Inat. J. Mh. Math. S.
Vol. I, (1978) 21-30
21
EXPLICIT L 2 INEQUALITIES FOR PARABOLIC AND
PSEUDOPARABOLIC EQUATIONS
WITH NEUMANN BOUNDARY CONDITIONS
J. R. KUTTLER and V. G. SIGILLITO
Applled Physics Laboratory
The Johns Hopkins University
Laurel, Maryland 20810
(Received February 17, 1977 and in flnal revised form November 1, 1977)
ABSTRACT.
Expllcit L 2 inequalltles are derived for second and third order
diffusion equations with Neumann boundary conditions.
Such inequalities are
useful in approximating solutlons to partlal differential equations by the
method of a priori inequalltles.
1.
INTRODUCTION.
In this paper we derive explicit a priori inequalities which are useful in
yielding approximate solutions, with norm or polntwlse error bounds, of the
Neumann initial-boundary value problem associated with the diffusion operator
Lu
u
u
t
and the related third order operator
LlU
A(u +
ut)
u
t.
These
inequalities complete a series of a priori inequalitles which are applicable to
the Dirichlet and Robin boundary value problems for parabolic and pseudoparabolic
operators
[5], [6], [7], [8].
A priori inequalities for second order elliptic
operators with Dirichlet, Neumann or Robin boundary conditions have appeared in
J.R. KUTTLER & V. G. SIGILLITO
22
[], [2], [3].
A comprehensive treatment of explicit a priori inequalities and their appllcations is given in [ii] (also see
[9], [i0]).
Recently a priori inequalities
have been shown to be useful in yielding upper and lower bounds in classical
and Steklov eigenvalue problems [4].
Inequalities using more general parabolic and pseudoparabollc operators than
L or L
we have chosen these
I can be derived by the method presented here but
simpler cases to keep the derivations from becoming unnecessarily cluttered.
In the next section we introduce notation and then briefly describe the use
of the a priori inequalities to approximate solutions of boundary value problems.
The inequalities are then derived in the final section.
2.
USE OF THE
INEQUALITIES.
Our motivation for developing the a priori inequalities is the standard
Neumann problem
where
LI,
denotes either L or
G
in
R,
u
F
on
B,
u
n
H
on
S,
B is a region in n-dimenslonal Euclidean space
with boundary B, R is the time cylinder B )< (0,T], and
normal derivative on S
on S.
(2.1)
B K (0,T] where
(nl)
u_
n
u, ln i is the
is the unit outer normal vector
A comma denotes differentiation and the summation convention is used so
that repeated in dlees are to be summed over that index from i to n.
Suppose for definiteness
thatffi L,
then problem (2.1) can be solved
approximately by the method of a priori inequalities providing the inequality
v 2 dxdt <
R
i
v 2 dx
B
+ 2
S
(n)
dsdt
+ 3 (Lv)2dxdt’
R
(2.2)
EXPLICIT INEQUALITIES FOR EQUATIONS
can be obtained with explicit constants a I,
a2,
a
23
3 which depend on R but not
on the function v which is an arbitrary smooth function.
The method of a priori inequalities puts
v
N
E
i=l
u
into (2.2) for some set of test functions
(2. i).
R
u- u
ai i
{i }.
a
Here u denotes the solution of
This leads to
(U-Ua) 2dxdt
<
al
(F-Ua(X’0))2dx + a2
B
(H
S
---)2dsdt + a3
R
(G_Lua) 2 dxd t
(2.3)
in which the right-hand side is in terms of known quantities and the undetermined coefficients
ai, i=l,’’’,N.
The right-hand side is now minimized with
respect to the a i yielding an L bound on the error.
2
obtainable from this L
detail in
3.
2
Pointwise bounds are
The procedure for computing them is given in
bound.
[5], [I0].
THE INEQUALITIES.
A.
THE INEQUALITY FOR L.
<
R
Cl
u2
The desired inequality for the operator L is
dx)1/2 + c2
S
()2 dsdt)1/2 + c3
(Lu)2dxdt)1/2,
R
(3.1)
for an arbitrary function u e R, which is once continuously differentiable in
and twice continuously differentiable in x.
