Mathematical and Numerical Analysis of Some FSI Problems - Fluid-Structure Interaction and Biomedical Applications - page 30

where the extension v 7! v is defined by ( 1.54 )andwhere R Ǜ is a lifting operator

defined by

LJ LJ LJ LJ

.0;0;b/ T for z Ǜ;
.0;0; Ǜ b/ T

R Ǜ .b/ D

C w Ǜ in C Ǜ ; for a.e. t;

(1.62)

with w Ǜ such that div . w Ǜ / D b and w Ǜ 2 H 0 .C Ǜ /, k w Ǜ k H 0 .C Ǜ / Ckbk L 2 .0;L/ ,for
a.e. t. Note that w Ǜ exists because b has a zero mean. The first term of has a trace
equal to zero on the interface, the second term matches the structure test function
at the interface. Note moreover that a space regularization of v D u R Ǜ .@ t /,
denoted by v has been introduced in order to have bounded in H 1 .0;T I H 1 .B//
independently of LJ 2 . It verifies div .v / D 0, v 2 L 2 .0;T I H 0 . .t/// and

kv v

k L 2 .0;T I L 2 . .t/// ! 0; uniformly in LJ 2 ; as goes to zero;

(1.63)

kv

k L 2 .0;T I H 1 . .t/// C :

The construction of v relies on the fact that the elastic interface does not touch
the bottom of the fluid cavity. Moreover, the uniform c o nvergence of .v / as
! 0 in L 2 .0;T I L 2 . .t/// is made possible since v is uniformly bounded
in L 2 .0;T I H s .B//;0 < s < 1=4. With this choice,

k@ t N 0 k@ t N 0

k L 1 .0;T I L 2 .0;L// C; k L 2 .0;T I H s .0;L// C;

kbk W 1; 1 .0;T I L 2 .0;L// C; kbk H 1 .0;T I H s .0;L// C;

k L 2 .0;T I H s ..B// C;8s 0 <s<1=4;

kv k L 1 .0;T I L 2 .B// C;kv

and

k@ t N 0 k L 1 .0;T I H 1 .0;L// C N 0 ; kbk W 1; 1 .0;T I H 1 .0;L// C N 0 ; kv k L 2 .0;T I H 1 .B// C ;

where C denotes a strictly positive constant that depends only on the data and not
on LJ 2 and N 0 ,andC N 0 (resp. C ) denotes and will denote a strictly positive constant
that depends on the data and not on LJ 2 but may depend on N 0 (resp. ). The integer
N 0 (resp. the real ) is chosen large enough (resp. small enough). Then for well-
chosen , .;b/are admissible test functions. Indeed, is divergence free t h anks
to the definitions of the lifting operator R Ǜ , the extension operator v 7! v,the

operator v 7! v and the definition of the regularization v 7! v . Moreover
belongs to H 1 .0;T I H 1 .B//. The function b belongs to H 1 .0;T I H 0 .0;L//.Both
of them are bounded in the previous spaces independently of LJ 2 but not of N 0 and .