Taylor M.E. Partial Differential Equations III: Nonlinear Equations
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598 17. Euler and Navier–Stokes Equations for Incompressible Fluids
The representation (6.82)–(6.83) is effective on a neighborhood of fx D 0g,
for example the disk
D
1=2
Dfx 2 R
2
Wjxj1=2g, and it shows that s
.t; r/ !
s
0
.r/ uniformly on r 1=2 provided v
.t; / ! v
0
.t; / in C
1
.D
1=2
/. Results
from Chap. 6, 8 (cf. Propositions 8.1–8.2) imply one has such convergence if
v
0
2 C
1
.D/, and in particular if v
0
2 C
1
.D/.
Furthermore, the maximum principle implies
(6.84) e
t
jQs
0
Qs
jh
t
jQs
0
Qs
j;
where h
t
is the free space heat kernel, given (with n D 2)by
(6.85) h
t
.x/ D .4t/
n=2
e
jxj
2
=4t
;
and jQs
0
Qs
j is extended by 0 outside D. We hence have the following boundary
layer estimates on R
3
.
Proposition 6.8. Assume v
0
;w
0
2 C
1
.D/. Then, given T 2 .0; 1/, we have a
uniform bound
(6.86) jR
3
.;t;x/jC;
for t 2 Œ0; T ; 2 .0; 1; x 2
D.Furthermore,as ! 0,
(6.87) R
3
.;t;x/ ! 0 uniformly on D n
!./
;
as long as
(6.88)
!./
p
!1:
We recall
ı
Dfx 2 D W dist.x; @D/ ıg.
Among other results established in [MT1]–[MT2], we mention one here. For
k 2 N,set
(6.89)
V
k
.D/ Dff 2 L
2
.D/ W X
j
1
X
j
`
f 2 L
2
.D/; 8` k; X
j
m
2 X
1
.D/g;
where X
1
.D/ denotes the space of smooth vector felds on D that are tangent to
@D . After establishing that
(6.90) f 2 V
k
.D/ H) lim
t!0
e
t
f D f; in V
k
-norm;
and
(6.91) f 2 V
k
.D/ H) lim
&0
e
t.X/
f D e
tX
f; in V
k
-norm;
these works proved the following (cf. [MT2], Proposition 3.10).
7. From velocity field convergence to flow convergence 599
Proposition 6.9. Assume v
0
2 C
1
.D/ and w
0
2 C
1
.D/.Takek 2 N and also
assume w
0
2 V
k
.D/. Then, for each t>0,as & 0,
(6.92) v
.t; / ! v
0
and w
.t; / ! w
0
.t; /; in V
k
-norm:
Such a result is consistent with Prandtl’s principle, that in the boundary layer
it is normal derivatives of the velocity field, not tangential derivatives, that blow
up as ! 0. We mention that the convergence of R
2
.;t;x/ to 0 in V
k
-norm
follows from the analysis described in (6.50), and the convergence of R
1
.;t;x/
to 0 in V
k
-norm, given w
0
2 C
1
.D/, follows from a parallel analysis, carried
out in [MT2], but not here. The convergence of R
3
.;t;x/to 0 in V
k
-norm, given
v
0
;w
0
2 C
1
.D/, does not follow from the results on R
3
established here; this
requires further arguments.
The two cases analyzed above are much simpler than the general cases, which
might involve turbulent boundary layers and boundary layer separation. Another
issue is loss of stability of a solution as decreases. One can read more about such
problems in [Bat, Sch, ChM, OO], and references given there. We also mention
[VD], which has numerous interesting illustrations of fluid phenomena, at various
viscosities.
Exercises
1. Verify the characterization (6.12)ofvectorfieldsoftheform(6.11).
2. Verify the calculation (6.13)–(6.14).
3. Produce a proof of (6.90), at least for k D 1. Try for larger k.
7. From velocity field convergence to flow convergence
In 6 we have given some results on convergence of the solutions u
to the
Navier–Stokes equations
(7.1)
@u
@t
Cr
u
u
Crp
D u
on I ;
div u
D 0; u
ˇ
ˇ
I @
D 0; u
.0/ D u
0
;
to the solution u to the Euler equation
(7.2)
@u
@t
Cr
u
u Crp D 0 on I ;
div u D 0; u k@; u.0/ D u
0
;
as & 0 (given divu
0
D 0; u
0
k@).