To obtain this inequality write u
f
+
g
- -
+
h where
Lf
0,
Lg
Lu,
Lh
0,
in
R,
f
u,
g
0,
h
0,
on
B,
_9_u
n’
on
S.
8f
0,
g
n
0
8h
2
J. E. KITTLER & V. G. SIGLLITO
Now substitute f, g, and h successively into the identity
2
Bt
dx
g
2
L
dx- 2
+2
0
II
0
where B
’t ’t
dxd’r- 2
B
0
-dsdT
)n
%
S
ls the lntersectlon of R wlth t
B
z and S
(3.2)
B X (0,T].
Putting f into (3.2) yields
f2
B
u 2 dx,
<_
B
t
T gives
0 to t
and integrating from t
dx
f2
<_ T
dxdt
R
I
u 2 dx.
(3.3)
B
Puttlng g into (3.2) yields
g2
B
g2
g Lu dxdT < a
dx <- 2
B
0
t
0
T
B
dxdT
+
(Lu) 2 dxdx,
a
0
T
B
T
by the arithmetlc-geometric mean inequality for arbitrary positive a.
lng by e
dt
Multiply-
and rearranging glves
g2 dxdT)
0
B
T
< a
(Lu) 2 dxdT.
e
0
B
T
Integration wlth respect to t from 0 to T and multlpllcatlon by e
aT then
gives
EXPLICIT INEQUALITIES FOR EQUATIONS
25
t
g2
(Lu)
e
dxdt < a
0
0
R
(Lu) 2 dxd dt
e
e
a
0
0
B
(Lu) 2 dxdz
a
B
0
(e
a
a
(Lu) 2 dxdt
i)
B
0
< a
-2
dxdx dt
B
(eaT
(Lu) 2 dxdt,
i)
R
where integration by parts was used in going from the first to the second llne
above.
8T
Setting a
g2
dxdt <
-i
gives
B-2(eB
i) T 2 |
(Lu) 2 dxdt
<
(Lu) 2 dxdt,
1.544138653 T 2 |
JR
R
#
R
(3.4)
1.59362.
with the optimal choice of 8
Finally, putting h into (3.2) yields
h 2 dx
B
2
0
t
h,i h,i
B
<- 2
B
0
h,i
dxdz
h, i dxd
T
+
2
S
0
+
a
hn dsdz
u
-i
0
S
2
dsdz
T
t
+
If
0
h 2 dsd.
$-r
(3.5)
J. R. KUTTLER & V. G. SlGILLITO
26
Now introduce a continuously differential vector field
that min
S
S
I
b > 0 (see [i] and [2] for methods of constructing such vector
flnl
fields).
the property
fl which has
Then,
h2dx
<
b-I
S
I h2finlds
I
b-i h2fi
B
+
i
b-1
B
2b-I
B
I
I (h2fl)’i
h
h,i fi
dE
dx
(3.6)
<_b -I
If i il MB
l
h 2dx+
if iil MB h2dx
< b
2b-I (Ififil
M
+ 2
B
T
h,ih,ldx
B
] h2dx
+
B
I h,lh,idx)1/2
Iflfll MB
T
h2dx.
T
where the subscript M denotes the maximum value of the function.
Using this inequality in (3.5) gives
t
h2dx
B
t
(n)
<
0
S
dsdT
+ (If i
il M
T
b
-I
+ Ififil
b
-2)
h2dxdT,(3.7)
M
B
0
T
t
where in (3.5) we have chosen
1 to cancel out the term -2
II
0
with that in (3.6).
If we now define K
Ifi,il M b
+ 1
B
h,ih,idxd
x
ififi[ M b-
2
-Kt we can
multiply (3.7) by e
write
t
-6d (e-Kt f
0
BI
t
e-Kt f "u)2[n
I--
h2dxdz)<
S
0
dsd.
z
Integrating with respect to t from 0 to T and multiplying by e
T
yields
and
27
EXPLICIT INEQUALITIES FOR EQUATIONS
T
t
K.T
h 2 dxdt < e
dsdz dt
R
0
0
K-1
S
I
T
(eK" (T-t)
8u 2
(n)
i)
S
0
dsdt
t
(3.8)
<
K-I
(e
K.T
u 2
(-gin)
)
dsdt
S
C
2
2
I (8u’2
dsdt,
S
where we have again used integration by parts in a similar manner to go from
the first to the second line above.
