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Continuous dependence on spatial geometry for solutions of the Navier-Stokes equations backward in time
Nonlrnear Anolym, Theory, Printed m Great Britain.
Methods
& Apphcafrons,
Vol.
No.9,
21,
pp.
651-664,
0362-546X/93 $6.00+ .OO C 1993 Pergamon Press Ltd
1993.
CONTINUOUS DEPENDENCE ON SPATIAL GEOMETRY FOR SOLUTIONS OF THE NAVIER-STOKES EQUATIONS BACKWARD IN TIME?_ L. E. PAYNE Department
of Mathematics,
Cornell
University,
Ithaca,
NY 14853, U.S.A.
(Received 30 June 1992; received for publication 18 May 1993) Key words and phrases: Navier-Stokes
equations,
continuous
dependence.
1. INTRODUCTION
WHEN ATTEMPTING to investigate the past history of an evolutionary process one is typically led to the study of an ill-posed problem for the governing system of partial differential equations. As we know, solutions to such problems seldom exist in the usual sense and when solutions do exist they are unstable in the sense that they do not depend continuously on initial data, boundary data, coefficients, space and time geometry and in fact on any quantities which are subject to error in the modeling of the physical problem in question. We usually avoid nonexistence difficulties by altering our concept of what we accept as a “solution” and we attempt to recover stability by appropriately restricting solutions to lie in some constraint set. Many of the physical theories we study are modeled by a coupled system, a basic ingredient of which is the Navier-Stokes system. Before we can deal with these more complicated systems we need first to investigate the Navier-Stokes equations themselves and since we are interested in past history we wish to “solve” them backward in time. The first work that the author is aware of, for solutions of the Navier-Stokes equations backward in time, is the early uniqueness proof of Serrin [l] for the solution of the so-called “final value” problem. The first attempt at stabilizing the final value problem against errors in the final data was that of Knops and Payne [2] who showed that solutions to the problem defined on a bounded region and appropriately constrained do depend continuously on the data. Using a different measure of continuous dependence Payne [3] was able to relax somewhat the constraint restrictions. At the same time he was able to stabilize the problem against errors in the final time geometry. Other work for exterior problems has been carried out by Straughan [4] and by Galdi and Straughan [5]. In this paper we investigate the question of continuous dependence on the spatial geometry. If we have any hope of solving this inherently unstable problem numerically we need to stabilize the problem against geometric errors since the elements will seldom fit the domain exactly. Instead of investigating the forward Navier-Stokes equations backward in time we shall change the time variable t to -t and study instead the backward Navier-Stokes equation forward in time. Our final value problem then becomes an initial value problem for the backward Navier-Stokes equations. Specifically let D, and D, be two bounded regions in R3 with C’ boundaries aD, and dD2, respectively. We assume that these domains have nonzero intersection
t This research was supported
by NSF Grant
# DMS-91.00786. 651
652
L. E. PAYNE
D := D, n D, and that classical solutions problems exist on the time interval (0, T)
u,(x, t) and
Ui,t- UjUi,j+ VAU~ = P,i
ui(x, t), i = 1,2, 3 of the following
(1.1)
in D, x (0, T)
Uj,j= 0 1 with x E D,
Ui(x9 O) = L(X),
(1.2) t)
=
0
V~ Vi,j
+
VAVi = q,i
Ui(Xy
on aD, x [0, T],
and Ui,t-
in D, x (0, T)
yi,j= 0
(1.3)
with x E D,
ui(x7O) = gi (x)Y
(1.4)
on aD, x [0,T].
U,(X, t> = 0
Here v is the coefficient of kinematic viscosity, p and q are unspecified pressure terms and x = (xi, x2, x3). We have used the summation convention and a comma indicates partial differentiation. We wish to compare Ui and Ui assuming that (DI U D2) fl DC is in some sense small. In fact we seek appropriate constraint sets for Ui and Ui such that Ui will remain “close” to Ui (in a sense to be made precise) in some appropriate measure on the interval 0 < r < T provided the measure of (Dl U D,) flDC is small. We emphasize that the constraint restrictions which we impose are sufficient to stabilize the problem, but are probably somewhat conservative. Since we shall be considering the difference function Wi(X, t) given by Wi =
Uj
-
(1.5)
Vi,
even though the investigations would be somewhat more complicated, we could treat the more general problem in which (1.1 J and (1.2J have inhomogeneous forcing terms as well as the case in which Ui and ui have inhomogeneous boundary data (see, e.g. Crooke and Payne [6]). Also to keep an already complicated problem as simple as possible we shall assume that fi(x)
= gi(x)
in D
(1.6)
and that D, and D, are each star-shaped with respect to a point in D which we take as the origin in R3. We introduce two sets of functions 312, and 3n,, the set 9X, consisting of all functions oi which satisfy I M,2 SUP (3i(Ti DI x (0,T) for some constant
M, and the set %I, consisting
of all differentiable
SUP [XiXi+ Xi,jXi,j + Xi,tXi,tls M,’ 4 x (0,r)
(1.7) functions
Xi satisfying (1.8)
653
Continuous dependence on spatial geometry
for some constant M,. In what follows we shall constrain dependence ui E 312Z, and use as a measure of continuous
ui and vi by requiring
Ui E 311, and
where llWl)2
=
i
WiWidx
(1.9)
.D and wi is defined by (1.5). This problem function as described in Section 2.
is simplified
2. AN AUXILIARY
The function
by the introduction
FUNCTION
wi(x, t) is easily seen to satisfy the partial
aw _
-
UjWi,j
-
WjVi,j
f
at
of an auxiliary
differential
equation
in D x (0,T)
VAWi = R,i
(2.1)
where
(2.2)
R-p-q. The initial
and boundary
conditions
become Wi(X, 0) = 0
Wi(X,
t)
=
Ui(X,
t)
-
in D
Vj(X,
t)
Since it is difficult to work with the inhomogeneous function Hi which for each t E (0, T) is a solution for some harmonic function S
on
aD x [0,T].
(2.3)
boundary data we introduce an auxiliary of the following well-posed linear problem
VAHi = S,i in D
(2.4)
Hj,j = 0 ) with on aD.
