Taylor M.E. Partial Differential Equations III: Nonlinear Equations
Подождите немного. Документ загружается.
568 17. Euler and Navier–Stokes Equations for Incompressible Fluids
J
"
D ."/,where./ is the characteristic function of Œ1; 1.ThenJ
"
com-
mutes with and with P . We also require u
"
.0/ D J
"
u
0
;thenu
"
.t/ D J
"
u
"
.t/.
Now from (4.9), which holds here also, we have
(4.32) fu
"
W " 2 .0; 1g is bounded in L
1
.R
C
;L
2
/:
This follows from (4.8), further use of which yields
(4.33) 4
Z
T
0
kDef u
"
.t/k
2
L
2
dt DkJ
"
u
0
k
2
L
2
ku
"
.T /k
2
L
2
;
as in (1.39) of Chap.15. Hence, for each bounded interval I D Œ0; T ,
(4.34) fu
"
g is bounded in L
2
.I; H
1
.M //:
Now, as in (4.18), we write our PDE for u
"
as
(4.35)
@u
"
@t
C PJ
"
div.u
"
˝ u
"
/ D u
"
;
since J
"
J
"
u
"
D u
"
.From(4.32) we see that
(4.36) fu
"
˝ u
"
W " 2 .0; 1g is bounded in L
1
.R
C
;L
1
.M //:
We use the inclusion L
1
.M / H
n=2ı
.M /. Hence, by (4.35), for each ı>0,
(4.37) f@
t
u
"
g is bounded in L
2
.I; H
n=21ı
.M //;
so
(4.38) fu
"
g is bounded in H
1
.I; H
n=21ı
.M //:
As in the proof of Proposition 1.7 in Chap. 15, we now interpolate between
(4.34)and(4.38), to obtain
(4.39) fu
"
g is bounded in H
s
.I; H
1ss.n=2C1Cı/
.M //;
and hence, as in (1.45)there,
(4.40) fu
"
g is compact in L
2
.I; H
1
.M //;
for all >0.
Now the rest of the argument is easy. We can pick a sequence u
k
D u
"
k
("
k
! 0) such that
4. Navier–Stokes equations 569
(4.41) u
k
! u in L
2
.Œ0; T ; H
1
.M //; in norm,
arranging that this hold for all T<1, and from this it is easy to deduce that u is
a desired weak solution to (4.6).
Solutions of (4.6) obtained as limits of u
"
as in the proof of Theorem 4.6 are
called Leray–Hopf solutions to the Navier–Stokes equations. The uniqueness and
smothness of a Leray–Hopf solution so constructed remain open problems if dim
M 3. We next show that when dim M D 3, such a solution is smooth except
for at most a fairly small exceptional set.
Proposition 4.7. If dim M D 3 and u is a Leray–Hopf solution of (4.6), then
there is an open dense subset J of .0; 1/ such that R
C
nJ has Lebesgue measure
zero and
(4.42) u 2 C
1
.J M/:
Proof. For T>0arbitrary, I D Œ0; T ,use(4.40). With u
k
D u
"
k
, passing to a
subsequence, we can suppose
(4.43) ku
kC1
u
k
k
E
2
k
;ED L
2
.I; H
1
.M //:
Now if we set
(4.44) .t/ D sup
k
ku
k
.t/k
H
1
;
we have
(4.45) .t/ ku
1
.t/k
H
1
C
1
X
kD1
ku
kC1
.t/ u
k
.t/k
H
1
;
hence
(4.46) 2 L
2
.I /:
In particular, .t/ is finite almost everywhere.Let
(4.47) S Dft 2 I W .t/ < 1g:
For small >0;H
1
.M / L
p
.M / with p close to 6 when dim M D 3,
and products of two elements in H
1
.M / belong to H
1=2
0
.M /, with
0
>0
small. Recalling that u
"
satisfies (4.35), we now apply the analysis used in the
proof of Proposition 4.3 to u
k
, concluding that, for each t
0
2 S, there exists
T.t
0
/>0, depending only on .t
0
/, such that, for small
0
>0,wehave
fu
k
g bounded in C
Œt
0
;t
0
C T.t
0
/; H
1
.M /
\ C
1
.t
0
;t
0
C T.t
0
// M
:
570 17. Euler and Navier–Stokes Equations for Incompressible Fluids
Consequently, if we form the open set
(4.48) J
T
D
[
t
0
2S
t
0
;t
0
C T.t
0
/
;
then any weak limit u of fu
k
g has the property that u 2 C
1
.J
T
M/. It remains
only to show that I n J
T
has Lebesgue measure zero; the denseness of J
T
in I
will automatically follow. To see this, fix ı
1
>0. Since meas.I n S/ D 0,there
exists ı
2
>0such that if S
ı
2
Dft 2 S W T.t/ ı
2
g, then meas.I n S
ı
2
/<ı
1
.
