Fluid–Structure Interaction in Hemodynamics: Modeling, Analysis, and Numerical Simulation - Fluid-Structure Interaction and Biomedical Applications - page 150

Fig. 2.11 Left panel —Case 1: t h. The graph of v N is shown in solid line , while the
graph of the shifted function h v N is shown in dashed line .The shaded area denotes the nonzero
contributions to the norm k h v N v N k

2
L 2 . Right panel —Case2: t < h D t C s;0 < s < t.
The graph of v N is shown in solid line , while the graph of the shifted function h v N is shown in
the dashed line .The shaded areas denote nonzero contributions to the norm k h v N v N k

2
L 2 .The

two colors represent the contributions to the first and second integral in ( 2.160 ) separately

The last inequality follows from jhj <ı "=.2C/.
Case 2: t < h . In this case we can write h D lt C s for some ł 2 N

,

0<s t. Similarly, as in the first case, we get (see Fig. 2.11 , right):

Z .j C 1/t s

N l 1

X

kv N v j C l

2 2

L 2 .0;1/

k h v N v N k

L 2 .! I L 2 .0;1// D

k

N

jt

j D 1

! : (2.160)

Z .j C 1/t
.j C 1/t s kv N v j C l C 1

2
L 2 .0;1/

C k

N

Now we use the triangle inequality to bound each term under the two integrals
from above by P l C 1

i D 1 kv j C i 1

v j C i
N

2

k

L 2 .0;1/ : After combining the two terms

N

together one obtains

N l 1 l C 1

X X

i D 1 kv j C i 1

v j C i
N

2 2

k h v N v N k

L 2 .! I L 2 .0;1// t

k

L 2 .0;1/ : (2.161) N

j D 1

Lemma 2.1 now implies that the right-hand side of ( 2.161 ) is bounded by t.l C
1/C.Now,sinceh D lt C s we see that t h=l, and so the right-hand side
of ( 2.161 ) is bounded by l C l hC.Sincejhj <ıand from the form of ı we get

l C 1
l

l C 1
l

"
2 <":

2
L 2 .! I L 2 .0;1// t.l C 1/C

k h v N v N k hC

Thus, if we set ! D Œı=2;T ı=2 we have shown:

2
L 2 .ı=2;T ı=2 I L 2 .0;1// <"; N2 N

k ı=2 v N v N k :