Hi(X, t) = Wi(Xy t) It follows then
from the uniqueness
of (2.4), (2.5), that since Wi(X, 0) = 0, I
of solutions
i = 1,2, 3.
0) = 0,
Hi(X,
(2.5) 132, 3 (2.6)
We now form vi = wi - H.I. It follows
(2.7)
that t+visatisfies avi
at
UjWi,j -
‘//jV;,j
+
VAW~ = P,i + Li(H,
U,
V)
(2.8)
where Li(H, U,
V)
=
-~
+
UjH;,j + HjVi,j
(2.9)
654
L. E. PAYNE
and P=R-S. The boundary
and initial
conditions
now become
Witx,
t,
=
on D x [0, T]
O
in D.
ry;(x, 0) = 0 Since
we derive the desired continuous dependence inequality by bounding the two expressions on the right of (2.12). A bound for the first expression will be given in the next section. Bounds for various norms of Hi are known, but for completeness are derived explicitly in the Appendix. Thus, in Section 3 we establish the following theorem. THEOREM
a function
1. If Ui(X, t) E 311, and Ui(X, t) E im, then it is possible to compute an explicit K and s(t) (0 < 6 5 1) independent of Ui and ui such that for 0 5 c < T
where r is the maximum
distance
along
a ray between
3. PROOF
To bound
i?D, and 6'D, .
OF THEOREM
the first term on the right of (2.12) we form
where & is a constant
to be determined
Q, =
later and
I (CT-
r1)211ff,,l12 + ilffl121 drl.
(3.2)
.O
An appropriate bound for Q, is given in the Appendix. We derive a second order differential inequality for F which will yield the desired bound for F. Since wi(X, 0) = 0 we have ‘)I F’(t)
= 2
= 2
II 0 ‘f
(f - ul)lM2dv
(t - ~r)~(w>w,,) dv I
= 2v “(t / .O
(t - Y/)~(w, I+Y* Vu + L) dq.
- ~)211v~I12 dq + 2 I .O
(3.3)
Continuous
We have used the differential (l/I,
dependence
equation ly
on spatial
for I+Yand the notation
vv + L) =
*
655
geometry
Vi(Wjvi,j
dXn
Li)
+
\ .D A further
differentiation F’(t)
(3 -4)
gives
= 4v “(f - u)~(vIJ/, vw,J drj + 4 ” (t - q)(w, IV. Vv + L) drl / / .O .0 ‘I = 4 (t - rj)2(~,,, (//,? - u . VW - w. Vu -- L) dy !0 +
‘(t - ~)(I,u, y * Vv + L) dv.
4
(3.5)
0
Setting @i =
Wi,t
-
and using the fact that ui and vi are elements F”(t)
2 4
”(t -
t”j
of 3n, and 3n,,
(t - r1)211vv1i2drl - 4
‘- f
rl~211~vllIlvll dv + 4~~ ! 0
‘0 - r)(W,L)drl i ,0
use of the definition
4
(t - a)2(@, v/.
Vu)
dv
I’ t
(t ?0
Making that
we have
!0
- 2M,M,
4
respectively,
‘t
r/)211Q’112 dq - M:
0
+
(3.6)
Wi,j
*t I!
- 4
(t - r)llwll”drl
‘0 - r1)2(W,,,L)drl. 1 .,O
(2.9) we now examine
(3.7)
the last two terms in detail.
We note first
(t
-0
5 4(.i,:llIl12dnjl/2r!‘:(T
- )1)211H,~\12d~j”2
t + 4M,
(t - r1)21/Vw112 drl ]
“2(.i:
lIfd12
dvj
“’
0
,/ + 4M2
(t - rl)211v/112 dq ]
“‘ii:
llHl12
drlj
1’2.
(3.8)
0
Since wi = 0 when t = 0 it follows
12
from Schwarz’s
(t - r1)211Qt12dr7
inequality
that
(t - r1)211Vw112 drl 1’;‘].
(3.9)
656
L.
E.
PAYNE
In the last step we have used the triangle
inequality.
Thus, l/2
(t - r1)21\Q)\\2drl
(t - r7)2\\H,,?
drl
I
l/2 +
4M1
(t - ~)211Vw112drl
(1 - @21/H,#
drl 1
t/2
or 4
+
4M,
+
41%f2
(t
-
vl)*IiVy/l12
dv
(3.10)
t Ii
(t - rl)(lv> L) drl
.O
for computable constants yI. To bound the last term in (3.7) we note that ‘*f (t - q)2(y/,,
L) dyj =
I0 <, We examine
”(t
- ~)~(w,,,
-H,,
+ u . VH + H * vu) drl.
(3.12)
/ .O
each term on the right separately
‘t
II
vr)*W,, , ~4. VW dv
(f -
.O
-1
sI
=I!
(t -
10
q)(u . vy/, -2H + (t - q)H,,) drl +
I
0 - ~)2(~,, . VW, H) dv
I0
s(
+
II
(t -
.O
r~)~(vly - iv,, + u,,l, H) drl .
(3.13)
Continuous
To bound
dependence
on spatial
geometry
the last term of (3.13) we note first that
=
!‘t
(t -
q)2(q/ VW, u - v) drl. ,‘I
(3.14)
*
,O
Thus, ” t
I!
(t - v)~(w,,* VW, H) drl
.O
‘(t 0
-
~)2~~Vv/~~2
drl
1 3
(3.15) for computable
constants
y4 and ys . But in (3.13) ‘f
‘lf (t
I!
-
rl12(Vw
* u,,,,
W
5
dv
I
0
l/2
(t - r1)21b’y/i12dv
M2T
ii
0
Inserting (3.16) and (3.15) back into (3.13) and the result obvious inequalities to
i
.O
’ iffii2
drl
.
(3.16)
1
into (3.12) leads after the use of
or 4
+
~9
“(I / .O
for computable
yi.