But J
T
contains the translate of S
ı
2
by ı
2
=2, so meas.I nJ
T
/ ı
1
Cı
2
=2.This
completes the proof.
There are more precise results than this. As shown in [CKN], when M D R
3
,
the subset of R
C
M on which a certain type of Leray–Hopf solution, called
“admissible,” is not smooth, must have vanishing one-dimensional Hausdorff
measure. In [CKN] it is shown that admissible Leray–Hopf solutions exist.
We now discuss some results regarding the uniqueness of weak solutions to the
Navier–Stokes equations (4.6). Thus, let I D Œ0; T , and suppose
(4.49) u
j
2 L
1
I;L
2
.M /
\ L
2
I;H
1
.M /
;jD 1; 2;
are two weak solutions to
(4.50)
@u
j
@t
C P div.u
j
˝ u
j
/ D u
j
; u
j
.0/ D u
0
;
where u
0
2 L
2
.M /; div u
0
D 0.Thenv D u
1
u
2
satisfies
(4.51)
@v
@t
C P div.u
1
˝ v C v ˝ u
2
/ D v; v.0/ D 0:
We will estimate the rate of change of kv.t/k
2
L
2
, using the following:
Lemma 4.8. Provided
(4.52) v 2 L
2
.I; H
1
.M // and
@v
@t
2 L
2
.I; H
1
.M //;
then kv.t/k
2
L
2
is absolutely continuous and
d
dt
kv.t/k
2
L
2
D 2.v
t
;v/
L
2
2 L
1
:
Furthermore, v 2 C.I;L
2
/.
Proof. The identity is clear for smooth v, and the rest follows by approximation.
4. Navier–Stokes equations 571
By hypothesis (4.49), the functions u
j
satisfy the first part of (4.52). By (4.50),
the second part of (4.52) is satisfied provided u
j
˝ u
j
2 L
2
.I M/,thatis,
provided
(4.53) u
j
2 L
4
.I M/:
We now proceed to investigate the L
2
-norm of v, solving (4.51). If u
j
satisfy
both (4.49)and(4.53), we have
(4.54)
d
dt
kv.t/k
2
L
2
D2.r
v
u
1
;v/ 2.r
u
2
v; v/ 2krvk
2
L
2
D 2.u
1
; r
v
v/ 2krvk
2
L
2
;
since r
v
Dr
v
and r
u
2
Dr
u
2
for these two divergence-free vector fields.
Consequently, we have
(4.55)
d
dt
kv.t/k
2
L
2
2ku
1
k
L
4
kvk
L
4
krvk
L
2
2krvk
2
L
2
:
Our goal is to get a differential inequality implying kv.t/k
L
2
D 0;thisre-
quires estimating kv.t/k
L
4
in terms of kv.t/k
L
2
and krvk
L
2
.SinceH
1=2
.M
2
/
L
4
.M
2
/ and H
1
.M
3
/ L
6
.M
3
/, we can use the following estimates when
dim M D 2 or 3:
(4.56)
kvk
L
4
C kvk
1=2
L
2
krvk
1=2
L
2
C C kvk
L
2
; dim M D 2;
kvk
L
4
C kvk
1=4
L
2
krvk
3=4
L
2
C C kvk
L
2
; dim M D 3:
With these estimates, we are prepared to prove the following uniqueness result:
Proposition 4.9. Let u
1
and u
2
be weak solutions to (4.6), satisfying (4.49) and
(4.53). Suppose dim M D 2 or 3;ifdimM D 3, suppose furthermore that
(4.57) u
1
2 L
8
I;L
4
.M /
:
If u
1
.0/ D u
2
.0/,thenu
1
D u
2
on I M .
