A Strategy for Bioremediation of Marine Shorelines by Using Several Nutrient Release Points - Mathematical Modelling and Numerical Simulation of Oil Pollution Problems - page 30

ˆμ ∂ˆ
∂

2

ˆ ∇· (μ ∇ ˆ) dr = dS − μ |∇ ˆ | dr ,

n

D ∂ D D

1
2

ˆ ∇· ˆ s dr = ˆˆ s · n dS .

D ∂ D

S + , S − Finally, dividing each integral over ∂ D into the four integrals over S T ,

and S B , and applying the conditions ( 2.6 )-( 2.9 ) and ( 2.12 ), we get

2 dr 2 2 k

( ˆ,ˆ) = ˃ˆ + μ |∇ ˆ | +

S T ʶˆ

·

A dr n dS

D D

n dS

1
2

2 dS 2 dS 2 k 2 k

+

U n ˆ

−

U n ˆ

+

v s ˆ

· −

v s ˆ

·

n dS

S + S −

S T S B

(2.15)

0in S − , k Since U n < · n > 0at S T and k · n < 0at S B ,Eq.( 2.15 ) can be

rewritten as

2 dr 2 dr 2 k

μ | ∇ ˆ |

( A ˆ,ˆ) = ˃ˆ + +

S T ʶˆ

· n dS

D D

⊧
⊨

⊫
⊬

1
2

2 dS 2

|

U n | ˆ

| |

+ + v s ˆ k · n dS .

⊩ ⊭

S + ∪ S −

S T ∪ S B

Thus, operator A is positive semidefinite: 0.

Taking the inner product of every term of Eq. ( 2.4 ) with

( A ˆ,ˆ) ≥

ˆ we obtain

∂ˆ
∂

N

t ,ˆ

= ( f ,ˆ) − ( A ˆ,ˆ), f ( r , t ) = Q i ( t )ʴ( r − r i ).

i = 1

Using the condition ( A ˆ,ˆ) ≥ 0 and the Schwarz inequality [ 17 ], the last equation

implies the inequality

ˆ, ∂ˆ
∂

≤ ˆ f , ˆ = (ˆ, ˆ).

t

Further,

ˆ, ∂ˆ
∂

1

2 ∂

∂
∂

2

=

t ˆ

= ˆ

t ˆ t ∂