Taylor M.E. Partial Differential Equations III: Nonlinear Equations
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588 17. Euler and Navier–Stokes Equations for Incompressible Fluids
with
(6.14) p
0
.x/ DQp
0
.jxj/; Qp
0
.r/ D
Z
1
r
s
0
./
2
d:
Consequently, in this case the vanishing viscosity problem is to show that the
solution u
to (6.1)–(6.3) satisfies u
.t/ ! u
0
as ! 0. The following is the key
to the analysis of the solution u
.
Proposition 6.1. Given that u
0
has the form (6.11), the solution u
to (6.1)–(6.3)
(with F
0) is circularly symmetric for each t>0,oftheform
(6.15) u
.t; x/ D s
.t; jxj/x
?
;
and it coincides with the solution to the linear PDE
(6.16)
@u
@t
D u
;
with boundary condition (6.2) and initial condition (6.3).
Proof. Let u
solve (6.16), (6.2), and (6.3), with u
0
as in (6.11). We claim (6.15)
holds. In fact, for each unit vector ! 2 R
2
, ˆ
!
u
.t; ˆ
!
x/ also solves (6.16),
with the same initial data and boundary conditions as u
, so these functions must
coincide, and (6.15) follows. Hence div u
D 0 for each t>0.Alsowehavean
analogue of (6.13)–(6.14):
(6.17)
r
u
u
Drp
;p
.t; x/ DQp
.t; jxj/;
Qp
.t; r/ D
Z
1
r
s
.t; /
2
d:
Hence this u
is the solution to (6.1)–(6.3).
To restate matters, for D D, the solution to (6.1)–(6.3) is in this case simply
(6.18) u
.t; x/ D e
t
u
0
.x/:
The following is a simple consequence.
Proposition 6.2. Assume u
0
,oftheform(6.11), belongs to a Banach space X of
R
2
-valued functions on D.Iffe
t
W t 0g is a strongly continuous semigroup on
X,thenu
.t; / ! u
0
in X as ! 0, locally uniformly in t 2 Œ0; 1/.
As seen in Chap. 6, fe
t
W t 0g is strongly continuous on the following
spaces:
(6.19) L
p
.D/; 1 p<1;C
o
.D/ Dff 2 C.D/ W f D 0 on @Dg:
6. Vanishing viscosity limits 589
Also, it is strongly continuous on D
s
D D ../
s=2
/ for all s 2 R
C
. We recall
from Chap. 5 that
(6.20) D
2
D H
2
.D/ \ H
1
0
.D/; D
1
D H
1
0
.D/;
and
(6.21) D
s
D ŒL
2
.D/; H
1
0
.D/
s
; 0<s<1:
In particular, by interpolation results given in Chap. 4,
(6.22) D
s
D H
s
.D/; 0 < s <
1
2
:
We also mention Proposition 7.4 of Chap. 13, which implies this heat semigroup
is strongly continuous on
(6.23) C
1
b
.D/ Dff 2 C
1
.D/ W f D 0 on @D g:
On the other hand, if u
0
2 C.D/ but does not vanish on @D,thene
t
u
0
does not
converge uniformly to u
0
on D,ast ! 0, though as shown in Corollary 8.2 of
Chap. 6, we do have convergence of e
t
u
0
to u
0
uniformly on compact subsets
of D. Thus there is a boundary layer attached to @D where uniform convergence
fails. We recall Proposition 8.3 of Chap. 6 in the current context.
Proposition 6.3. Given u
0
2 C
1
.D/, we have, as & 0, locally uniformly in
t 2 R
C
,
(6.24)
e
t
u
0
u
0
.x/ C
X
k1
.t/
k
kŠ
k
u
0
.x/
X
j 0
2b
j
.x/.4t/
j=2
E
j
'.x/
p
4t
:
Here, b
j
2 C
1
.D/; '.x/ D dist.x; @D/ D 1 jxj, and the special functions
E
j
.y/ are given by
(6.25) E
j
.y/ D
1
p
Z
1
y
e
s
2
.s y/
j
ds:
We mention that b
0
D u
0
on @D and b
j
j
@D
D 0 for j odd. Also, E
0
.0/ D 1=2.
