
Q
D 0:
It is also useful to consider vorticity in another form. Namely, to
Q
we associate
a section of ƒ
n2
T (n D dim ), so that the identity
(5.25)
Q
^ ˛ Dh; ˛i!
holds, for every .n 2/-form ˛,where! is the volume form on ,whichwe
assume to be oriented. We have
L
v
Q
^ ˛ D L
v
.
Q
^ ˛/
Q
^ L
v
˛
DhL
v
; ˛i! Ch; L
v
˛i! C .div v/h; ˛i!
Q
^ L
v
˛
DhL
v
; ˛i! C .div v/h; ˛i!;
so (5.24) implies
(5.26)
@
@t
C L
v
C .div v/ D 0:
This takes a neater form if we consider vorticity divided by :
(5.27) w D
:
452 16. Nonlinear Hyperbolic Equations
Then the left side of (5.26) is equal to
@w=@t C L
v
w
C
@=@t Cr
v
C
.div v/
w, and if we use (5.13), we see that
(5.28)
@w
@t
C L
v
w D 0:
This vorticity equation takes special forms in two and three dimensions,
respectively. When dim D n D 2; w is a scalar field, often denoted as
(5.29) w D
1
rot v;
and (5.28) becomes the
2-D vorticity equation.
(5.30)
@w
@t
C v grad w D 0;
which is a conservation law.
If n D 3; w is a vector field, denoted as
(5.31) w D
1
curl v;
and (5.28) becomes the
3-D vorticity equation.
(5.32)
@w
@t
C Œv; w D 0;
or equivalently,
(5.33)
@w
@t
Cr
v
w r
w
v D 0:
The first form (5.23) of the vorticity equation implies
(5.34)
Q
.0/ D .F
t
/
Q
.t/;
where F
t
.x/ D F.t;x/;
Q
.t/.x/ D
Q
.t;x/. Similarly, (5.28) yields
(5.35) w.t; y/ D ƒ
n2
DF
t
.x/ w.0; x/; y D F.t;x/;
where DF
t
.x/ W T
x
! T
y
is the derivative. In case n D 2,thisissimply
w.t; y/ D w.0; x/, the conservation law mentioned after (5.30).
5. Compressible fluid motion 453
One implication of (5.34) is the following. Let S be an oriented surface in ,
with boundary C ;letS.t/ be the image of S under F
t
,andC.t/ the image of C ;
then (5.34) yields
(5.36)
Z
S.t/
Q
.t/ D
Z
S
Q
.0/:
Since
Q
D d Qv, this implies the following:
Kelvin’s circulation theorem.
(5.37)
Z
C.t/
Qv.t/ D
Z
C
Qv.0/:
We take a look at some phenomena special to the case dim D n D 3,where
the vorticity is a vector field on , for each t.Fixt
0
and consider D .t
0
/.
Let S be an oriented surface in , transversal to .Avortex tube T is defined
to be the union of orbits of through S, to a second transversal surface S
2
.For
simplicity we will assume that none of these orbits ends at a zero of the vorticity
field, though more general cases can be handled by a limiting argument.
Since d
Q
D d
2
Qv D 0, we can use Stokes’ theorem to write
(5.38) 0 D
Z
T
d
Q
D
Z
@T
Q
:
Now @T consists of three pieces: S and S
2
(with opposite orientations) and the
lateral boundary L the union of the orbits of from @S to @S
2
. Clearly, the pull-
back of
Q
to L is 0,so(5.38) implies
(5.39)
Z
S
Q
D
Z
S
2
Q
:
Applying Stokes’ theorem again, for
Q
D d Qv,wehave
Helmholtz’ theorem. For any two curves C;C
2
enclosing a vortex tube,
(5.40)
Z
C
Qv D
Z
C
2
Qv:
This common value is called the strength of the vortex tube T .
Also, note that if T is a vortex tube at t
0
D 0, then, for each t; T .t/,theimage
of T under F
t
, is a vortex tube, as a consequence of (5.35), with n D 3,since
and w D = have the same integral curves. Furthermore, (5.37) implies that the
strength of T .t/ is independent of t. This conclusion is also part of Helmholtz’
theorem.
454 16. Nonlinear Hyperbolic Equations
If we write L
v
Qv in terms of exterior derivatives, we obtain from (5.22)the
equivalent formula
(5.41)
@ Qv
@t
C .d Qv/cv Dd
1
2
jvj
2
C X
0
./
:
We can use this to obtain various results known collectively as Bernoulli’s law.
First, taking the inner product of (5.41) with v, we obtain
(5.42)
1
2
@
@t
L
v
jvj
2
DL
v
X
0
./:
Now, consider the special case when the flow v is irrotational (i.e., d Qv D 0).
The vorticity equation (5.24) implies that if this holds for any t, then it holds for
all t.If is simply connected, we can pick x
0
2 and define a velocity potential
'.t; x/ by
(5.43) '.t; x/ D
Z
x
x
0
Qv;
the integral being independent of path. Thus d' DQv,and(5.41) implies
(5.44) d
@'
@t
C
1
2
jvj
2
C X
0
./
D 0
on , for an irrotational flow on a simply connected domain .Inotherwords,
in such a case,
(5.45)
@'
@t
C
1
2
jvj
2
C X
0
./ D H.t/
is a function of t alone. This is Bernoulli’s law for irrotational flow.
Another special type of flow is steady flow, for which v
t
D 0 and
t
D 0.