Proof. For v D u
1
u
2
, we have the estimate (4.55). Using (4.56), we have
(4.58)
2ku
1
k
L
4
kvk
L
4
krvk
L
2
krvk
2
L
2
C C
3
kvk
2
L
2
ku
1
k
4
L
4
C C
1
kvk
2
L
2
ku
1
k
2
L
4
when dim M D 2,and
(4.59)
2ku
1
k
L
4
kvk
L
4
krvk
L
2
krvk
2
L
2
C C
7
kvk
2
L
2
ku
1
k
8
L
4
C C
1
kvk
2
L
2
ku
1
k
2
L
4
572 17. Euler and Navier–Stokes Equations for Incompressible Fluids
when dim M D 3. Consequently,
(4.60)
d
dt
kv.t/k
2
L
2
C
n
./kv.t/k
2
L
2
ku
1
k
p
L
4
Cku
1
k
2
L
4
;
where p D 4 if dim M D 2 and p D 8 if dim M D 3. Then Gronwall’s inequality
gives
kv.t/k
2
L
2
ku
1
.0/ u
2
.0/k
2
L
2
exp
n
C
n
./
Z
t
0
ku
1
.s/k
p
L
4
Cku
1
.s/k
2
L
4
ds
o
;
proving the proposition.
We compare the properties of the last proposition with properties that Leray–
Hopf solutions can be shown to have:
Proposition 4.10. IfuisaLeray–Hopfsolutionto(4.1) and I D Œ0; T ,then
(4.61) u 2 L
4
.I M/ if dim M D 2;
and
(4.62) u 2 L
8=3
I;L
4
.M /
if dim M D 3:
Also,
(4.63) u 2 L
2
I;L
4
.M /
if dim M D 4:
Proof. Since u 2 L
1
.I; L
2
/ \ L
2
.I; H
1
/,(4.61) follows from the first part of
(4.56), and (4.62) follows from the second part. Similarly, (4.63) follows from the
inclusion
H
1
.M
4
/ L
4
.M
4
/:
In particular, the hypotheses of Proposition 4.9 are seen to hold for Leray–Hopf
solutions when dim M D 2, so there is a uniqueness result in that case. On the
other hand, there is a gap between the conclusion (4.62) and the hypothesis (4.57)
when dim M D 3.
Exercises
In the exercises below, assume for simplicity that Ric D 0,so(4.5) holds with c
0
D 0.
1. One place dissipated energy can go is into heat. Suppose a “temperature” function
T D T.t;x/satisfies a PDE
(4.64)
@T
@t
Cr
u
T D ˛T C 4jDef uj
2
;
Exercises 573
coupled to (4.6), where ˛ is a positive constant. Show that the total energy
E.t/ D
Z
M
n
ju.t; x/j
2
C T.t;x/
o
dx
is conserved, provided u and T possess sufficient smoothness. Discuss local existence
of solutions to the coupled equations (4.1)and(4.64).
2. Show that under the hypotheses of Theorem 4.1,
u
! v; as ! 0;
v being the solution to the Euler equation (i.e., the solution to the D 0 case of (4.6)).
In what topology can you demonstrate this convergence?
3. Give the details of the interpolation argument yielding (4.39).
4. Combining Propositions 4.3 and 4.5, show that if div u
0
D 0; p > n,andku
0
k
L
p
is
small enough, then (4.6) has a global solution
u 2 C
Œ0; 1/; L
p
\ C
1
.0; 1/ M
:
In Exercises 5–10, suppose dim M D 3.Letu solve (4.6), with vorticity w.
5. Show that the vorticity satisfies
(4.65)
d
dt
kw.t/k
2
L
2
D 2.r
w
u;w/ 2krwk
2
L
2
:
6. Using .r
w
u;w/ D.u; r
w
w/
u;.div w/w
, deduce that
d
dt
kw.t/k
2
L
2
C kuk
L
3
kwk
L
6
krwk
L
2
2krwk
2
L
2
:
Show that
(4.66) kwk
L
6
C krwk
L
2
C C kuk
L
2
;
and hence
d
dt
kw.t/k
2
L
2
C kuk
L
3
krwk
2
L
2
Ckuk
2
L
2
2krwk
2
L
2
:
7. Show that
kuk
L
3
C kuk
1=2
L
2
kwk
1=2
L
2
C C kuk
L
2
;
and hence, if ku
0
k
L
2
D ˇ;
d
dt
kw.t/k
2
L
2
C
ˇ
1=2
kwk
1=2
L
2
C ˇ
krwk
2
L
2
C ˇ
2
2krwk
2
L
2
:
8. Show that there exist constants A; B 2 .0; 1/, depending on M , such that
(4.67) krwk
2
L
2
Akwk
2
L
2
Bˇ
2
;
574 17. Euler and Navier–Stokes Equations for Incompressible Fluids
and hence that y.t/ Dkw.t/k
2
L
2
satisfies
dy
dt
C
ˇ
1=2
y
1=4
C ˇ
ˇ
2
Ay C Bˇ
2
as long as
(4.68) C
ˇ
1=2
y
1=4
C ˇ
<:
9. As long as (4.68) holds, dy=d t ˇ
2
.1 C B/ Ay.Asin(4.30), this gives
y.t/ max
˚
y.t
0
/; ˇ
2
.1 C B/A
1
;; for t t
0
:
Thus (4.68) persists as long as C
ˇ.1 C B/
1=4
A
1=4
C ˇ
<. Deduce a global
existence result for the Navier–Stokes equations (4.1)whendimM D 3 and
(4.69)
C
ku
0
k
1=2
L
2
kw.0/k
1=2
L
2
Cku
0
k
L
2
<;
C ku
0
k
L
2
1 C .1 C B/
1=4
A
1=4
<:
For other global existence results, see [Bon]and[Che1].