600 17. Euler and Navier–Stokes Equations for Incompressible Fluids
We now tackle the question of what can be said about convergence of fluid
flows generated by the t-dependent velocity fields u
to the flow generated by
u. Given the convergence results of 6, we are motivated to see what sort of flow
convergencecan be deduced from fairly weak hypotheses on u; u
, and the nature
of the convergence u
! u. We obtain some such results here; further results can
be found in [DL].
We will make the following hypotheses on the t-dependent vector fields
u and u
.
u 2 Lip.Œ0; T
/; div u.t/ D 0; u.t/ k@;(7.3)
u
2 Lip.Œ"; T /; 8">0; div u
.t/ D 0; u
.t/ k@;(7.4)
u
2 L
1
.Œ0; T /:(7.5)
Say 2 .0; 1.Here
is a smoothly bounded domain in R
n
, or more generally
it could be a compact Riemannian manifold with smooth boundary @. We do not
assume any uniformity in on the estimates associated to (7.4)–(7.5).
The field u defines volume preserving bi-Lipschitz maps
(7.6) '
t;s
W ! ; s; t 2 Œ0; T ;
satisfying
(7.7)
@
@t
'
t;s
.x/ D u.t; '
t;s
.x//; '
s;s
.x/ D x:
Similarly the fields u
define volume preserving bi-Lipschitz maps
(7.8) '
t;s
W ! ; s; t 2 .0; T ;
satisfying
(7.9)
@
@t
'
t;s
.x/ D u
.t; '
t;s
.x//; '
s;s
.x/ D x:
Note that
(7.10)
'
t;s
ı '
s;r
D '
t;r
; r;s;t 2 Œ0; T ;
'
t;s
ı '
s;r
D '
t;r
; r;s;t 2 .0; T :
Our convergence results will be phrased in terms of strong operator conver-
gence on L
p
./ of operators S
t;0
to S
t;0
,where
(7.11)
S
t;s
f
0
.x/ D f
0
.'
s;t
.x//; s; t 2 Œ0; T ;
S
t;s
f
0
.x/ D f
0
.'
s;t
.x//; s; t 2 .0; T ;
7. From velocity field convergence to flow convergence 601
and S
t;0
(as well as '
0;t
) will be constructed below. These operators are also
characterized as follows. For f D f.t;x/satisfying
(7.12)
@f
@t
Dr
u.t/
f.t/; t 2 Œ0; T ;
we set
(7.13) S
t;s
f.s/ D f.t/; s;t 2 Œ0; T :
Note that f is advected by the flow generated by u.t/. Clearly
(7.14)
S
t;s
W L
p
./ ! L
p
./; isometrically isomorphically; 8s; t 2 Œ0; T :
Similarly, for f
D f
.t; x/ solving
(7.15)
@f
@t
Dr
u
.t/
f
.t/;
we set
(7.16) S
t;s
f
.s/ D f
.t/; s; t 2 .0; T ;
and again
(7.17)
S
t;s
W L
p
./ ! L
p
./; isometrically isomorphically; 8s; t 2 .0; T :
Note that
(7.18)
S
t;s
S
s;r
D S
t;r
; r;s;t 2 Œ0; T ;
S
t;s
S
s;r
D S
t;r
; r;s;t 2 .0; T :
We will extend the scope of (7.16) to the case s D 0. Then we will show that,
given (7.3)–(7.5), p 2 Œ1; 1/; t 2 Œ0; T ; f
0
2 L
p
./,
(7.19)
u
! u in L
1
.Œ0; T ; L
p
.//
H) S
t;0
f
0
! S
t;0
f
0
in L
p
-norm:
In light of the relationship
(7.20) S
t;0
f
0
.x/ D f
0
.'
0;t
.x//;
which will be established below, this convergence amounts to some sort of con-
vergence
(7.21) '
0;t
! '
0;t
for the backward flows '
0;t
.
