Topics in the Mathematical Theory of Interactions of Incompressible Viscous Fluid with Rigid Bodies - Fluid-Structure Interaction and Biomedical Applications - page 292

where 2 D 2 and 3 D . On the other hand, we rewrite:
Z Z 2 Z

nl d 2 d 0 dx d : nd D

@ l n .@B 1 [ @B 2 / 1 j .x 1 ;:::;x d 1 / jD l

Here “d 0 D ı x 1 D l C ı x 1 D l ” in the two-dimensional case and “d 0 =d ”
corresponding to cylindrical coordinates .r;; z / associated with .x 1 ;x 2 ;x 3 / in the
three-dimensional case. We note that:
Z

nd 0 D 0:

j .x 1 ;:::;x d 1 / jD l;x d D 1 .l/

Consequently, we bound from above the lateral flux by combining a Hölder
inequality with a Poincaré inequality:

LJ LJ LJ LJ

nd LJ LJ LJ LJ

Z

@ l n .@B 1 [ @B 2 /

. 2 .l/ 1 .l// 2 p l .d 2/.1 p / Z

p d

1
p

j .x 1 ;:::;x d 1 / jD l jrj

: (4.73)

Finally, introducing ( 4.72 )and( 4.73 )into( 4.71 ) yields:

j d l d 1 Œ. 2 1 / e d j

p / Z

p d

1
p

1
p l .d 2/.1

1

. 2 .l/ 1 .l// 2

j .x 1 ;:::;x d 1 / jD l jrj

:

We can now integrate this identity between l D 0 and l D r 2 .0;ı/.As 2 1 is
non-decreasing, this entails that:

1
p r .d 1/.1

1
p / d

jŒ. 2 1 / e d jC. 2 .r/ 1 .r// 2 krI L p . R d /k;

w ith a constant C depending on d.Forh sufficiently small, we might choose r D

p h to obtain:

1
2 .3 .d C 1/ p /

jŒ. 2 1 / e d j Ch krI L p . R d /k;

C depends now on d and ı.

where t

We conclude the proof of Theorem 4.3 , item (i), as follows. Assume that the
distance dŒf B i .t/g i D 0;:::;n goes to 0 for a sequence of time t ! T . Then, by
a compactness argument, there exist X 1 and X 2 such that B.X 1 ;ı/ B

i and