-
r1)2/IVwii2dv
(3.18)
658
L. E.
Returning
to (3.7) we may, thus, write for computable ‘>f
F”(t)
PAYNE
2 4
(t - ~)211@l12dq I
,O
x
I/2
\”(f t
lj)211@(DI12d~ 1
<, 0
‘I Y13 i
II
yi
l/2
,I0
(f
-
~i)~llwll~
drl
+
[(YMT
+
d1;11”2 1
MiIH112
1
l/2
-
Y16
‘k I ,O
-
rl)211Vv112drl
-
Y17
(t
-
v)211Vv112
drl
” Cf i 0
-
d2t/
‘f -
YlS
” (t \ .“O
-
vl)llwI12
dv
-
Y19
] (t
i.i
-
vl)‘llvwll”
drl
-
rl)21/VIc/I~2
dv
“Cl
-
~)2~~~,,t~2
1
*t
l/2 (t
Y21
““1
dv
I,/ 0
ii ,O
-
1
l/2
l/2 (t
Y20
llff12
drl
0
,I -
1:
w/I2
-
r1)2/bd12
dv
il ,O
”
llH1/2
dv
,O !
(3.19)
. 1
Now from (3.3) we note that ”I 2~
I,I 0
(f - d21/Vv//12drl ‘f
I
F’ + 2M2
(f - v)~IIwII~ dv + 2
”(t ii
”l r1)2~~v~~2 drl \ (t - ~)2~~ff,T~~2 drl
I0 c_ ‘f f 2M2T (f - r)211vl12 dvl .c llH;l’dvl ‘I2 (1.0 f’f + 2M, (t - v)2/lVly1/2dv “0 - v)211H112dvl i [I,O 1 .O from which it is clear that for constants
l/2
“’ l/2
(3.20)
3
yz2 and yz3
“I I
(t - r1)2i/VWI12drl 5
~22F’
+
(3.21)
Y23F.
.O
Combining
terms as in [2] we conclude
that for constants
K, and K2 (3.22)
FF” - (F’)’ L -K, FF’ - K,F”. But this inequality
implies
(see [7, p. 261) F(t)
5 [F(O)]““‘[F(T)]‘-““’
eKZK;‘P(o
(3.23)
where a(t)
= (1 - eeKl’)(l
- emKl ‘)-’
(3.24)
and p(t)
= ((T - f) + temKIT - TePKI’)(l
- eeK17))l
< T.
(3.25)
659
Continuous dependence on spatial geometry
If we can now show that (3.26)
F(0) = ,kJr 5 K,s then the theorem
is proved.
In the Appendix
we show that for computable
‘7[CT- v~)~W,,>>~+ W))21 drl ! ,O
Q, I&
K4 (see (A.16)),
(3.27)
where W,,>>2
=
Hi,vHi,q
(3.28)
ds-
f?D
To bound
the right-hand
side of (3.27) in terms of r we recall that on t!lH Hi = ui aHi
hi
at 37 Now define
extensions
(3.29)
_--!au. at’
(3.30)
of ui and ui as follows L(l” I =
vi* ZZ Thus,
u;
ui
in 0,
x [0, T]
0
in 0;
x [0, T]
vi
in D2 x [0, T]
0
in 0;
(3.31)
(3.32) x [0, T].
from (3.27) we have Q, 5 &
[k
- v~)~((w,;>>~ +
,O
Gw*>>ldul
(3.33)
where wi” = u* - vi*. Let the boundary
and the boundary
of D be described
in spherical
(3.34)
coordinates
(r, Q) by
r = r,(Q),
(3.35)
r = r2(Cl).
(3.36)
of D, U D, by
Then ‘3 awlc wi”(r,(Q), and using Schwarz’s
inequality
Q t) = -
\
-L
ar
(3.37)
dr,
we find
w*w: 5
’ r2
max(r, R
- rr)
!
I rl
wC~ W~,i
dr.
(3.38)
L. E. PAYNE
660
It then follows
that
WiTiWil;r2 dr da
where n, is the radial
component
of the outward r,
unit normal
= max DI”4
(3.39)
cXI and
r
(3.40)
ho = min m,. 8D
But 1
1r&7
I!
WlTjWil;r2 dr dL2 5 2
ui,j”i,jdx
+
I,,
(3.41)
vi,jvi,jdX]*
,R .i-l It then follows
that ^T
Ui,j u;,J
h dV +
! .i
.O
But "T
”
!!
.O
‘7
Ui,j
Ui,j
dx drl = - ~ V
Ui,j
Ui,j
dx dll
.
(3.42)
D2
n
!I 0
’
ui(-“i,q
+
uj”i,j
+
P,il
dX
CD,
’
1
I
%,,
LliUi
dX
- Mm <
(3.43)
2v
where ID,1 is the measure of D, . A similar result holds for the second (3.42). Replacing w* by WC, we derive in an analogous manner
integral
on the right of
‘T
I
CT
-
v)2W,;))2
dv
,O
(T - V)‘Ui,j,Ui,j,
h
dV +
(T - q)2Vi,jg Vi,jn dxdv
.
(3.44)
Continuous
dependence
on spatial
661
geometry
Integrating each term on the right by parts with respect to xj, and using the differential we find for instance
Using the differential equation second we find (after dropping
in the first term on the right and Schwarz’s negative terms)
equation
inequality
on the
l/2 13
Ui,j
U;,j
dx
drl +
[U;Ui12 dx drl
(T - V)2Ui,j, ui,j, ~JLdV
l/2 (T s!:0
This leads clearly to an inequality “T
-
V12uj,qUj,q
&dV
.
(3.46)
D
of the type
‘I
!!
CT
-
V)2ui,jvui,jq
&dV
5
(3.47)
K7
,O CD, the last result following from (3.43). Using the analogous bound for the second integral on the right of (3.45) and inserting into (3.44) together with the insertion of (3.43) and the analogous bounds for the second integral on the right of (3.41), into (3.41), we obtain the desired bound for Qi , i.e. for computable K, Q, I K,r. The theorems
then follow
(3.48)
from (3.23) and (3.26) since
T I
and 1; ((H’II dq I Qi
.
T3[M;lDll
We have, of course,
0 in (2.12).