602 17. Euler and Navier–Stokes Equations for Incompressible Fluids
To construct S
t;0
on L
p
./, we first note that
(7.22)
"; ı 2 .0; T ; f
0
2 Lip./
H) kS
ı;"
f
0
f
0
k
L
1
ku
k
L
1
kf
0
k
Lip
j" ıj;
which in turn implies
(7.23)
kS
t;ı
f
0
S
t;"
f
0
k
L
1
DkS
t;"
.S
";ı
f
0
f
0
/k
L
1
DkS
";ı
f
0
f
0
k
L
1
ku
k
L
1
kf
0
k
Lip
j" ıj:
Hence
(7.24) lim
"&0
S
t;"
f
0
D S
t;0
f
0
exists for all f
0
2 Lip./, convergence in (7.24) holding in sup-norm, and a for-
tiori in L
p
-norm. The uniform boundedness from (7.17) then implies that (7.24)
holds in L
p
-norm for each f
0
2 L
p
./, as long as p 2 Œ1; 1/,soLip./ is
dense in L
p
./. This defines
(7.25) S
t;0
W L
p
./ ! L
p
./; 1 p<1;t2 .0; T ;
and we have
(7.26) kS
t;0
f
0
k
L
p
D lim
"&0
kS
t;"
f
0
k
L
p
kf
0
k
L
p
;
so S
t;0
is an isometry on L
p
./ for each p 2 Œ1; 1/.
We note that, parallel to (7.22), for "; ı 2 .0; T ; x 2
,
(7.27) dist.'
ı;"
.x/; x/ ku
k
L
1
j" ıj;
and, parallel to (7.23), if also t 2 .0; T ,
(7.28)
dist.'
";t
.x/; '
ı;t
.x// D dist.'
";ı
.'
ı;t
.x//; '
ı;t
.x//
ku
k
L
1
j" ıj:
It follows that
(7.29) '
0;t
.x/ D lim
"&0
'
";t
.x/
exists and '
0;t
W ! continuously, preserving the volume. Furthermore, we
have
7. From velocity field convergence to flow convergence 603
(7.30) S
t;0
f
0
.x/ D f
0
.'
0;t
.x//;
first for f
0
2 Lip./, then, by limiting arguments, for all f
0
2 C./,andfur-
thermore, for all f
0
2 L
p
./, in which case (7.30) holds, for each t 2 .0; T ,for
a.e. x,i.e.,(7.30) is an identity in the Banach space L
p
./.
We derive some more properties of S
t;0
. Note from (7.23)–(7.24)that,when
f
0
2 Lip./; " 2 .0; T ,
(7.31) kS
t;0
f
0
S
t;"
f
0
k
L
1
ku
k
L
1
kf
0
k
Lip
";
and hence, by uniform operator boundedness, for each f
0
2 L
p
./; p 2
Œ1; 1/; s; t 2 .0; T ,
(7.32)
S
t;0
f
0
D lim
"&0
S
t;"
f
0
.in L
p
-norm/
D lim
"&0
S
t;s
S
s;"
f
0
.by (7.18)/
D S
t;s
S
s;0
f
0
:
Hence, S
s;t
S
t;0
D S
s;0
on L
p
./; 8s; t 2 .0; T , or equivalently,
(7.33) S
ı;t
S
t;0
D S
ı;0
on L
p
./:
We also have from (7.23)–(7.24)that
(7.34) f
0
2 Lip./ H) kS
ı;0
f
0
f
0
k
L
1
ku
k
L
1
kf
0
k
Lip
ı;
and hence, again by uniform operator boundedness and denseness of Lip.
