Topics in the Mathematical Theory of Interactions of Incompressible Viscous Fluid with Rigid Bodies - Fluid-Structure Interaction and Biomedical Applications - page 302

4.4.3 Contact vs No-Contact

We discuss now on the possibility of contacts in weak and classical solutions to
(FRBI). This question is tackled by Vázquez and Zuazua [ 61 ] on the following one-
dimensional baby-model introduced in [ 60 ]:
8
<

@ t u C u @ x u @ xx u D 0; in R nfh i .t/g i D 1;:::;n
u .t;h i .t// D h i .t/; for i D 1;:::;n;
Œ@ x u .t;h i .t// D m i h i .t/; for i D 1;:::;n:

(4.77)

:

In this system the h i 's stand for the position of the bodies (which are points in
the one-dimensional case). For simplicity, we assume that they are numbered in
increasing order: h i <h i C 1 . The two last equations mimic the no-slip boundary
conditions and body dynamics, respectively. We used the convention that:

Πu .t;h/ D

x ! h;x>h u .t;x/

lim

x ! h;x<h u .t;x/;

lim 8 t>0; 8 h 2 R :

In the first equation, of viscous Burgers type, we introduced the parameter >0
which stands for the viscosity of the fluid, and 2 R a dimensionless parameter.
For this system, J.L. Vázquez and E. Zuazua prove among other results that,
whatever the choice of the initial data:

2 L 2 . R /; .h i ; h i / 2 R 2n ; s.t. u 0

i 2f 1;:::;n 1 g jh i C 1 h i j >0;

inf

the system ( 4.77 ) admits a unique global solution in which no contact between rigid
bodies occurs in finite time:

i ¤ j jh i C 1 .t/ h i .t/j >0;

inf 8 t>0:

In this one-dimensional case, the regularity estimate for classical solutions yields
the bound:
Z T

" n 1

2 # dt<C 0 .T/;

Z h i C 1 .t/

X X

n

m i j h i j

2 d z C

j@ xx u .t; z /j (4.78)

0 h i .t/

i D 1 i D 1

where C 0 depends on initial data and T only. Applying classical arguments to
the no-slip boundary condition seen as a differential equation, this control on
@ xx u induces that h i C 1 might not collide h i in finite time. The result extends
to the case of L 2 initial data due to the classical smoothening properties of the
viscous Burgers equations. In the multi-dimensional case, the regularity estimate
for solutions to (FRBI) relies on ellipticity of the Stokes problem which depends
itself on the minimal distance between two rigid boundaries (see [ 15 , Sect. 4.1]).