+ M;ID,I]
+ T2
.O i
llHl12dv +
Q1
(3.49)
again used the fact that
(t - rl)211ffl12 drl 5 T2Q,
(3.50)
662
L. E. PAYNE 4. REMARKS
Instead of the measure (see (3.34))
of continuous
dependence
used in (2.13) we could have used
f li
zI a
i’ I Jo .JD,ffD,
(t - #~$v~dxdy
=
‘0 - r1)2b*i/2dljl JO f
+ i
(t - #uiui
i
dxdr
Jo ,lD,/D
P
(t -
+
r/)%;u; dx dq.
(4.1)
.ii0 ~ DZ/D
But the second and third integrals are clearly seen to be O(r) since letting r = F(O)
(4.2)
denote the boundary of D, U D2 we have (t - a)“tr - F(Sl)]uiuin, ds ”
=3
f
Ii
(f - ri)‘~iZ~idx dil + 2
(r - ~0% - WW.W,,
%0 * D,/D
dx drl
(4.3)
which can be solved to yield (t -
q)‘uj ui dx dq (4.4)
The result then follows from the constraint requirement
on ui and the fact that
(4.5) The second step makes use of the differential equation satisfied by ui . The last integral in (4.1) may be treated in the same manner. REFERENCES SERRINJ., The initial value problem for the Navier-Stokes University of Wisconsin Press, Madison (1963). KNOPSR. J. & PAYNE L. E., On the stability of solutions
equations,
Pruc. Symp. .~on~jne~r Problems, pp. 69-98.
of the Navier-Stokes
equations
backward
in time, Archs
ration. Mech. Analysis 29, 331-335 (1968). PAYNE L. E., Some remarks on ill-posed problems for viscous STRAUUGHAN B., Backward uniqueness and unique continuation exterior domain, J. ma/h. pures appl. 62, 49-62 (1983).
fluids, Inf. J. Eng. Sci. 30, 1341-1347 (1992). for solutions to the Navier-Stokes equations on an
Continuous 5. 6. I. 8. 9.
dependence
on spatial
663
geometry
G. P. & STRAUGHAN B., Stability of solutions to the Navier-Stokes equations backward in time, Archs ration. Mech. Analysis 101, 107-l 14 (1988). CROOKE P. S. & PAYNE L. E., Continuous dependence on geometry for the backward heat equation, Math. Mefh. Appl. Sci. 6, 433-448 (1984). PAYNE L. E., Improperly posed problems in partial differential equations, Regional Conf. Series in Applied Mathematics, No. 22. SIAM, Philadelphia (1975). PAYNE L. E., Uniqueness criteria for steady state solutions of the Navier-Stokes equations, Simp. Int. SuNe Applic. de/i' Analisi alla Fisica Mat., pp. 131-153, Cagliari-Sassari (1964). HORCAN C. 0. & PAYNE L. E., Lower bounds for free membrane and related eigenvalues, R. Mat. Rorna Ser. 7(10), 324-357 (1990).
GALDI
APPENDIX We now derive (3.27) where Qi is given by (3.2). We need merely show that for 0 5
t5 T
IIHIV 2 &WZD.
(A.1)
Then since H,, satisfies the same differential equation as H (except that S is replaced by S,,) (3.27) will follow integration with respect to t. To accomplish this we introduce the auxiliary function cp,(x, t) which for 0 5 t 5 T satisfies A(o, = H, + n[,, c 1,1 = 0
in D
upon
(~4.2)
1
with V, = 0 For some n, unspecified
a priori,
it is known
An integration
that the solution
H,n,(v,,, - (DJ,z) ds -
zz
on aD.
by parts and use of the differential
(A.3)
of (A.2),
nH,n,d.-
equation
(A.3) exists. Then
i co,,W,,~. (cp,, -
.D’
for H demonstrates
(A.4)
”
that the last integral
on the right is zero.
(A.5) To bound
the integral
in braces
on the right we make use of a Rellich identity
as in [8]. From the fact that
wJ,,,l(9,,, - (Pj,J,J- H, - ‘,,I b = 0 we conclude
upon integration
64.6)
by parts that
(A.7) We note that the last term vanishes
since on L?D
c
a
9,35 Since jD q,,, co,,, ti
= 0 it follows
=
n,--n,-
ax,
a ax,
)
cp,=O.
(A.@
that
(P,,j ~ CP,,,)P‘,, b + i IIV~OII’ 5 ~MIIVVII IIHII,
(A.9)
664
L. E. PAYNE
where h, and r, are given by (3.40).
Using the arithmetic-geometric - 9,,,)9,‘,
Returning to (A.5) we now derive a bound a function Y satisfying
A’? = 0
in D
ay -_=x
on aD.
upon integration
by parts
= (‘=,
and use of Schwarz’s
(A. 10) employed
in [8], i.e. we introduce
(A.ll) (A.2), (A.3) we choose this constant up to an arbitrary constant. Then
vy)
= Ii D?](9,.,
inequality
~ 9,,,),;
so that
(Y./r,
- Y,,n,)Y,,n,b
\
1,1an
.I
in [8] that for computable
(A.12)
~ H,]ds.
we have
\ I/Z
1
I I’ ((70Y 5 1 \
that
air
I
But it was shown
d.Y5 &lHIV.
up to an arbitrary additive constant by the problem we are assured of the existence of Y, defined again
((n)Y = , anR;ids Thus,
on the right we conclude
for C(n)). We again make use of techniques
an
Since rr was defined 430 z ds = 0. Thus,
mean inequality
(cp,,, - 9,,,)9,,,cls~ ao
+ I
IIW Ilfa.
K,
+o
5 K, ~ a.
W,,n, - y,,W’,,n,ds Also
ay 2
d.s =
an
K,(W2.
(A.13)
(A.14) where p2 is the first nonzero This leads to
Stekloff
eigenvalue
for D. For lower bounds
for pZ see [9] and the papers
cited therein.
I/Z ((jr)) 5 K:‘=
(i s ao
+ pi”*/lHl/
(cp,,, - 9,,,)9,,,d~
We have made use of (A.10) in the last step. Inserting IlHII 5 (T&“* Noting
the remark
following
(A.1) we observe
5 [IK,r;,h,‘)“2
+ p;“z]~IH~l.
(A. 15)
1 (A. IO) and (A.15) into (A.5) we conclude
+ {K,r,&h,‘)“’
+ p;“‘j((H))
= K;“((H)).
that (3.27) has been established.
that (A.16)
Methods
& Apphcafrons,
Vol.