The primary “boundary layer” term is
(6.26) 2u
0
.x/E
0
1 jxj
p
4t
;
and we see the boundary layer thickness is
p
4t.
590 17. Euler and Navier–Stokes Equations for Incompressible Fluids
We pass from this class of 2D problems to the following class of 3D problems.
We look for solutions to (6.1)–(6.3) with u
D u
.t; x; z/; p
D p
.t; x; z/,
.t; x; z/ 2 R
C
,where D D R, D being the 2D disk as above. Thus
is an infinitely long circular pipe. In this case, we consider external force fields
of the form
(6.27) F
.t; x; z/ D .0; f
.t//;
so F
is parallel to the z-axis, with z-component f
.t/. We take initial data of the
following form:
(6.28) u
.0; x; z/ D u
0
.x/ D .v
0
.x/; w
0
.x//;
where v
0
is a vector field on D and w
0
is the z-component of u
0
. We require the
conditions
(6.29) div u
0
D 0; u
0
k@; i:e:; divv
0
D 0; v
0
k@D;
and we require the vector field v
0
on D to be circularly symmetric, so, as in (6.11),
v
0
.x/ D s
0
.jxj/x
?
, hence
(6.30) u
0
.x/ D .s
0
.jxj/x
?
;w
0
.x//:
The fact that is infinite is inconvenient. To get the theoretical treatment
started, it is convenient to modify the set-up by requiring that solutions be pe-
riodic (say of period L)inz, so we replace by
L
D D .R=LZ/.Insuch
a case, results of 5 imply that, for each >0,(6.1)–(6.3) has a unique strong,
short time solution, given mild regularity hypotheses on v
0
.x/ and w
0
.x/ (a so-
lution that, as we will shortly see, persists for all time t>0under the current
hypotheses), and the solution is z-translation invariant, i.e.,
(6.31) u
D .v
.t; x/; w
.t; x//; p
D p
.t; x/:
Consequently,
(6.32) r
u
u
D .r
v
v
; r
v
w
/; div u
D div v
:
Hence, in the current setting, (6.1) is equivalent to the following system of equa-
tions on R
C
D:
@v
@t
Cr
v
v
Crp
D v
; div v
D 0;(6.33)
@w
@t
Cr
v
w
D w
C f
:(6.34)
6. Vanishing viscosity limits 591
Note that (6.33) is the Navier–Stokes equation for flow on D, which we have
just treated. Given initial data satisfying (6.30), we have
(6.35) v
.t; x/ D e
t
v
0
.x/;
where is the Laplace operator on D, with Dirichlet boundary condition. The
results of Proposition 6.2, complemented by (6.19)–(6.23) apply, as do those of
Proposition 6.3, taking care of (6.33).
It remains to investigate (6.34). For this, we have the initial and boundary con-
ditions
(6.36) w
.0/ D w
0
;w
ˇ
ˇ
R
C
@D
D 0;
and we ask whether, as & 0, w
converges to w
0
, solving
(6.37)
@w
0
@t
Cr
v
0
w
0
D f
0
.t/; w
0
.0/ D w
0
:
We impose no boundary condition on w
0
, which is natural since v
0
D v
0
is
tangent to @D.
Before pursuing this convergence question, we pause to observe a class of
steady solutions to (6.33)–(6.34) known as Poiseuille flows. Namely, given ˛ 2
R n 0,
(6.38) u
0
.x/ D ˛.0; 1 jxj
2
/
is such a steady solution, with
(6.39) p
.t; x/ D 0; f
.t/ D .0; 4˛/:
An alternative description is to set
(6.40) p
.t; x; z/ D4˛z;f
.t/ D 0:
This latter is a common presentation, and one refers to Poiseuille flow as “pressure
driven” However, this presentation does not fit into our set-up, since we passed
from the infinite pipe D R to the periodized pipe D .R=LZ/,andp
in (6.40)
is not periodic in z. These Poiseuille flows do fit into our set-up, but we need to
represent the force that maintains the flow as an external force.
