Topics in the Mathematical Theory of Interactions of Incompressible Viscous Fluid with Rigid Bodies - Fluid-Structure Interaction and Biomedical Applications - page 265

0

are smooth, there are other alternatives. In [ 26 ], the authors require B i .t/

to be connected. Applying Proposition 4.1 , this implies that the rigid velocity on
B i .t/ is uniquely defined, so that

B

i and

B i moves as one block and cannot split. In [ 51 ],

the condition that B i .t/ remains connected is obtained by construction. The authors

apply that, for any radial function W R d ! Œ0; 1/ of unitary mass there holds

D ,forall 2 R (where stands for the classical convolution operator).

Consequently, the authors choose to replace the unknown ' i with the indicator
function i of the ı-interior of B i .t/:

B i .t/Œ ı WD fx 2 B i .t/ s.t. B.x;ı/ B i .t/g: (4.35)

This function is a weak solution to

) (4.36)

@ t i Cr. i Œ u ı / D 0; on .0;T/ ;

i .0; / D 1 B

i Œ ı ; on :

0

where Œ u ı D u ı for a radial mollifier ı such that Supp. ı / B.0;ı/. Then,
' i is computed as the indicator of the ı-exterior of Supp. i /:

[

ŒSupp. i / ı WD B.x;ı/: (4.37)

x 2 Supp. i /

The parameter ı>0is fixed sufficiently small in order that the operation ı-exterior
is the converse operation of ı-interior on

i :

B

0
i ı Œ ı D Œ

0
i Œ ı ı D B

0
i ;

ΠB B 8 i 2f1;:::;ng: (4.38)

i has a sufficiently smooth boundary. When u is a rigid velocity-
field u i on Supp.' i /, there holds Œ u ı D

Such a ı exists if B

u i D u on Supp. i /. Moreover as Œ u ı 2

L 1 .0;T I C loc . R d //,theflowt 7! M

t i associated with Œ u ı is well defined and

i / is an open connected subset of ,forall
i 2f1;:::;ng and the difficulty is overcome.
Finally, once it is obtained that the body domain

t i .

lipschitzian. In particular B i .t/ D M B

.t/ is made of n rigid bodies,
it is classical to show that if u is smooth in the remaining fluid domain and
satisfies ( 4.32 ), then there exists a pressure p so that ( 4.9 ) holds true in a classical
sense. As long as no contact occurs and the solid boundaries are smooth, it is
also possible to lift any set of rigid velocities on the

S

B i .t/'s into a test-function

w Q S ; in order to prove that ( 4.12 )and( 4.13 ) are also satisfied with i

and ! i computed w.r.t. u j B i .t/ . If the body boundaries and container boundary are
smooth, we might also apply trace arguments to recover ( 4.10 )-( 4.11 ).
We proceed with detailing the construction of weak solutions. As classical, the
main points in such constructions are

2 K Œ

(i) Definition of approximate problems;
(ii) Existence of solutions to the approximate problems;