10. Deduce from (4.65)that
d
dt
kw.t/k
2
L
2
C
kwk
3
L
3
Ckwk
2
L
3
kuk
L
2
2krwk
2
L
2
:
Work on this, applying
kwk
L
3
C kwk
1=2
L
2
kwk
1=2
L
6
;
in concert with (4.66).
11. Generalize results of this section to the case where no extra hypotheses are made on
Ric. Consider also cases where some assumptions are made (e.g. Ric 0,orRic 0).
(Hint: Instead of (4.6)or(4.18), we have
@u
@t
D u P div.u ˝u/ C PBu;Bu D 2 Ric.u/:/
12. Assume u is a Killing field on M ,thatis,u generates a group of isometries of M .
According to Exercise 11 of 1, u provides a steady solution to the Euler equation
(1.11). Show that u also provides a steady solution to the Navier–Stokes equation
(4.1), provided L isgivenby(4.4). If M D S
2
or S
3
, with its standard metric, show
that such u (if not zero) does not give a steady solution to (4.1)ifL is taken to be either
the Hodge Laplacian or the Bochner Laplacian r
r. Physically, would you expect
such a vector field u to give rise to a viscous force?
13. Show that a t-dependent vector field u.t/ on Œ0; T / M satisfying
u 2 L
1
Œ0; T /; Lip
1
.M /
generates a well-defined flow consisting of homeomorphisms.
14. Let u be a solution to (4.1) with u
0
2 L
p
.M /; p > n, as in Proposition 4.3.Show
that, given s 2 .0; 2,
ku.t/k
H
s;p
Ct
s=2
; 0<t <T:
5. Viscous flows on bounded regions 575
Taking s 2 .1 Cn=p; 2/, deduce from Exercise 13 that u generates a well-defined flow
consisting of homeomorphisms.
For further results on flows generated by solutions to the Navier–Stokes equations, see
[ChL]and[FGT].
5. Viscous flows on bounded regions
In this section we let be a compact manifold with boundary and consider the
Navier–Stokes equations on R
C
,
(5.1)
@u
@t
Cr
u
u D Lu grad p; div u D 0:
We will assume for simplicity that is flat, or more generally, Ric D 0 on ,so,
by (4.4), L D .When@ ¤;, we impose the “no-slip” boundary condition
(5.2) u D 0; for x 2 @:
We also set an initial condition
(5.3) u.0/ D u
0
:
We consider the following spaces of vector fields on , which should be com-
pared to the spaces V
of (1.6)andV
k
of (3.4). First, set
(5.4) V Dfu 2 C
1
0
.; T / W div u D 0g:
Then set
(5.5) W
k
D closure of V in H
k
.; T /; k D 0; 1:
Lemma 5.1. We have W
0
D V
0
and
(5.6) W
1
Dfu 2 H
1
0
.; T / W div u D 0g:
Proof. Clearly, W
0
V
0
. As noted in 1, it follows from (9.79)–(9.80) of
Chap. 5 that
(5.7) .V
0
/
?
Dfrp W p 2 H
1
./g;
the orthogonal complement taken in L
2
.; T /. To show that V is dense in V
0
,
suppose u 2 L
2
.; T / and .u;v/ D 0 for all v 2 V. We need to conclude that
u Drp for some p 2 H
1
./. To accomplish this, let us make note of the
following simple facts. First,
576 17. Euler and Navier–Stokes Equations for Incompressible Fluids
(5.8) rWH
1
./ ! L
2
.; T / has closed range R
0
I R
?
0
D ker r
D V
0
:
The last identity follows from (5.7). Second, and more directly useful,
(5.9)
rWL
2
./ ! H
1
.; T / has closed range R
1
;
R
?
1
D ker r
Dfu 2 H
1
0
.; T / W div u D 0gDW
.1/
;
the last identity defining W
.1/
.
Now write as an increasing union
1
2
% , each
j
having smooth boundary. We claim u
j
D uj
j
is orthogonal to W
.1/
j
,defined
as in (5.9). Indeed, if v 2 W
.1/
j
(and you extend v to be 0 on n
j
), then
."
p
/v D v
"
belongs to V if O 2 C
1
0
.R/ and " is small, and v
"
! v in
H
1
-norm if .0/ D 1,so.u;v/ D lim.u;v
"
/ D 0.From(5.9) it follows that there
exist p
j
2 L
2
.
j
/ such that u Drp
j
on
j
I p
j
is uniquely determined up to
an additive constant (if
j
is connected) so we can make all the p
j
fit together,
giving u Drp.Ifu 2 L
2
.; T /; p must belong to H
1
./.
The same argument works if u 2 H
1
.; T / is orthogonal to V; we obtain
u Drp with p 2 L
2
./; one final application of (5.9) then yields (5.6), finishing
off the lemma.
Thus, if u
0
2 W
1
, we can rephrase (5.1), demanding that
(5.10)
d
dt
.u;v/
W
0
C .r
u
u;v/
W
0
D.u;v/
W
1
; for all v 2 V:
Alternatively, we can rewrite the PDE as
(5.11)
@u
@t
C P r
u
u DAu:
Here, P is the orthogonal projection of L
2
.; T / onto W
0
D V
0
, namely, the
same P as in (1.10)and(3.1), hence described by (3.5)–(3.6). The operator A is
an unbounded, positive, self-adjoint operator on W
0
, defined via the Friedrichs
extension method, as follows. We have A
0
W W
1
! .W
1
/
given by
(5.12) hA
0
u;viD.u;v/
W
1
D .du;dv/
L
2
;
the last identity holding because div u D div v D 0.Thenset
(5.13) D.A/ Dfu 2 W
1
W A
0
u 2 W
0
g;AD A
0
ˇ
ˇ
D.A/
;
using W
1
W
0
.W
1
/
. Automatically, D.A
1=2
/ D W
1
. The operator A is
called the Stokes operator. The following result is fundamental to the analysis of
(5.1)–(5.2):
5. Viscous flows on bounded regions 577
Proposition 5.2. D.A/ H
2
.; T /.Infact,D.A/ D H
2
.; T / \ W
1
.
In fact, if u 2 D.A/ and Au D f 2 W
0
,then.f; v/
L
2
D .u;v/
L
2
,for
all v 2 V. We know u 2 H
1
, so from Lemma 5.1 and (5.9) we conclude that
there exists p 2 L
2
./ such that
(5.14) u D f Crp:
Also we know that div u D 0 and u 2 H
1
0
.; T /. We want to conclude that
u 2 H
2
and p 2 H
1
. Let us identify vector fields and 1-forms, so
(5.15) u D f C dp; ı u D 0; u
ˇ
ˇ
@
D 0:
In order not to interrupt the flow of the analysis of (5.1)–(5.2), we will show in
Appendix A at the end of this chapter that solutions to (5.15) possess appropriate
regularity.
We will define
(5.16) W
s
D D.A
s=2
/; s 0:
Note that this is consistent with (5.5), for s D k D 0 or 1.
We now construct a local solution to the initial-value problem for the Navier–
Stokes equation, by converting (5.11) into an integral equation:
(5.17) u.t/ D e
tA
u
0
Z
t
0
e
.st/A
P div
u.s/ ˝ u.s/
ds D ‰u.t/:
We want to find a fixed point of ‰ on C.I;X/,forI D Œ0; T , with some T>0,
and X an appropriate Banach space. We take X to be of the form
(5.18) X D W
s
D D.A
s=2
/;
for a value of s to be specified below. As in the construction in 4, we need a
Banach space Y such that
(5.19) ˆ W X ! Y is Lipschitz, uniformly on bounded sets;
where
(5.20) ˆ.u/ D P div.u ˝u/;
and such that, for some <1,
(5.21) ke
tA
k
L.Y;X/
Ct
;