We return to the convergence problem. For notational convenience, we set
(6.41) X
Dr
v
D s
.t; jxj/
@
@
;XDr
v
0
D s
0
.jxj/
@
@
:
592 17. Euler and Navier–Stokes Equations for Incompressible Fluids
Thus we examine solutions to
(6.42)
@w
@t
D w
X
w
C f
.t/;
w
.0; x/ D w
0
.x/; w
ˇ
ˇ
R
C
@D
D 0;
compared to solutions to
(6.43)
@w
0
@t
DXw
0
C f
0
.t/; w
0
.0; x/ D w
0
.x/:
We do not assume w
0
j
@D
D 0. In order to separate the two phenomena that make
(6.42) a singular perturbation (6.43), namely the appearance of on the one
hand and the replacement of X by X
on the other hand, we rewrite (6.42)as
(6.44)
@w
@t
D . X/w
C .X X
/w
C f
.t/;
and apply Duhamel’s formula to get
(6.45) w
.t/ D e
t.X/
w
0
C
Z
t
0
e
.ts/.X/
.X X
/w
.s/ C f
.s/
ds:
By comparison, we can write the solution to (6.43)as
(6.46) w
.t/ D e
tX
w
0
C
Z
t
0
f
0
.s/ ds:
Consequently,
(6.47) w
.t; x/ w
0
.t; x/ D R
1
.;t;x/C R
2
.;t;x/C R
3
.;t;x/;
where
(6.48)
R
1
.;t;x/ D e
t.X/
w
0
e
tX
w
0
;
R
2
.;t;x/ D
Z
t
0
h
f
.s/e
.ts/.X/
1 f
0
.s/
i
ds;
R
3
.;t;x/ D
Z
t
0
e
.ts/.X/
.s
0
s
/
@w
@
ds:
The term R
2
is the easiest to treat. By radial symmetry,
e
.ts/.X/
1 D e
.ts/
1;
6. Vanishing viscosity limits 593
and we can write
(6.49)
R
2
.;t;x/ D
Z
t
0
f
.s/ f
0
.s/
ds
C
Z
t
0
f
.s/
h
e
.ts/
1 1
i
ds:
The uniform asymptotic expansion of the last integrand is a special case of (6.24):
(6.50) e
.ts/
1 1
X
j 0
2b
j
.x/..t s//
j=2
E
j
1 jxj
p
4.t s/
;
with b
j
2 C
1
.D/; b
0
j
@D
D 1,andE
j
as in (6.25). The principal contribution
giving the boundary layer effect for the term R
2
.;t;x/is
(6.51) 2
Z
t
0
f
.s/E
0
1 jxj
p
4.t s/
ds:
Methods initiated in [MT1] and carried out for this case in [MT2] produce
a uniform asymptotic expansion for R
1
.;t;x/ almost as explicit as that given
above for R
2
, but with much greater effort. Here we will be content to present
simpler estimates on R
1
. Our analysis of
(6.52) W
.t; x/ D e
t.X/
w
0
.x/
starts with the following. Recall that X is divergence free and tangent to @D .
Lemma 6.4. Given >0,
(6.53) D.. X/
j
/ D D.
j
/; j D 1; 2:
Proof. We have, for >0,
(6.54) D. X/ Dff 2 H
2
.D/ W f
ˇ
ˇ
@D
D 0g;
and
(6.55) D.. X/
2
/ Dff 2 H
4
.D/ W f
ˇ
ˇ
@D
D f Xf
ˇ
ˇ
@D
D 0g:
The first space is clearly equal to D./.SinceX is tangent to @D , f j
@D
D
0 ) Xf j
@D
D 0, so the second space coincides with ff 2 H
4
.D/ W f j
@D
D
f j
@D
D 0g,whichisD.
2
/.
Remark: The analogous identity of domains typically fails for larger j .
594 17. Euler and Navier–Stokes Equations for Incompressible Fluids
To proceed, since W
in (6.52) satisfies @
t
W
DXW
C W
, we can
use Duhamel’s formula to write
(6.56) W
.t/ D e
tX
w
0
C
Z
t
0
e
.ts/X
W
.s/ ds;
hence
(6.57) ke
t.X/
w
0
e
tX
w
0
k
L
p
Z
t
0
kW
.s/k
L
p
ds:
The following provides a useful estimate on the right side of (6.57)whenp D 2.
