Topics in the Mathematical Theory of Interactions of Incompressible Viscous Fluid with Rigid Bodies - Fluid-Structure Interaction and Biomedical Applications - page 266

(iii) Analysis of cluster points of solutions to approximate problems.

In the end of this section, we sketch methods to tackle the points (i) and (iii).

Construction of Weak Solutions I: Designing Approximate Problems

We recall that we do not consider contact for now and that we do not make any
assumption on the regularity of body boundaries (having in mind the construction
of global-in-time weak solutions without addressing the contact issue). In order to
introduce an approximate problem that one can tackle either by a Fadeo-Galerkin
method or by a semi-group approach, it is necessary to handle with care the
following difficulties in Definition 4.2 :

• the space of test-functions depends on the solution itself through the constraint

that test-functions must be equal to a rigid velocity-field on body domains;

• the weak formulation of Navier Stokes equations contains a nonlinear term:

Z T Z

u ǝ u W D. w / I

0

• the shapes of the rigid bodies do not have smooth boundaries a priori.

The most intriguing part of the construction is to deal with the first nonlinearity.
Classical methods enable to handle the nonlinear convective term. As for the regu-
larity of the bodies and container boundaries, we argue by compactness introducing
smoothened rigid bodies and container. We shall detail the regularization process in
the compactness argument. Hence, we assume for now that the body shapes have
smooth boundaries.
Concerning the convective term, a classical method is to linearize by replacing
one u either with a regularized velocity-field (see [ 35 , 39 ]) or with a previous guess
of solution (in a fixed-point approach). We remark that this should be done with
care. For instance, let us denote by this other velocity-field and assume that the
linearized problem is given by the following system:
8
<

F

@ t u f C r u f Dr T . u f ; p f / in .t/;

F

r u f D 0; .t/;

u f D Q i C! i .x G i /; on @ B i .t/;
u f D 0;

in

:

on @;

B i .t/ (and thus also the fluid domain F

where the body domains .t/) are computed

with respect to the body motion prescribed by u f . The unknowns in this new system
are .. B i .t/; Q i ; ! i / i D 1;:::;n ; u f ; p f /. Multiplying the linearized momentum equation
by u f yields the following a priori estimate: