Topics in the Mathematical Theory of Interactions of Incompressible Viscous Fluid with Rigid Bodies - Fluid-Structure Interaction and Biomedical Applications - page 259

4.2.3 Outline of the Paper

The Cauchy problem for (FRBI) has been the subject of many studies in the last
15 years. Several existence results have been obtained introducing little by little
complexity in the geometrical configurations under consideration. As classical when
dealing with Navier Stokes equations, two families of solutions are constructed:
classical solutions for which ( 4.9 ) is satisfied almost everywhere, weak solutions
for which a specific definition of solving ( 4.9 )-( 4.13 ) is introduced. In Sect. 4.3 ,we
first describe the construction of weak solutions up to contact between bodies in the
case of several bodies moving inside a bounded container. The second part of this
section is devoted to the construction of classical solutions in the case of one rigid
body inside a bounded container.
All the results of Sect. 4.3 concern solutions prior to contact between bodies
or bodies and the container boundary. We analyze how contacts are handled by
solutions to (FRBI) in the last section. Several questions are discussed. First, in
the frame of classical solutions, we show that contact implies blow-up and discuss
which norms of the solution blows up in case of contact. We envisage then the
extension of classical solutions by weak solutions after contact. We show that such
an extension is possible with the method described in Sect. 4.3 and reduces to
completing (FRBI) with a sticky contact law. Finally, we discuss the possibility
of contact occurrence in finite time for weak and strong solutions to (FRBI).

4.3 Existence and Uniqueness for the Initial Boundary-Value

Problems

The first modern contributions to the study of the Cauchy problem associated with
(FRBI) are the references [ 47 , 52 ] which tackle the free-fall of one rigid body in
an unlimited container ( fills the whole three-dimensional space). Following these

seminal works, the (two-dimensional and three-dimensional) case of one rigid body
moving inside a bounded container is considered in [ 5 , 33 - 35 , 43 , 44 ]: in [ 33 , 34 ],
existence of classical solutions is obtained (under restrictive assumptions on the
body densities), while the same problem is solved in a weak setting in [ 5 , 35 , 43 , 44 ]
(see also [ 4 ]). We point out that existence of solutions is obtained up to contact
between the body and the container boundary in [ 5 , 34 , 35 ], while [ 43 , 44 ] are the first
occurrences of existence results without this restriction. The case of several rigid
bodies moving inside a bounded container is then tackled in [ 15 , 16 , 20 , 51 , 56 , 57 ].
First, existence up to contact is proven in [ 15 , 16 ], for weak solutions, and in [ 56 , 57 ],
for classical solutions. In these latter references, the method of [ 33 , 34 ] is improved
yielding existence of classical solutions without restriction on the magnitude of the
body densities. Then, global existence of weak solutions regardless contact is also
obtained for the “several rigid body” case [ 20 , 51 ]. Uniqueness of weak solutions in
the two-dimensional case is tackled in [ 30 ]. Finally, the configuration of one body

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