Lemma 6.5. Ta ke w
0
2 D.
2
/ D D ..X/
2
/, and construct W
as in (6.52).
Then there exists K 2 .0; 1/, independent of >0, such that
(6.58) kW
.t/k
2
L
2
e
2Kt
kw
0
k
2
L
2
:
Proof. We have
(6.59)
d
dt
kW
.t/k
2
L
2
D 2 Re.@
t
W
;W
/
D 2 Re.
2
W
;W
/ 2 Re.XW
;W
/
2 Re.X W
;W
/
D2 Re.XW
;W
/ 2 Re.Œ; X W
;W
/
2KkW
k
2
L
2
;
with K independent of . The last estimate holds because
(6.60) g 2 D./ )j.Xg; g/jK
1
kgk
2
L
2
;
and
(6.61)
W
.t/ 2 D.
2
/ ) Œ; XW
.t/ 2 L
2
.D/; and
kŒ; XW
.t/k
L
2
e
K
2
kW
.t/k
H
2
K
2
kW
.t/k
L
2
:
Theestimate(6.58) follows.
We can now prove the following.
Proposition 6.6. Given p 2 Œ1; 1/; w
0
2 L
p
.D/, we have
(6.62) e
t.X/
w
0
! e
tX
w
0
; as & 0;
with convergence in L
p
-norm.
6. Vanishing viscosity limits 595
Proof. We know that e
t
is a contraction semigroup on L
p
.D/ and e
tX
is a
group of isometries on L
p
.D/, and we have the Trotter product formula:
(6.63) e
t.X/
w
0
D lim
n!1
e
.t=n/
e
.t=n/X
n
w
0
;
in L
p
-norm, hence e
t.X/
is a contraction semigroup on L
p
.D/.By(6.58)and
(6.57), we have L
2
convergence for w
0
2 D.
2
/, which is dense in L
2
.D/.This
gives (6.62)forp D 2, by the standard approximation argument, a second use of
which gives (6.62)forallp 2 Œ1; 2.
Suppose next that p 2 .2; 1/, with dual exponent p
0
2 .1; 2/. The previous
results work with X replaced by X, yielding e
t.Cx/
g ! e
tX
g,as & 0,in
L
p
0
-norm, for all g 2 L
p
0
.D/. This implies that for w
0
2 L
p
.D/, convergence
in (6.62) holds in the weak
topology of L
p
.D/.Now,sincee
tX
is an isometry
on L
p
.D/,wehave
(6.64) ke
tX
w
0
k
L
p
lim sup
!0
ke
t.X/
w
0
k
L
p
;
for each w
0
2 L
p
.D/.SinceL
p
.D/ is a uniformly convex Banach space for
such p, this yields L
p
-norm convergence in (6.62).
To produce higher order Sobolev estimates, we have from (6.58) the estimate
(6.65) ke
t.X/
w
0
k
D./
e
Kt
kw
0
k
D./
;
first for each w
0
2 D.
2
/, hence for each w
0
2 D./. Interpolation with the
L
2
-estimate then gives
(6.66) ke
t.X/
w
0
k
D../
s=2
/
e
Kt
kw
0
k
D../
s=2
/
;
for each s 2 Œ0; 2; w
0
2 D../
s=2
/ D D
s
. As noted in (6.22), D
s
D H
s
.D/
for 0 s<1=2,sowehave
(6.67) ke
t.X/
w
0
k
H
s
.D/
Ce
Kt
kw
0
k
H
s
.D/
;0 s<
1
2
;
with C and K independent of 2 .0; 1. We can interpolate the estimate (6.67)
with
(6.68) ke
t.X/
w
0
k
L
p
.D/
kw
0
k
L
p
.D/
;1 p<1:
Using
(6.69) ŒH
s
.D/; L
p
.D/
D H
.1/s;q./
.D/;
1
q./
D
1
2
C
p
;
596 17. Euler and Navier–Stokes Equations for Incompressible Fluids
which follows from material in Chap. 13, 6,wehave
(6.70) ke
t.X/
w
0
k
H
;q
.D/
C
;q
e
Kt
kwk
H
;q
.D/
;
valid for
(6.71) 2 q<1;q2 Œ0; 1/:
Similar arguments give such operator bounds on e
tX
. We have the following
convergence result.