In such a case, the equation (5.42) becomes
(5.46) L
v
1
2
jvj
2
C X
0
./
D 0;
that is, the function .1=2/jvj
2
C X
0
./ is constant on the integral curves of v,
called streamlines. For steady flow, the equation (5.13) becomes
(5.47) div.v/ D 0; i.e., d. Qv/ D 0:
If dim D 2 and is simply connected, this implies that there is a function
on , called a stream function, such that
(5.48) Qv D d ; i.e., Qv D
1
d :
5. Compressible fluid motion 455
In particular, v is orthogonal to r , so the stream function is also constant on
the integral curves of v, namely, the streamlines. One is temped to deduce from
(5.46) that, for some function H W R ! R,
(5.49)
1
2
jvj
2
C X
0
./ D H. /
in this case, and certainly this works out in some cases.
If a flow is both steady and irrotational, then from (5.44)weget
(5.50) d
1
2
jvj
2
C X
0
./
D 0;
which is stronger than (5.46).
We next discuss conservation of energy in compressible fluid flow. The total
energy
(5.51) E.t/ D K.t/ C V.t/ D
Z
n
1
2
jv.t; x/j
2
C W
.t; x/
o
.t; x/ dx
is constant, for smooth solutions to (5.12)–(5.13). In fact, a calculation gives
(5.52) E
0
.t/ D
Z
@
t
e.t; x/ dx D
Z
div ˆ.t; x/ dx D 0;
where
(5.53) e.t; x/ D
1
2
jvj
2
C X./
is the total energy density and
(5.54) ˆ.t; x/ D
1
2
jvj
2
C X
0
./
v D .e C p/v:
One passes from the first integral in (5.52) to the second via
(5.55) @
t
e.t; x/ C div ˆ.t; x/ D 0;
which is a consequence of (5.12)–(5.13), for smooth solutions.
As we will see in 8 in the special case n D 1, the equation (5.55) can break
down in the presence of shocks. “Entropy satisfying” solutions with shocks then
have the property that E.t/ is a nonincreasing function of t.
Now any equation of physics in which energy is not precisely conserved must
be incomplete. Dissipated energy always goes somewhere. Energy dissipated by
456 16. Nonlinear Hyperbolic Equations
shocks acts to heat up the fluid. Say the heat energy density of the fluid is h.One
way to extend (5.18) is to couple a PDE of the form
(5.56) @
t
.h/ C div.hv/ D@
t
e div ˆ:
In such a case, solutions preserve the total energy
(5.57)
Z
.e C h/ dx:
For smooth solutions, the left side of (5.56) is equal to
.h
t
Cr
v
h/ C h
t
C div.v/
;
so in that case we are equivalently adjoining the equation
(5.58) .h
t
Cr
v
h/ De
t
div ˆ:
The right side of (5.58) vanishes for smooth solutions, recall, so we simply have
h
t
Cr
v
h D 0, describing the transport of heat along the fluid trajectories. (We
are neglecting the diffusion of heat here.)
If we consider the total energy intensity
(5.59) E D
1
2
jvj
2
C
1
X./ C h;
so E D e C h, we obtain
@
t
.E / C div.Ev/ C div.pv/
D @
t
e C div
.e C p/v
C @
t
.h/ C div.hv/;
whose vanishing is equivalent to (5.56). Using this, we have the augmented system
(5.60)
.v/
t
C div.v ˝ v/ Crp D 0;
t
C div.v/ D 0;
.E /
t
C div.Ev/ C div.pv/ D 0:
As in (5.20), this is supplemented by an equation of state, which in this context
can take a more general form than p D p./, namely p D p.; E/. Compare
with (5.62)below.
We mention another extension of the system (5.18), based on ideas from ther-
modynamics. Namely, a new variable, denoted as S, for “entropy,” is introduced,
and one adjoins .S/
t
C div.Sv/ D 0,to(5.18), so the augmented system takes
the form
6. Weak solutions to scalar conservation laws; the viscosity method 457
(5.61)
.v/
t
C div.v ˝ v/ Crp D 0;
t
C div.v/ D 0;
.S /
t
C div.S v/ D 0:
For smooth solutions, the left side of the last equation is equal to
.S
t
Cr
v
S/ C S
t
C div.v/
;
so in that case we are equivalently adjoining the equation S
t
Cr
v
S D 0.
Adjoining the last equation in (5.61) apparently does not affect the system
(5.18) itself, but, as in the case of (5.60), it opens the door for a significant change,
for it is now meaningful, and in fact physically realistic, to consider more general
equations of state,
(5.62) p D p.; S/:
In particular, one often generalizes (5.20)to
(5.63) p D A.S/
:
Brief discussions of the thermodynamic concepts underlying (5.61) can be
found in [CF]and[LL]. In [CF] there is a discussion of how the system (5.60)
leads to (5.61), while [LL] discusses how (5.61) leads to (5.60).
It must be mentioned that certain aspects of the behavior of gases, related to
interpenetration, are not captured in the model of a fluid as described in this sec-
tion. Another model, involving the “Boltzmann equation,” is used. We say no
more about this, but mention the books [CIP]and[RL] for treatments and further
references.
Exercises
1. Write down the equations of radially symmetric compressible fluid flow, as a system in
one “space” variable.
6. Weak solutions to scalar conservation laws;
the viscosity method
For real-valued u D u.t; x/, we will obtain global weak solutions to PDE of the
form
(6.1)
@u
@t
D
X
@
j
F
j
.u/; u.0/ D f;