We address a fluid-structure interaction model describing the motion of an elastic body immersed in an incompressible fluid. We establish a prioriestimates for the local existence of solutions for a class of initial data which also guarantees uniqueness.

1.
G.
Avalos
,
I.
Lasiecka
, and
R.
Triggiani
, “
Higher regularity of a coupled parabolic-hyperbolic fluid-structure interaction system
,”
Georgian Math. J.
2
(
3
),
403
437
(
2008
).
2.
G.
Avalos
and
R.
Triggiani
, “
The coupled PDE system arising in fluid/structure interaction. I. Explicit semigroup generator and its spectral properties
,” in
Fluids and Waves
,
Contemporary Mathematics
Vol.
440
(
American Mathematical Society
,
Providence, RI
,
2007
), pp.
15
54
.
3.
M.
Boulakia
, “
Existence of weak solutions for the three-dimensional motion of an elastic structure in an incompressible fluid
,”
J. Math. Fluid Mech.
9
(
2
),
262
294
(
2007
).
4.
M.
Boulakia
and
S.
Guerrero
, “
Regular solutions of a problem coupling a compressible fluid and an elastic structure
,”
J. Math. Pures Appl.
94
(
4
),
341
365
(
2010
).
5.
M.
Boulakia
and
S.
Guerrero
, “
A regularity result for a solid-fluid system associated to the compressible Navier-Stokes equations
,”
Ann. Inst. Henri Poincare, Anal. Non Lineaire
26
(
3
),
777
813
(
2009
).
6.
V.
Barbu
,
Z.
Grujić
,
I.
Lasiecka
, and
A.
Tuffaha
, “
Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model
,” in
Fluids and Waves
,
Contemporary Mathematics
Vol.
440
(
American Mathematical Society
,
Providence, RI
,
2007
), pp.
55
82
.
7.
V.
Barbu
,
Z.
Grujić
,
I.
Lasiecka
, and
A.
Tuffaha
, “
Smoothness of weak solutions to a nonlinear fluid-structure interaction model
,”
Indiana Univ. Math. J.
57
(
3
),
1173
1207
(
2008
).
8.
F.
Bucci
and
I.
Lasiecka
, “
Optimal boundary control with critical penalization for a PDE model of fluid-solid interactions
,”
Calculus Var. Partial Differ. Equ.
37
(
1–2
),
217
235
(
2010
).
9.
D.
Coutand
and
S.
Shkoller
, “
Motion of an elastic solid inside an incompressible viscous fluid
,”
Arch. Ration. Mech. Anal.
176
(
1
),
25
102
(
2005
).
10.
D.
Coutand
and
S.
Shkoller
, “
The interaction between quasilinear elastodynamics and the Navier-Stokes equations
,”
Arch. Ration. Mech. Anal.
179
(
3
),
303
352
(
2006
).
11.
Q.
Du
,
M. D.
Gunzburger
,
L. S.
Hou
, and
J.
Lee
, “
Analysis of a linear fluid-structure interaction problem
,”
Discrete Contin. Dyn. Syst.
9
(
3
),
633
650
(
2003
).
12.
G.
Grubb
and
V.
Solonnikov
, “
Boundary value problems for the nonstationary Navier-Stokes equations treated by pseudo-differential methods
,”
Math. Scand.
69
,
217
290
(
1991
).
13.
G.
Guidoboni
,
R.
Glowinski
,
N.
Cavallini
, and
S.
Canic
, “
Stable loosely-coupled-type algorithm for fluid-structure interaction in blood flow
,”
J. Comput. Phys.
228
(
18
),
6916
6937
(
2009
).
14.
G.
Guidoboni
,
R.
Glowinski
,
N.
Cavallini
,
S.
Canic
, and
S.
Lapin
, “
A kinematically coupled time-splitting scheme for fluid-structure interaction in blood flow
,”
Appl. Math. Lett.
22
(
5
),
684
688
(
2009
).
15.
T. J. R.
Hughes
and
J. E.
Marsden
, “
Classical elastodynamics as a linear symmetric hyperbolic system
,”
J. Elast.
8
(
1
),
97
110
(
1978
).
16.
I.
Kukavica
and
A.
Tuffaha
, “
Solutions to a fluid-structure interaction free boundary problem
,”
Discrete Contin. Dyn. Syst.
32
(
4
),
1355
1389
(
2012
).
17.
I.
Kukavica
and
A.
Tuffaha
, “
Solutions to a free boundary problem of fluid-structure interaction
,” (submitted).
18.
I.
Kukavica
and
A.
Tuffaha
, “
Well-posedness for the compressible Navier-Stokes-Lamé system with a free interface
,”
Nonlinearity
25
(
11
),
3111
3137
(
2012
).
19.
I.
Kukavica
,
A.
Tuffaha
, and
M.
Ziane
, “
Strong solutions to a nonlinear fluid structure interaction system
,”
J. Differ. Equations
247
(
5
),
1452
1478
(
2009
).
20.
I.
Kukavica
,
A.
Tuffaha
, and
M.
Ziane
, “
Strong solutions to a Navier-Stokes-Lamé system on a domain with a non-flat boundary
,”
Nonlinearity
24
(
1
),
159
176
(
2011
).
21.
I.
Kukavica
,
A.
Tuffaha
, and
M.
Ziane
, “
Strong solutions for a fluid structure interaction system
,”
Adv. Differ. Equ.
15
(
3–4
),
231
254
(
2011
).
22.
J.-L.
Lions
,
Quelques méthodes de résolution des problèmes aux limites non linéaires
(
Dunod
,
1969
).
23.
J.-L.
Lions
, “
Hidden regularity in some nonlinear hyperbolic equations
,”
Mat. Apl. Comput.
6
(
1
),
7
15
(
1987
).
24.
J.-L.
Lions
and
E.
Magenes
,
Non-Homogeneous Boundary Value Problems and Applications
(
Springer-Verlag
,
New York
,
1972
), Vol.
II
, Translated from French by P. Kenneth, Die Grundlehren der mathematischen Wissenschaften, Band 182.
25.
I.
Lasiecka
,
J.-L.
Lions
, and
R.
Triggiani
, “
Nonhomogeneous boundary value problems for second order hyperbolic operators
,”
J. Math. Pures Appl.
65
(
2
),
149
192
(
1986
).
26.
I.
Lasiecka
and
D.
Toundykov
, “
Semigroup generation and “hidden” trace regularity of a dynamic plate with non-monotone boundary feedbacks
,”
Commun. Math. Anal.
8
(
1
),
109
144
(
2010
).
27.
I.
Lasiecka
and
R.
Triggiani
, “
Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions
,”
Appl. Math. Optim.
25
(
2
),
189
224
(
1992
).
28.
I.
Lasiecka
and
R.
Triggiani
, “
Sharp regularity theory for elastic and thermoelastic Kirchoff equations with free boundary conditions
,”
Rocky Mt J. Math.
30
(
3
),
981
1024
(
2000
).
29.
I.
Lasiecka
and
A.
Tuffaha
, “
Riccati theory and singular estimates for a Bolza control problem arising in linearized fluid-structure interaction
,”
Syst. Control Lett.
58
(
7
),
499
509
(
2009
).
30.
J. E.
Marsden
and
T. J. R.
Hughes
,
Mathematical Foundations of Elasticity
(
Dover
,
New York
,
1994
).
31.
J.
Prüss
and
G.
Simonett
, “
On the two-phase Navier-Stokes equations with surface tension
,”
Interfaces Free Boundaries
12
,
311
345
(
2010
).
32.
R.
Temam
,
Infinite-Dimensional Dynamical Systems in Mechanics and Physics
, 2nd ed.,
Applied Mathematical Sciences
Vol.
68
(
Springer
,
New York
,
1997
).
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