/,
(7.35) lim
ı&0
S
ı;0
f
0
D f
0
in L
p
-norm; 8f
0
2 L
p
./:
We now want to compare S
t;0
f
0
with S
t;0
f
0
.Tobegin,take
(7.36) f
0
2 Lip./; f .t/ D S
t;0
f
0
:
Then f.t/ satisfies
(7.37)
@f
@t
Dr
u
.t/
f.t/Cr
u
.t/u.t /
f.t/; f.0/ D f
0
;
so Duhamel’s formula gives
(7.38) f.t/ D S
t;0
f
0
C
Z
t
0
S
t;s
r
u
.s/u.s/
f.s/ds:
604 17. Euler and Navier–Stokes Equations for Incompressible Fluids
Now, by hypothesis (7.3)onu, we see that, for s 2 Œ0; T ,
(7.39)
f
0
2 Lip./ H) kf.s/k
Lip
Akf
0
k
Lip
H) jr
u
.s/u.s/
f.s/jAkf
0
k
Lip
ju
.s/ u.s/j:
Hence
(7.40)
f
0
2 Lip./ H) kS
t;0
f
0
S
t;0
f
0
k
L
p
Z
t
0
kS
t;s
r
u
.s/u.s/
f.s/k
L
p
ds
Akf
0
k
Lip
Z
t
0
ku
.s/ u.s/k
L
p
ds:
Hence, given p 2 Œ1; 1/; t 2 Œ0; T ,
(7.41)
u
! u in L
1
.Œ0; T ; L
p
.//
H) S
t;0
f
0
! S
t;0
f
0
in L
p
-norm;
for all f
0
2 Lip./, and hence, by the uniform operator bounds (7.26) and dense-
ness of Lip.
/ in L
p
./,wehave:
Proposition 7.1. Under the hypotheses (7.3)–(7.5), given p 2 Œ1; 1/; t 2 Œ0; T ,
convergence in (7.41) holds for all f
0
2 L
p
./.
In fact, we can improve Proposition 7.1, as follows. (Compare [DL],
Theorem II.4.)
Proposition 7.2. Given p 2 .1; 1/; t 2 Œ0; T ,
(7.42)
f
0
2 L
p
./; u
! uinL
1
.Œ0; T ; L
1
.//
H) S
t;0
f
0
! S
t;0
f
0
in L
p
-norm.
Proof. By Proposition 7.1, the hypotheses of (7.42)imply
(7.43) S
t;0
f
0
! S
t;0
f
0
in L
1
-norm. We also know that
(7.44) kS
t;0
f
0
k
L
p
DkS
t;0
f
0
k
L
p
Dkf
0
k
L
p
;
for each 2 .0; 1; t 2 Œ0; T . These bounds imply weak
compactness in
L
p
./, and we see that convergence in (7.43) holds weak
in L
p
./.Then
another use of (7.44), together with the uniform convexity of L
p
./ for each
p 2 .1; 1/ gives convergence in L
p
-norm in (7.43).
A. Regularity for the Stokes system on bounded domains 605
A. Regularity for the Stokes system on bounded domains
The following result is the basic ingredient in the proof of Proposition 5.2.
Assume that
is a compact, connected Riemannian manifold, with smooth
boundary, that
u 2 H
1
.; T
/; f 2 L
2
.; T
/; p 2 L
2
./;
and that
(A.1) u D f C dp ; ı u D 0; u
ˇ
ˇ
@
D 0:
We claim that u 2 H
2
.; T
/. More generally, we claim that, for s 0,
(A.2) f 2 H
s
.; T
/ H) u 2 H
sC2
.; T
/:
Indeed, given any 2 Œ0; 1/, it is an equivalent task to establish the implication
(A.2) when we replace (A.1)by
(A.3) . /u D f C dp ; ıu D 0; u
ˇ
ˇ
@
D 0:
In this appendix we prove this result. We also treat the following related prob-
lem. Assume v 2 H
1
.; T
/; p 2 L
2
./,and
(A.4) . /v D dp; ı v D 0; v
ˇ
ˇ
@
D g:
Then we claim that, for s 0,
(A.5) g 2 H
sC3=2
.@; T
/ H) v 2 H
sC2
.; T
/:
Here, for any x 2
(including x 2 @), T
x
D T
x
./ D T
x
M ,wherewetake
M to be a compact Riemannian manifold without boundary, containing as an
open subset (with smooth boundary @). In fact, take M to be diffeomorphic to
the double of
.
