Environmental Engineering Reference
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and hence,
∂
∂
t
ˆ
≤
f
.
Finally, the integration over time interval
(
0
,
T
)
leads to
ˆ
.
0
ˆ
≤
T
max
0
T
f
(
r
,
t
)
+
(
r
)
(2.16)
≤
t
≤
Since the dispersion model (
2.4
)-(
2.11
) is linear with respect to
, estimation (
2.16
)
assures that the solution of problem (
2.4
)-(
2.11
) is unique and continuously depends
on the initial conditions and forcing. Also, using the method described by Skiba and
Parra-Guevara [
43
], it is possible to prove the existence of generalized solution of
problem (
2.4
)-(
2.11
), that is the model (
2.4
)-(
2.11
) is well posed in the sense of
Hadamard [
13
]. Also note that the positive semidefiniteness of operator
A
allows
us to split the operator
A
in coordinate directions, and with the help of numerical
schemes by Marchuk [
22
] and Crank-Nicolson [
8
] construct unconditionally stable
and efficient numerical algorithm of second approximation order in space and time
for the solution of problem (
2.4
)-(
2.11
)[
41
].
ˆ
2.3 Adjoint Functions and the Duality Principle
It is rather difficult to analyse and solve the variational problem (
2.1
)-(
2.3
) because
the constraints in (
2.2
) are related with the solutions
Q
i
of the control problem
implicitly through the solution
of the dispersion model (
2.4
)-(
2.11
). In order to
establish an explicit dependence of the constraints on the control functions
Q
i
,we
now introduce one more model which is adjoint to the dispersion model (
2.4
)-(
2.11
).
It means that the operator
A
∗
is adjoint to the operator
A
of the model (
2.4
)-(
2.11
)
in the sense of the Lagrange identity
ˆ
A
∗
g
(
A
ˆ,
g
)
=
(ˆ,
),
where
[
22
]. Solutions of
this adjoint model will be used to establish a duality principle for the mean con-
centration of the released nutrient in the marine environment. Let us construct the
operator
A
∗
. The inner product
(
·
,
·
)
denotes the inner product in the Hilbert space
L
2
(
D
)
(
ˆ,
)
A
g
is
(
A
ˆ,
g
)
=
g
U
·∇
ˆ
dr
+
˃
g
ˆ
dr
−
g
∇·
μ
∇
ˆ
dr
+
g
∇·
ˆ
s
dr
.
D
D
D
D
The integrals in the last expression can be rewritten with the divergence theorem
as follows
g
U
·∇
ˆ
dr
=
g
ˆ
U
·
n
dS
−
ˆ
U
·∇
gdr
,
D
∂
D
D
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