Taylor M.E. Partial Differential Equations III: Nonlinear Equations
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528 16. Nonlinear Hyperbolic Equations
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17
Euler and Navier–Stokes Equations
for Incompressible Fluids
Introduction
This chapter deals with equations describing motion of an incompressible fluid
moving in a fixed compact space M , which it fills completely. We consider two
types of fluid motion, with or without viscosity, and two types of compact space,
a compact smooth Riemannian manifold with or without boundary. The two types
of fluid motion are modeled by the Euler equation
(0.1)
@u
@t
Cr
u
u Dgrad p; divu D 0;
for the velocity field u, in the absence of viscosity, and the Navier–Stokes equation
(0.2)
@u
@t
Cr
u
u D Lu grad p; div u D 0;
in the presence of viscosity. In (0.2), is a positive constant and L is the second-
order differential operator
(0.3) Lu D div Def u;
which on flat Euclidean space is equal to u,whendivu D 0. If there is a bound-
ary, the Euler equation has boundary condition n u D 0,thatis,u is tangent to the
boundary, while for the Navier–Stokes equation one poses the no-slip boundary
condition u D 0 on @M .
In 1 we derive (0.1) in several forms; we also derive the vorticity equation
for the object that is curl u when dim M D 3. We discuss some of the classical
physical interpretations of these equations, such as Kelvin’s circulation theorem
and Helmholtz’ theorem on vortex tubes, and include in the exercises other topics,
such as steady flows and Bernoulli’s law. These phenomena can be compared with
analogues for compressible flow, discussed in 5 of Chap. 16.
M.E. Taylor, Partial Differential Equations III: Nonlinear Equations,
Applied Mathematical Sciences 117, DOI 10.1007/978-1-4419-7049-7
5,
c
Springer Science+Business Media, LLC 1996, 2011
531
532 17. Euler and Navier–Stokes Equations for Incompressible Fluids
Sections 2–5 discuss the existence, uniqueness and regularity of solutions to
(0.1)and(0.2), on regions with or without boundary. We have devoted separate
sections to treatments first without boundary and then with boundary, for these
equations, at a cost of a small amount of redundancy. By and large, different
analytical problems are emphasized in the separate sections, and their division
seems reasonable from a pedagogical point of view.
The treatments in 2–5 are intended to parallel to a good degree the treatment
of nonlinear parabolic and hyperbolic equations in Chaps.15 and 16. Among the
significant differences, there is the role of the vorticity equation, which leads to
global solutions when dim M D 2. For dim M 3, the question of whether
smooth solutions exist for all t 0 is still open, with a few exceptions, such
as small initial data for (0.2). These problems, as well as variants, such as free
boundary problems for fluid flow, remain exciting and perplexing.
In 6 we tackle the question of how solutions to the Navier–Stokes equations
on a bounded region behave when the viscosity tends to zero. We stick to two spe-
cial cases, in which this difficult question turns out to be somewhat tractable. The
first is the class of 2D flows on a disk that are circularly symmetric. The second is
a class of 3D circular pipe flows, whose detailed description can be found in 6.
These cases yield convergence of the velocity fields to the fields solving associ-
ated Euler equations, though not in a particularly strong norm, due to boundary
layer effects. Section 7 investigates how such velocity convergence yields infor-
mation on the convergence of the flows generated by such time-varying vector
fields.
In Appendix A we discuss boundary regularity for the Stokes operator, needed
for the analysis in 5.
1. Euler’s equations for ideal incompressible fluid flow
An incompressible fluid flow on a region defines a one-parameter family of
volume-preserving diffeomorphisms
(1.1) F.t;/ W ! ;
where is a Riemannian manifold with boundary; if @ is nonempty, we suppose
it is preserved under the flow. The flow can be described in terms of its velocity
field
(1.2) u.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 u.t; y/ is tangent to
@. We want to derive Euler’s equation, a nonlinear PDE for u describing the
dynamics of fluid flow. We will assume the fluid has uniform density.