No.9,
21,
pp.
651-664,
0362-546X/93 $6.00+ .OO C 1993 Pergamon Press Ltd
1993.
CONTINUOUS DEPENDENCE ON SPATIAL GEOMETRY FOR SOLUTIONS OF THE NAVIER-STOKES EQUATIONS BACKWARD IN TIME?_ L. E. PAYNE Department
of Mathematics,
Cornell
University,
Ithaca,
NY 14853, U.S.A.
(Received 30 June 1992; received for publication 18 May 1993) Key words and phrases: Navier-Stokes
equations,
continuous
dependence.
1. INTRODUCTION
WHEN ATTEMPTING to investigate the past history of an evolutionary process one is typically led to the study of an ill-posed problem for the governing system of partial differential equations. As we know, solutions to such problems seldom exist in the usual sense and when solutions do exist they are unstable in the sense that they do not depend continuously on initial data, boundary data, coefficients, space and time geometry and in fact on any quantities which are subject to error in the modeling of the physical problem in question. We usually avoid nonexistence difficulties by altering our concept of what we accept as a “solution” and we attempt to recover stability by appropriately restricting solutions to lie in some constraint set. Many of the physical theories we study are modeled by a coupled system, a basic ingredient of which is the Navier-Stokes system. Before we can deal with these more complicated systems we need first to investigate the Navier-Stokes equations themselves and since we are interested in past history we wish to “solve” them backward in time. The first work that the author is aware of, for solutions of the Navier-Stokes equations backward in time, is the early uniqueness proof of Serrin [l] for the solution of the so-called “final value” problem. The first attempt at stabilizing the final value problem against errors in the final data was that of Knops and Payne [2] who showed that solutions to the problem defined on a bounded region and appropriately constrained do depend continuously on the data. Using a different measure of continuous dependence Payne [3] was able to relax somewhat the constraint restrictions. At the same time he was able to stabilize the problem against errors in the final time geometry. Other work for exterior problems has been carried out by Straughan [4] and by Galdi and Straughan [5]. In this paper we investigate the question of continuous dependence on the spatial geometry. If we have any hope of solving this inherently unstable problem numerically we need to stabilize the problem against geometric errors since the elements will seldom fit the domain exactly. Instead of investigating the forward Navier-Stokes equations backward in time we shall change the time variable t to -t and study instead the backward Navier-Stokes equation forward in time. Our final value problem then becomes an initial value problem for the backward Navier-Stokes equations. Specifically let D, and D, be two bounded regions in R3 with C’ boundaries aD, and dD2, respectively. We assume that these domains have nonzero intersection
t This research was supported
by NSF Grant
# DMS-91.00786. 651
652
L. E. PAYNE
D := D, n D, and that classical solutions problems exist on the time interval (0, T)
u,(x, t) and
Ui,t- UjUi,j+ VAU~ = P,i
ui(x, t), i = 1,2, 3 of the following
(1.1)
in D, x (0, T)
Uj,j= 0 1 with x E D,
Ui(x9 O) = L(X),
(1.2) t)
=
0
V~ Vi,j
+
VAVi = q,i
Ui(Xy
on aD, x [0, T],
and Ui,t-
in D, x (0, T)
yi,j= 0
(1.3)
with x E D,
ui(x7O) = gi (x)Y
(1.4)
on aD, x [0,T].
U,(X, t> = 0
Here v is the coefficient of kinematic viscosity, p and q are unspecified pressure terms and x = (xi, x2, x3). We have used the summation convention and a comma indicates partial differentiation. We wish to compare Ui and Ui assuming that (DI U D2) fl DC is in some sense small. In fact we seek appropriate constraint sets for Ui and Ui such that Ui will remain “close” to Ui (in a sense to be made precise) in some appropriate measure on the interval 0 < r < T provided the measure of (Dl U D,) flDC is small. We emphasize that the constraint restrictions which we impose are sufficient to stabilize the problem, but are probably somewhat conservative. Since we shall be considering the difference function Wi(X, t) given by Wi =
Uj
-
(1.5)
Vi,
even though the investigations would be somewhat more complicated, we could treat the more general problem in which (1.1 J and (1.2J have inhomogeneous forcing terms as well as the case in which Ui and ui have inhomogeneous boundary data (see, e.g. Crooke and Payne [6]). Also to keep an already complicated problem as simple as possible we shall assume that fi(x)
= gi(x)
in D
(1.6)
and that D, and D, are each star-shaped with respect to a point in D which we take as the origin in R3. We introduce two sets of functions 312, and 3n,, the set 9X, consisting of all functions oi which satisfy I M,2 SUP (3i(Ti DI x (0,T) for some constant
M, and the set %I, consisting
of all differentiable
SUP [XiXi+ Xi,jXi,j + Xi,tXi,tls M,’ 4 x (0,r)
(1.7) functions
Xi satisfying (1.8)
653
Continuous dependence on spatial geometry
for some constant M,. In what follows we shall constrain dependence ui E 312Z, and use as a measure of continuous
ui and vi by requiring
Ui E 311, and
where llWl)2
=
i
WiWidx
(1.9)
.D and wi is defined by (1.5). This problem function as described in Section 2.
is simplified
2. AN AUXILIARY
The function
by the introduction
FUNCTION
wi(x, t) is easily seen to satisfy the partial
aw _
-
UjWi,j
-
WjVi,j
f
at
of an auxiliary
differential
equation
in D x (0,T)
VAWi = R,i
(2.1)
where
(2.2)
R-p-q. The initial
and boundary
conditions
become Wi(X, 0) = 0
Wi(X,
t)
=
Ui(X,
t)
-
in D
Vj(X,
t)
Since it is difficult to work with the inhomogeneous function Hi which for each t E (0, T) is a solution for some harmonic function S
on
aD x [0,T].
(2.3)
boundary data we introduce an auxiliary of the following well-posed linear problem
VAHi = S,i in D
(2.4)
Hj,j = 0 ) with on aD.