As an example of a choice of the vector field
fi
assume that B is star-
shaped with respect to an interior point which we take as the origin.
could choose f
xi’
i
since then
nlxi
> 0 on the boundary of B.
The quantity
is the distance from the origin to the tangent line at the point
xln i
this case then
r21 M
fifll M
Then we
for r the distance to the origin and
In
(xl).
If i
i
n.
M
More specifically suppose B is a rectangle centered at the origin with sides of
length 2a and 2b.
a
fi,i
-I
Choose
+ b-
x/a,
fl
f2
y/b.
Then
Iflfll M =2,
t
i and
Thus
K
1
+ i/a + i/b.
Combining (3.3), (3.4) and (3.8) now yields the desired inequality (3.1)
with
c
I
1/2,
T
c
2
[K-l(eKY
i)]%,
C
3
1.242633757 T.
J. R. KUTTLER & V. G. SIGILLITO
28
(I
For the operator L I the a priori inequality
THE INEQUALITY FOR L I.
B.
dxdt)1/2
u2
< c
R
l(I
(u2 +
B
u’iu’i)dx)1/2 + c2(I S
+
c3(
I
R
(LIu) 2
(u +
2
ut)
dsdt
)1/2
dxdt)
The starting point is the
can be developed in an entirely analogous manner.
identity
(2 + ,i,i)dx
2
II
0
B
I
t
B
(2 + ,i#,i)dx
#,i#,idxdT +
2
2
II
0
B
L1
dxdT
T
dsdT.
0
S
The derivation then exactly parallels that of (3.1) and we omit the details.
ACKNOWLEDGMENT.
This work was supported by the Department, Naval Sea Systems
Command under Contract No. N00017-72-4401.
REFERENCES
i.
Bramble, J. and L. E. Payne.
Bounds for Solutions of Second Order Elliptic
Partial Differential Equations, Contributions to Differential Equations
I (1963) 95-127.
2.
Bramble, J. H. and L. E. Payne. Bounds in the Neumann Problem for Second
Order Uniformly Elliptic Operators, Pacific J. Math. 12 (1962) 823-833.
3.
Bramble, J. H. and L. E. Payne. Some Integral Inequalities for Uniformly
Elliptic Operators, Contributions to Differential Equations i (1963)
129-135.
4.
Kuttler, J. R. and V. G. Slgillito. Bounding Eigenvalues of Elliptic
Operators, SlAM J. Math. Anal. (to appear).
29
EXPLICIT INEQUALITIES FOR EQUATIONS
5.
Sigillito, V. G. Pointwlse Bounds for Solutions of the First InitialBoundary Value Problem for Parabolic Equations, SlAM J. Appl. Math.
1__4
(1966) 1038-1056.
6.
Sigillito, V. G. Pointwise Bounds for Solutions of Semilinear Parabolic
Equations, SIAM Rev. 9 (1967) 581-585.
7.
Sigillito, V. G. A Priori Inequalities and Dirichlet Problem for a
Pseudoparabolic Equation, SlAM J. Math. Anal. 7 (1976) 222-229.
8.
Sigillito, V. G. and J. C. Pirkle. A Priori Inequalities and Norm Error
Bounds for Solutions of a Third Order Diffusion-Like Equation, SlAM
J. Appl. Math. 2__5 (1973) 69-71.
9.
Sigillito, V. G. On a Continuous Method of Approximating Solutions of the
Heat Equation, J. ACM. 14 (1967) 732-741.
i0.
Sigillito, V. G. A Priori Inequalities and Approximate Solutions of the
First Boundary Value Problem for A2u
f, SIAM J. Numer. Anal. 1__3
(1976) 251-260.
ii.
Sigillito, V. G. Explicit a Priori Inequalities with Applications to
Boundary Value Problems, Research Notes in Mathematics Series No. 13,
Pitman Publishing Limited, London, 1977.
KEY WORDS AND PHRASES. Parabolic and pseudopaabolic operators, a_
rnequaities, Neumann boundary value problem.
AMS(MOS) SUBJECT CLASSIFICATIONS (1970).
riori
36 K 20, 36 S 15, 35 B 45.