1. Euler’s equations for ideal incompressible fluid flow 533
If we suppose there are no external forces acting on the fluid, the dynamics are
determined by the constraint condition, that F.t;/ preserve volume, or equiva-
lently, that div u.t; / D 0 for all t . The Lagrangian involves the kinetic energy
alone, so we seek to find critical points of
(1.3) L.F / D
1
2
Z
I
Z
˝
F
t
.t; x/; F
t
.t; x/
˛
dV dt;
on the space of maps F W I ! (where I D Œt
0
;t
1
/, with the volume-
preserving property.
For simplicity, we first treat the case where is a domain in R
n
. A variation
of F is of the form F.s;t;x/, with @F =@s D v.t;F.t;x//,ats D 0, where div
v D 0; v is tangent to @,andv D 0 for t D t
0
and t D t
1
.Wehave
(1.4)
DL.F /v D
“
D
F
t
.t; x/;
d
dt
v
t;F.t;x/
E
dV dt
D
“
D
u
t;F.t;x/
;
d
dt
v
t;F.t;x/
E
dV dt
D
“
D
@u
@t
C u r
x
u;v
E
dV dt:
The stationary condition is that this last integral vanish for all such v, and hence,
for each t,
(1.5)
Z
D
@u
@t
C u r
x
u;v
E
dV D 0;
for all vector fields v on (tangent to @), satisfying div v D 0.
To restate this as a differential equation, let
(1.6) V
D
˚
v 2 C
1
.; T / W div v D 0; v tangent to @
;
and let P denote the orthogonal projection of L
2
.; T / onto the closure of
the space V
. The operator P is often called the Leray projection. The stationary
condition becomes
(1.7)
@u
@t
C P.u r
x
u/ D 0;
in addition to the conditions
(1.8) div u D 0 on
534 17. Euler and Navier–Stokes Equations for Incompressible Fluids
and
(1.9) u tangent to @:
For a general Riemannian manifold , one has a similar calculation, with
u r
x
u in (1.5) generalized simply to r
u
u,wherer is the Riemannian connection
on . Thus (1.7) generalizes to
Euler’s equation, first form.
(1.10)
@u
@t
C P.r
u
u/ D 0:
Suppose now
is compact. According to the Hodge decomposition, the or-
thogonal complement in L
2
.; T / of the range of P is equal to the space
˚
grad p W p 2 H
1
./
:
This fact is derived in the problem set following 9 in Chap. 5, entitled “Exercises
on spaces of gradient and divergence-free vector fields”; see (9.79)–(9.80). Thus
we can rewrite (1.10)as
Euler’s equation, second form.
(1.11)
@u
@t
Cr
u
u Dgrad p:
Here, p is a scalar function, determined uniquely up to an additive constant
(assuming is connected). The function p is identified as “pressure.”
It is useful to derive some other forms of Euler’s equation. In particular, let Qu
denote the 1-form corresponding to the vector field u via the Riemannian metric
on .Then(1.11) is equivalent to
(1.12)
@Qu
@t
Cr
u
Qu Ddp :
We will rewrite this using the Lie derivative. Recall that, for any vector field X,
r
u
X D L
u
X Cr
X
u;
by the zero-torsion condition on r. Using this, we deduce that
(1.13) hL
u
Qu r
u
Qu;XiDhQu; r
X
ui:
In case hQu;viDhu;vi (the Riemannian inner product), we have
(1.14) hQu; r
X
uiD
1
2
hd juj
2
;Xi;
using the notation juj
2
Dhu; ui,so(1.12) is equivalent to
1. Euler’s equations for ideal incompressible fluid flow 535
Euler’s equation, third form.