Hi(X, t) = Wi(Xy t) It follows then
from the uniqueness
of (2.4), (2.5), that since Wi(X, 0) = 0, I
of solutions
i = 1,2, 3.
0) = 0,
Hi(X,
(2.5) 132, 3 (2.6)
We now form vi = wi - H.I. It follows
(2.7)
that t+visatisfies avi
at
UjWi,j -
‘//jV;,j
+
VAW~ = P,i + Li(H,
U,
V)
(2.8)
where Li(H, U,
V)
=
-~
+
UjH;,j + HjVi,j
(2.9)
654
L. E. PAYNE
and P=R-S. The boundary
and initial
conditions
now become
Witx,
t,
=
on D x [0, T]
O
in D.
ry;(x, 0) = 0 Since
we derive the desired continuous dependence inequality by bounding the two expressions on the right of (2.12). A bound for the first expression will be given in the next section. Bounds for various norms of Hi are known, but for completeness are derived explicitly in the Appendix. Thus, in Section 3 we establish the following theorem. THEOREM
a function
1. If Ui(X, t) E 311, and Ui(X, t) E im, then it is possible to compute an explicit K and s(t) (0 < 6 5 1) independent of Ui and ui such that for 0 5 c < T
where r is the maximum
distance
along
a ray between
3. PROOF
To bound
i?D, and 6'D, .
OF THEOREM
the first term on the right of (2.12) we form
where & is a constant
to be determined
Q, =
later and
I (CT-
r1)211ff,,l12 + ilffl121 drl.
(3.2)
.O
An appropriate bound for Q, is given in the Appendix. We derive a second order differential inequality for F which will yield the desired bound for F. Since wi(X, 0) = 0 we have ‘)I F’(t)
= 2
= 2
II 0 ‘f
(f - ul)lM2dv
(t - ~r)~(w>w,,) dv I
= 2v “(t / .O
(t - Y/)~(w, I+Y* Vu + L) dq.
- ~)211v~I12 dq + 2 I .O
(3.3)
Continuous
We have used the differential (l/I,
dependence
equation ly
on spatial
for I+Yand the notation
vv + L) =
*
655
geometry
Vi(Wjvi,j
dXn
Li)
+
\ .D A further
differentiation F’(t)
(3 -4)
gives
= 4v “(f - u)~(vIJ/, vw,J drj + 4 ” (t - q)(w, IV. Vv + L) drl / / .O .0 ‘I = 4 (t - rj)2(~,,, (//,? - u . VW - w. Vu -- L) dy !0 +
‘(t - ~)(I,u, y * Vv + L) dv.
4
(3.5)
0
Setting @i =
Wi,t
-
and using the fact that ui and vi are elements F”(t)
2 4
”(t -
t”j
of 3n, and 3n,,
(t - r1)211vv1i2drl - 4
‘- f
rl~211~vllIlvll dv + 4~~ ! 0
‘0 - r)(W,L)drl i ,0
use of the definition
4
(t - a)2(@, v/.
Vu)
dv
I’ t
(t ?0
Making that
we have
!0
- 2M,M,
4
respectively,
‘t
r/)211Q’112 dq - M:
0
+
(3.6)
Wi,j
*t I!
- 4
(t - r)llwll”drl
‘0 - r1)2(W,,,L)drl. 1 .,O
(2.9) we now examine
(3.7)
the last two terms in detail.
We note first
(t
-0
5 4(.i,:llIl12dnjl/2r!‘:(T
- )1)211H,~\12d~j”2
t + 4M,
(t - r1)21/Vw112 drl ]
“2(.i:
lIfd12
dvj
“’
0
,/ + 4M2
(t - rl)211v/112 dq ]
“‘ii:
llHl12
drlj
1’2.
(3.8)
0
Since wi = 0 when t = 0 it follows
12
from Schwarz’s
(t - r1)211Qt12dr7
inequality
that
(t - r1)211Vw112 drl 1’;‘].
(3.9)
656
L.
E.
PAYNE
In the last step we have used the triangle
inequality.
Thus, l/2
(t - r1)21\Q)\\2drl
(t - r7)2\\H,,?
drl
I
l/2 +
4M1
(t - ~)211Vw112drl
(1 - @21/H,#
drl 1
t/2
or 4
+
4M,
+
41%f2
(t
-
vl)*IiVy/l12
dv
(3.10)
t Ii
(t - rl)(lv> L) drl
.O
for computable constants yI. To bound the last term in (3.7) we note that ‘*f (t - q)2(y/,,
L) dyj =
I0 <, We examine
”(t
- ~)~(w,,,
-H,,
+ u . VH + H * vu) drl.
(3.12)
/ .O
each term on the right separately
‘t
II
vr)*W,, , ~4. VW dv
(f -
.O
-1
sI
=I!
(t -
10
q)(u . vy/, -2H + (t - q)H,,) drl +
I
0 - ~)2(~,, . VW, H) dv
I0
s(
+
II
(t -
.O
r~)~(vly - iv,, + u,,l, H) drl .
(3.13)
Continuous
To bound
dependence
on spatial
geometry
the last term of (3.13) we note first that
=
!‘t
(t -
q)2(q/ VW, u - v) drl. ,‘I
(3.14)
*
,O
Thus, ” t
I!
(t - v)~(w,,* VW, H) drl
.O
‘(t 0
-
~)2~~Vv/~~2
drl
1 3
(3.15) for computable
constants
y4 and ys . But in (3.13) ‘f
‘lf (t
I!
-
rl12(Vw
* u,,,,
W
5
dv
I
0
l/2
(t - r1)21b’y/i12dv
M2T
ii
0
Inserting (3.16) and (3.15) back into (3.13) and the result obvious inequalities to
i
.O
’ iffii2
drl
.
(3.16)
1
into (3.12) leads after the use of
or 4
+
~9
“(I / .O
for computable
yi.