Proposition 6.7. Let ; q satisfy (6.71). Then, for each t 2 .0; 1/,
(6.72) w
0
2 H
;q
.D/ H) lim
!0
e
t.X/
w
0
D e
tX
w
0
;
in H
;q
-norm.
Proof. Given w
0
2 H
;q
.D/,(6.70) implies fe
t.X/
w
0
W 2 .0; 1g is
bounded in H
;q
.D/ for each t 2 .0; 1/, so there is a weak
limit point. But
Proposition 6.6 yields convergence to e
tX
w
0
in L
q
-norm, so e
tX
w
0
is the only
possible weak
limit point. Norm convergence in H
;q
.D/, for each <,then
follows from the compactness of the inclusion H
;q
.D/ ,! H
;q
.D/.Taking
0
>such that
0
q<1, the argument above yields e
t.X/
w
0
! e
tX
w
0
in H
;q
-norm for each w
0
2 H
0
;q
.D/. The conclusion follows by denseness of
H
0
;q
.D/ in H
;q
.D/, plus the uniform operator bound (6.70).
This concludes our treatment of R
1
.;t;x/. As mentioned, more precise re-
sults, including boundary layer analyses, are given in [MT1]and[MT2].
We move to an analysis of R
3
.;t;x/in (6.48), i.e.,
(6.73) R
3
.;t;x/ D
Z
t
0
e
.ts/.X/
.s
0
s
/
@w
@
ds;
where w
solves (6.34)and(6.36). Note that @=@ commutes with X; X
,and,
so z
.t; x/ D @w
=@ solves
(6.74)
@z
@t
D . X
/z
; z
ˇ
ˇ
ˇ
R
C
@D
D 0; z
.0; x/ D
@w
0
@
:
The maximum principle gives
(6.75)
@w
@
.s/
L
1
.D/
@w
0
@
L
1
.D/
:
6. Vanishing viscosity limits 597
Since the semigroup e
t.X/
is positivity preserving, we have
(6.76) jR
3
.;t;x/jk@
w
0
k
L
1
Z
t
0
e
.ts/.X/
ˇ
ˇ
s
0
.jxj/ s
.s; jxj/
ˇ
ˇ
ds:
Also, by radial symmetry,
(6.77) e
.ts/.X/
js
0
s
jDe
.ts/
js
0
s
j;
so
(6.78) jR
3
.;t;x/jk@
w
0
k
L
1
Z
t
0
e
.ts/
jQs
0
Qs
jds;
where
(6.79) Qs
0
.x/ D s
0
.jxj/; Qs
.t; x/ D s
.t; jxj/;
and, we recall from (6.41),
(6.80) v
.t; x/ D s
.t; jxj/x
?
;v
0
.x/ D s
0
.jxj/x
?
:
Turning these around, we have
(6.81) s
.t; jxj/ D
1
jxj
2
v
.t; x/ x
?
;s
0
.jxj/ D
1
jxj
2
v
0
.x/ x
?
;
and also, if fe
1
;e
2
g denotes the standard orthonormal basis of R
2
,
(6.82)
s
.t; r/ D
1
r
v
.t; re
1
/ e
2
D
1
r
Z
1
0
d
d
v
.t; re
1
/ e
2
d
D
Z
1
0
e
2
r
e
1
v
.t; re
1
/d;
and similarly
(6.83) s
0
.r/ D
Z
1
0
e
2
r
e
1
v
0
.t; re
1
/d:
The representation (6.81) is effective away from a neighborhood of fx D 0g,
especially near @D , where one reads off the uniform convergence of s
.t; r/ to
s
0
.r/ except on the boundary layer discussed above in the analysis of v
.t; / !
v
0
./,givenv
0
2 C
1
.D/.