(1.15)
@Qu
@t
C L
u
Qu D d
1
2
juj
2
p
:
Writing the Lie derivative in terms of exterior derivatives, we obtain
(1.16)
@Qu
@t
C .dQu/cu Dd
1
2
juj
2
C p
:
Note also that the condition div u D 0 can be rewritten as
(1.17) ıQu D 0:
In the study of Euler’s equation, a major role is played by the vorticity,which
we proceed to define. In its first form, the vorticity will be taken to be
(1.18) Qw D dQu;
for each t a 2-form on . The Euler equation leads to a PDE for vorticity; indeed,
applying the exterior derivative to (1.15) gives immediately the
Vorticity equation, first form.
(1.19)
@ Qw
@t
C L
u
Qw D 0;
or equivalently, from (1.16),
(1.20)
@ Qw
@t
C d
Qwcu
D 0:
It is convenient to express this in terms of the covariant derivative. In analogy
to (1.13), for any 2-form ˇ and vector fields X and Y ,wehave
(1.21)
.r
u
ˇ L
u
ˇ/.X; Y / D ˇ.r
X
u;Y/C ˇ.X; r
Y
u/
D .ˇ#ru/.X; Y /;
where the last identity defines ˇ#ru. Thus we can rewrite (1.19)as
Vorticity equation, second form.
(1.22)
@ Qw
@t
Cr
u
Qw Qw#ru D 0:
It is also useful to consider vorticity in another form. Namely, to Qw we associate
a section w of ƒ
n2
T (n D dim ), so that the identity
536 17. Euler and Navier–Stokes Equations for Incompressible Fluids
(1.23) Qw ^ ˛ Dhw; ˛i!
holds, for every .n 2/-form ˛,where! is the volume form on . (We assume
is oriented.) The correspondence Qw $ w given by (1.23) depends only on the
volume element !. Hence
(1.24) div u D 0 H) L
u
Qw D
e
L
u
w;
so (1.19) yields the
Vorticity equation, third form.
(1.25)
@w
@t
C L
u
w D 0:
This vorticity equation takes special forms in two and three dimensions, re-
spectively. When dim D n D 2; w is a scalar field, often denoted as
(1.26) w D rot u;
and (1.25) becomes the
2-D vorticity equation.
(1.27)
@w
@t
C u grad w D 0:
This is a conservation law. As we will see, this has special implications for two-
dimensional incompressible fluid flow. If n D 3; w is a vector field, denoted as
(1.28) w D curl u;
and (1.25) becomes the
3-D vorticity equation.
(1.29)
@w
@t
C Œu;w D 0;
or equivalently,
(1.30)
@w
@t
Cr
u
w r
w
u D 0:
Note that (1.28) is a generalization of the notion of the curl of a vector field on
flat R
3
. Compare with material in the second exercise set following 8 in Chap. 5.
1. Euler’s equations for ideal incompressible fluid flow 537
The first form of the vorticity equation, (1.19), implies
(1.31) Qw.0/ D
F
t
Qw.t/;
where F
t
.x/ D F.t;x/; Qw.t /.x/ DQw.t; x/. Similarly, the third form, (1.25),
yields
(1.32) 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, this last identity
is simply w.t; y/ D w.0; x/, the conservation law mentioned after (1.27).
One implication of (1.31) 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 (1.31) yields
(1.33)
Z
S.t/
Qw.t/ D
Z
S
Qw.0/:
Since Qw D d Qu, this implies the following:
Kelvin’s circulation theorem.
(1.34)
Z
C.t/
Qu.t/ D
Z
C
Qu.0/:
We take a look at some phenomena special to the case dim D n D 3,where
the vorticity w is a vector field on , for each t.Fixt
0
, and consider w D w.t
0
/.
Let S be an oriented surface in , transversal to w. A vortex tube T is defined
to be the union of orbits of w through S, to a second transversal surface S
2
(see
Fig. 1.1). 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 Qw D d
2
Qu D 0, we can use Stokes’ theorem to write
(1.35) 0 D
Z
T
d Qw D
Z
@T
Qw:
FIGURE 1.1 Vortex Tube