-
r1)2/IVwii2dv
(3.18)
658
L. E.
Returning
to (3.7) we may, thus, write for computable ‘>f
F”(t)
PAYNE
2 4
(t - ~)211@l12dq I
,O
x
I/2
\”(f t
lj)211@(DI12d~ 1
<, 0
‘I Y13 i
II
yi
l/2
,I0
(f
-
~i)~llwll~
drl
+
[(YMT
+
d1;11”2 1
MiIH112
1
l/2
-
Y16
‘k I ,O
-
rl)211Vv112drl
-
Y17
(t
-
v)211Vv112
drl
” Cf i 0
-
d2t/
‘f -
YlS
” (t \ .“O
-
vl)llwI12
dv
-
Y19
] (t
i.i
-
vl)‘llvwll”
drl
-
rl)21/VIc/I~2
dv
“Cl
-
~)2~~~,,t~2
1
*t
l/2 (t
Y21
““1
dv
I,/ 0
ii ,O
-
1
l/2
l/2 (t
Y20
llff12
drl
0
,I -
1:
w/I2
-
r1)2/bd12
dv
il ,O
”
llH1/2
dv
,O !
(3.19)
. 1
Now from (3.3) we note that ”I 2~
I,I 0
(f - d21/Vv//12drl ‘f
I
F’ + 2M2
(f - v)~IIwII~ dv + 2
”(t ii
”l r1)2~~v~~2 drl \ (t - ~)2~~ff,T~~2 drl
I0 c_ ‘f f 2M2T (f - r)211vl12 dvl .c llH;l’dvl ‘I2 (1.0 f’f + 2M, (t - v)2/lVly1/2dv “0 - v)211H112dvl i [I,O 1 .O from which it is clear that for constants
l/2
“’ l/2
(3.20)
3
yz2 and yz3
“I I
(t - r1)2i/VWI12drl 5
~22F’
+
(3.21)
Y23F.
.O
Combining
terms as in [2] we conclude
that for constants
K, and K2 (3.22)
FF” - (F’)’ L -K, FF’ - K,F”. But this inequality
implies
(see [7, p. 261) F(t)
5 [F(O)]““‘[F(T)]‘-““’
eKZK;‘P(o
(3.23)
where a(t)
= (1 - eeKl’)(l
- emKl ‘)-’
(3.24)
and p(t)
= ((T - f) + temKIT - TePKI’)(l
- eeK17))l
< T.
(3.25)
659
Continuous dependence on spatial geometry
If we can now show that (3.26)
F(0) = ,kJr 5 K,s then the theorem
is proved.
In the Appendix
we show that for computable
‘7[CT- v~)~W,,>>~+ W))21 drl ! ,O
Q, I&
K4 (see (A.16)),
(3.27)
where W,,>>2
=
Hi,vHi,q
(3.28)
ds-
f?D
To bound
the right-hand
side of (3.27) in terms of r we recall that on t!lH Hi = ui aHi
hi
at 37 Now define
extensions
(3.29)
_--!au. at’
(3.30)
of ui and ui as follows L(l” I =
vi* ZZ Thus,
u;
ui
in 0,
x [0, T]
0
in 0;
x [0, T]
vi
in D2 x [0, T]
0
in 0;
(3.31)
(3.32) x [0, T].
from (3.27) we have Q, 5 &
[k
- v~)~((w,;>>~ +
,O
Gw*>>ldul
(3.33)
where wi” = u* - vi*. Let the boundary
and the boundary
of D be described
in spherical
(3.34)
coordinates
(r, Q) by
r = r,(Q),
(3.35)
r = r2(Cl).
(3.36)
of D, U D, by
Then ‘3 awlc wi”(r,(Q), and using Schwarz’s
inequality
Q t) = -
\
-L
ar
(3.37)
dr,
we find
w*w: 5
’ r2
max(r, R
- rr)
!
I rl
wC~ W~,i
dr.
(3.38)
L. E. PAYNE
660
It then follows
that
WiTiWil;r2 dr da
where n, is the radial
component
of the outward r,
unit normal
= max DI”4
(3.39)
cXI and
r
(3.40)
ho = min m,. 8D
But 1
1r&7
I!
WlTjWil;r2 dr dL2 5 2
ui,j”i,jdx
+
I,,
(3.41)
vi,jvi,jdX]*
,R .i-l It then follows
that ^T
Ui,j u;,J
h dV +
! .i
.O
But "T
”
!!
.O
‘7
Ui,j
Ui,j
dx drl = - ~ V
Ui,j
Ui,j
dx dll
.
(3.42)
D2
n
!I 0
’
ui(-“i,q
+
uj”i,j
+
P,il
dX
CD,
’
1
I
%,,
LliUi
dX
- Mm <
(3.43)
2v
where ID,1 is the measure of D, . A similar result holds for the second (3.42). Replacing w* by WC, we derive in an analogous manner
integral
on the right of
‘T
I
CT
-
v)2W,;))2
dv
,O
(T - V)‘Ui,j,Ui,j,
h
dV +
(T - q)2Vi,jg Vi,jn dxdv
.
(3.44)
Continuous
dependence
on spatial
661
geometry
Integrating each term on the right by parts with respect to xj, and using the differential we find for instance
Using the differential equation second we find (after dropping
in the first term on the right and Schwarz’s negative terms)
equation
inequality
on the
l/2 13
Ui,j
U;,j
dx
drl +
[U;Ui12 dx drl
(T - V)2Ui,j, ui,j, ~JLdV
l/2 (T s!:0
This leads clearly to an inequality “T
-
V12uj,qUj,q
&dV
.
(3.46)
D
of the type
‘I
!!
CT
-
V)2ui,jvui,jq
&dV
5
(3.47)
K7
,O CD, the last result following from (3.43). Using the analogous bound for the second integral on the right of (3.45) and inserting into (3.44) together with the insertion of (3.43) and the analogous bounds for the second integral on the right of (3.41), into (3.41), we obtain the desired bound for Qi , i.e. for computable K, Q, I K,r. The theorems
then follow
(3.48)
from (3.23) and (3.26) since
T I
and 1; ((H’II dq I Qi
.
T3[M;lDll
We have, of course,
0 in (2.12).
+ M;ID,I]
+ T2
.O i
llHl12dv +
Q1
(3.49)
again used the fact that
(t - rl)211ffl12 drl 5 T2Q,
(3.50)
662
L. E. PAYNE 4. REMARKS
Instead of the measure (see (3.34))
of continuous
dependence
used in (2.13) we could have used
f li
zI a
i’ I Jo .JD,ffD,
(t - #~$v~dxdy
=
‘0 - r1)2b*i/2dljl JO f
+ i
(t - #uiui
i
dxdr
Jo ,lD,/D
P
(t -
+
r/)%;u; dx dq.
(4.1)
.ii0 ~ DZ/D
But the second and third integrals are clearly seen to be O(r) since letting r = F(O)
(4.2)
denote the boundary of D, U D2 we have (t - a)“tr - F(Sl)]uiuin, ds ”
=3
f
Ii
(f - ri)‘~iZ~idx dil + 2
(r - ~0% - WW.W,,
%0 * D,/D
dx drl
(4.3)
which can be solved to yield (t -
q)‘uj ui dx dq (4.4)
The result then follows from the constraint requirement
on ui and the fact that
(4.5) The second step makes use of the differential equation satisfied by ui . The last integral in (4.1) may be treated in the same manner. REFERENCES SERRINJ., The initial value problem for the Navier-Stokes University of Wisconsin Press, Madison (1963). KNOPSR. J. & PAYNE L. E., On the stability of solutions
equations,
Pruc. Symp. .~on~jne~r Problems, pp. 69-98.
of the Navier-Stokes
equations
backward
in time, Archs
ration. Mech. Analysis 29, 331-335 (1968). PAYNE L. E., Some remarks on ill-posed problems for viscous STRAUUGHAN B., Backward uniqueness and unique continuation exterior domain, J. ma/h. pures appl. 62, 49-62 (1983).
fluids, Inf. J. Eng. Sci. 30, 1341-1347 (1992). for solutions to the Navier-Stokes equations on an
Continuous 5. 6. I. 8. 9.
dependence
on spatial
663
geometry
G. P. & STRAUGHAN B., Stability of solutions to the Navier-Stokes equations backward in time, Archs ration. Mech. Analysis 101, 107-l 14 (1988). CROOKE P. S. & PAYNE L. E., Continuous dependence on geometry for the backward heat equation, Math. Mefh. Appl. Sci. 6, 433-448 (1984). PAYNE L. E., Improperly posed problems in partial differential equations, Regional Conf. Series in Applied Mathematics, No. 22. SIAM, Philadelphia (1975). PAYNE L. E., Uniqueness criteria for steady state solutions of the Navier-Stokes equations, Simp. Int. SuNe Applic. de/i' Analisi alla Fisica Mat., pp. 131-153, Cagliari-Sassari (1964). HORCAN C. 0. & PAYNE L. E., Lower bounds for free membrane and related eigenvalues, R. Mat. Rorna Ser. 7(10), 324-357 (1990).
GALDI
APPENDIX We now derive (3.27) where Qi is given by (3.2). We need merely show that for 0 5
t5 T
IIHIV 2 &WZD.
(A.1)
Then since H,, satisfies the same differential equation as H (except that S is replaced by S,,) (3.27) will follow integration with respect to t. To accomplish this we introduce the auxiliary function cp,(x, t) which for 0 5 t 5 T satisfies A(o, = H, + n[,, c 1,1 = 0
in D
upon
(~4.2)
1
with V, = 0 For some n, unspecified
a priori,
it is known
An integration
that the solution
H,n,(v,,, - (DJ,z) ds -
zz
on aD.
by parts and use of the differential
(A.3)
of (A.2),
nH,n,d.-
equation
(A.3) exists. Then
i co,,W,,~. (cp,, -
.D’
for H demonstrates
(A.4)
”
that the last integral
on the right is zero.
(A.5) To bound
the integral
in braces
on the right we make use of a Rellich identity
as in [8]. From the fact that
wJ,,,l(9,,, - (Pj,J,J- H, - ‘,,I b = 0 we conclude
upon integration
64.6)
by parts that
(A.7) We note that the last term vanishes
since on L?D
c
a
9,35 Since jD q,,, co,,, ti
= 0 it follows
=
n,--n,-
ax,
a ax,
)
cp,=O.
(A.@
that
(P,,j ~ CP,,,)P‘,, b + i IIV~OII’ 5 ~MIIVVII IIHII,
(A.9)
664
L. E. PAYNE
where h, and r, are given by (3.40).
Using the arithmetic-geometric - 9,,,)9,‘,
Returning to (A.5) we now derive a bound a function Y satisfying
A’? = 0
in D
ay -_=x
on aD.
upon integration
by parts
= (‘=,
and use of Schwarz’s
(A. 10) employed
in [8], i.e. we introduce
(A.ll) (A.2), (A.3) we choose this constant up to an arbitrary constant. Then
vy)
= Ii D?](9,.,
inequality
~ 9,,,),;
so that
(Y./r,
- Y,,n,)Y,,n,b
\
1,1an
.I
in [8] that for computable
(A.12)
~ H,]ds.
we have
\ I/Z
1
I I’ ((70Y 5 1 \
that
air
I
But it was shown
d.Y5 &lHIV.
up to an arbitrary additive constant by the problem we are assured of the existence of Y, defined again
((n)Y = , anR;ids Thus,
on the right we conclude
for C(n)). We again make use of techniques
an
Since rr was defined 430 z ds = 0. Thus,
mean inequality
(cp,,, - 9,,,)9,,,cls~ ao
+ I
IIW Ilfa.
K,
+o
5 K, ~ a.
W,,n, - y,,W’,,n,ds Also
ay 2
d.s =
an
K,(W2.
(A.13)
(A.14) where p2 is the first nonzero This leads to
Stekloff
eigenvalue
for D. For lower bounds
for pZ see [9] and the papers
cited therein.
I/Z ((jr)) 5 K:‘=
(i s ao
+ pi”*/lHl/
(cp,,, - 9,,,)9,,,d~
We have made use of (A.10) in the last step. Inserting IlHII 5 (T&“* Noting
the remark
following
(A.1) we observe
5 [IK,r;,h,‘)“2
+ p;“z]~IH~l.
(A. 15)
1 (A. IO) and (A.15) into (A.5) we conclude
+ {K,r,&h,‘)“’
+ p;“‘j((H))
= K;“((H)).
that (3.27) has been established.
that (A.16)