Topics in the Mathematical Theory of Interactions of Incompressible Viscous Fluid with Rigid Bodies - Fluid-Structure Interaction and Biomedical Applications - page 301

for explicit formulas). Let then introduce, for all b 2 Œ0;R r/ W

8
<

r ? .x 2 .jG b .x/j//;

8 x 2 B.0;R/ n B.be 1 ;r/;

b .x/ D

:

e 1 ; 8 x 2 B.be 1 ;r/:

The vector-field satisfies the following straightforward properties:

b 2 V.B.0;R//\ C 1 .B.0;R/ n B.be 1 ;r//;

8 b 2 .0;R r/:

On the basis of explicit formulas for G, it satisfies also the supplementary ones:

Proposition 4.8. There exists an absolute constant C<1 , such that, for all b 2
Œ0;R r/ :

k b I L 2 .B.0;R//kC

C
.R r b/ 4

kr b I L 2 .B.0;R//k

Then, one looks for a solution of the form:

b.t/ b.t/ .x/;

u .t;x/ D 8 t 2 .0;2T/; 8 x 2 B.0;R/:

Given T>0,andb given by:

1 4

t
T

.t/ WD ; t/D .t/.R r/; 8 t 2 .0;2T/;

it yields that, . 1 B.b.t/e 1 ;r/ ; u / is a weak solution to (FRBI) with a source term

g 2 L 2 .0;T I ŒV.B.0;R// /:

We would also be able to construct a weak solution . '; u / to (FRBI) with the initial
condition . 1 B.0;r/ ; u .0/// and the same source term g 2 L 2 .0;T I ŒV.B.0;R// /
applying the method of the previous subsection. However both weak solutions
might not coincide. Indeed, in our construction the body

B 1 remains stuck to
the container boundary after contact whereas it splits from the boundary in the
case of the solution constructed by V. Starovoitov. Hence, in the two-dimensional
case, uniqueness of the weak solution including contact does not hold for arbitrary
initial condition without a supplementary rebound law. We point out that, in this
construction, existence of a contact is enforced by the introduction of a singular
source term g.