We will represent solutions to (A.4) in terms of layer potentials, in a fash-
ion parallel to constructions in 11 of Chap. 7. Such an approach is taken in
[Sol1]; see also [Lad]. A different sort of proof, appealing to the theory of sys-
tems elliptic in the sense of Douglis–Nirenberg, is given in [Tem]. An extension of
the boundary-layer approach to Lipschitz domains is given in [FKV]. This work
has been applied to the Navier–Stokes equations on Lipschitz domains in [DW].
Here the analysis was restricted to Lipschitz domains with connected boundary.
This topological restriction was removed in [MiT]. Subsequently, [Mon] produced
strong, short time solutions on 3D domains with arbitrarily rough boundary.
606 17. Euler and Navier–Stokes Equations for Incompressible Fluids
Pick 2 .0; 1/. We now define some operators on D
0
.M /,sothat
(A.6) . /ˆ dQ D I on D
0
.M; T
/; ıˆ D 0:
To get these operators, start with the Hodge decomposition on M :
(A.7) dıG C ıdG C P
h
D I on D
0
.M; ƒ
/;
where P
h
is the orthogonal projection onto the space H of harmonic forms on M ,
and G is
1
on the orthogonal complement of H.Then(A.6) holds if we set
(A.8)
ˆ D . /
1
.ıdG C P
h
/ 2 OP S
2
.M /;
Q DıG 2 OPS
1
.M /:
Let F.x;y/ and Q.x; y/ denote the Schwartz kernels of these operators. Thus
(A.9) .
x
/F .x; y/ d
x
Q.x; y/ D ı
y
.x/I; ı
x
F.x;y/ D 0:
Note that as dist.x; y/ ! 0,wehave(fordim D n 3)
(A.10)
F.x;y/ A
0
.x; y/ dist.x; y/
2n
C;
Q.x; y/ B
0
.x; y/ dist.x; y/
1n
C;
where A
0
.Exp
y
v; y/ and B
0
.Exp
y
v; y/ are homogeneous of degree zero in
v 2 T
y
M .
We now look for solutions to (A.4) in the form of layer potentials:
(A.11)
v.x/ D
Z
@
F.x; y/ w.y/ dS.y/ D Fw.x/;
p.x/ D
Z
@
Q.x; y/ w.y/ dS.y/ D Qw.x/:
The first two equations in (A.4) then follow directly from (A.9), and the last equa-
tionin(A.4) is equivalent to
(A.12) ‰w D g;
where
(A.13) ‰w.x/ D
Z
@
F.x; y/w.y/ dS.y/; x 2 @;
A. Regularity for the Stokes system on bounded domains 607
defines
(A.14) ‰ 2 OPS
1
.@; T
/:
Note that ‰ is self-adjoint on L
2
.@; T
/. The following lemma is incisive:
Lemma A.1. The operator ‰ is an elliptic operator in OPS
1
.@/.
We can analyze the principal symbol of ‰ using the results of 11 in Chap. 7,
particularly the identity (11.12) there. This implies that, for x 2 @; 2
T
x
.@/; the outgoing unit normal to @ at x,
(A.15)
‰
.x; / D C
n
Z
1
1
ˆ
.x; C / d:
From (A.8), we have
(A.16)
ˆ
.x; /ˇ Djj
4
^
ˇ; ; ˇ 2 T
x
M:
This is equal to jj
2
P
?
ˇ,whereP
?
is the orthogonal projection of T
x
onto
./
?
. Thus
(A.17)
ˆ
.x; /ˇ D A./ˇ B./ˇ;
with
A./ˇ Djj
2
ˇ; B./ˇ Djj
4
.ˇ /:
Hence
(A.18)
Z
1
1
A. C / d D
Z
1
1
.jj
2
C
2
/
1
d D
1
jj
1
;
with
1
D
Z
1
1
1
1 C
2
d:
Also
(A.19)
Z
1
1
B. C /ˇ d D
Z
1
1
jj
2
C
2
2
.ˇ / C
2
.ˇ /
d
D
2
jj
3
.ˇ / C
3
jj
1
.ˇ /;
with
(A.20)
2
D
Z
1
1
1
.1 C
2
/
2
d;
3
D
Z
1
1
2
.1 C
2
/
2
d: