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.
REFERENCES
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
).© 2012 American Institute of Physics.
2012
American Institute of Physics
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