Backwards uniqueness of the semigroup associated with a parabolichyperbolic StokesLamé partial differential equation system
Authors:
George Avalos and Roberto Triggiani
Journal:
Trans. Amer. Math. Soc. 362 (2010), 35353561
MSC (2010):
Primary 35B99, 35M30
Published electronically:
February 19, 2010
MathSciNet review:
2601599
Fulltext PDF
Abstract 
References 
Similar Articles 
Additional Information
Abstract: In this paper the ``backwarduniqueness property'' is ascertained for a two or threedimensional, fluidstructure interactive partial differential equation (PDE) system, for which an explicit semigroup formulation was recently given by Avalos and Triggiani (2007) on the natural finite energy space . (See also their Contemporary Mathematics article of 2007 for a preliminary, simplified, canonical model.) This system of coupled PDEs comprises the parabolic Stokes equations and the hyperbolic Lamé system of dynamic elasticity. Each dynamic evolves within its respective domain, while being coupled on the boundary interface between fluid and structure. In terms of said fluidstructure semigroup , posed on the associated finite energy space , the backwarduniqueness property can be stated in this way: If for given initial data , for some , then necessarily . The proof of this property hinges on establishing necessary PDE estimates for a certain static fluidstructure equation in order to invoke the abstract backwarduniqueness resolventbased criterion by Lasiecka, Renardy, and Triggiani (2001). The backwarduniqueness property for the coupled StokesLamé PDE is motivated by, and has positive implications to, the problem of exact controllability (in the hyperbolic state variables ) and, simultaneously, approximate controllability (in the parabolic state variable ) of the present coupled PDE model, under boundary control. A similar situation occurred for thermoelastic models as shown in papers by M. Eller, V. Isakov, I. Lasiecka, M. Renardy, and R. Triggiani.
 [A.1]
George
Avalos, The strong stability and instability of a fluidstructure
semigroup, Appl. Math. Optim. 55 (2007), no. 2,
163–184. MR 2305089
(2008b:93090), http://dx.doi.org/10.1007/s002450060884z
 [AT.1]
G.
Avalos and R.
Triggiani, The coupled PDE system arising in fluid/structure
interaction. I. Explicit semigroup generator and its spectral
properties, Fluids and waves, Contemp. Math., vol. 440, Amer.
Math. Soc., Providence, RI, 2007, pp. 15–54. MR 2359448
(2008k:35465), http://dx.doi.org/10.1090/conm/440/08475
 [AT.2]
George
Avalos and Roberto
Triggiani, Uniform stabilization of a coupled PDE system arising in
fluidstructure interaction with boundary dissipation at the
interface, Discrete Contin. Dyn. Syst. 22 (2008),
no. 4, 817–833. MR 2434971
(2009i:74025), http://dx.doi.org/10.3934/dcds.2008.22.817
 [AT.3]
G. Avalos and R. Triggiani, Wellposedness and stability analysis of a coupled StokesLamé PDE system, 2007.
 [AT.4]
George
Avalos and Roberto
Triggiani, Backward uniqueness of the s.c.\ semigroup arising in
parabolichyperbolic fluidstructure interaction, J. Differential
Equations 245 (2008), no. 3, 737–761. MR 2422526
(2010d:35270), http://dx.doi.org/10.1016/j.jde.2007.10.036
 [AT.5]
G. Avalos and R. Triggiani, Uniform stabilization of the coupled StokesLamé PDE system with boundary dissipation at the interface, 2007.
 [BS.1]
Susanne
C. Brenner and L.
Ridgway Scott, The mathematical theory of finite element
methods, Texts in Applied Mathematics, vol. 15, SpringerVerlag,
New York, 1994. MR 1278258
(95f:65001)
 [BGLT.1]
Viorel
Barbu, Zoran
Grujić, Irena
Lasiecka, and Amjad
Tuffaha, Existence of the energylevel weak solutions for a
nonlinear fluidstructure interaction model, Fluids and waves,
Contemp. Math., vol. 440, Amer. Math. Soc., Providence, RI, 2007,
pp. 55–82. MR 2359449
(2008m:35260), http://dx.doi.org/10.1090/conm/440/08476
 [CT.1]
S.
K. Chang and Roberto
Triggiani, Spectral analysis of thermoelastic plates with
rotational forces, Optimal control (Gainesville, FL, 1997) Appl.
Optim., vol. 15, Kluwer Acad. Publ., Dordrecht, 1998,
pp. 84–115. MR 1635994
(99g:35126), http://dx.doi.org/10.1007/9781475760958_5
 [CR.1]
H. Cohen and S. I. Rubinow, Some mathematical topics in biology, Proc. Symp. on System Theory, Polytechnic Press, New York (1965), 321337.
 [DGHL.1]
Q.
Du, M.
D. Gunzburger, L.
S. Hou, and J.
Lee, Analysis of a linear fluidstructure interaction problem,
Discrete Contin. Dyn. Syst. 9 (2003), no. 3,
633–650. MR 1974530
(2004c:74017), http://dx.doi.org/10.3934/dcds.2003.9.633
 [ELT.1]
M. Eller, I. Lasiecka, and R. Triggiani, Simultaneous exact/approximate boundary controllability of thermoelastic plates with variable coefficient, Marcel Dekker Lecture Notes Pure and Applied Mathematics 216 (February 2001), 109230, invited paper for the special volume entitled Shape Optimization and Optimal Designs, J. Cagnol and J. P. Zolesio (Editors). [The preliminary version is in invited paper in Semigroup of Operators and Applications, Birkhäuser (2000), 335351, A. V. Balakrishnan (editor).]
 [ELT.2]
M.
Eller, I.
Lasiecka, and R.
Triggiani, Unique continuation for overdetermined Kirchoff plate
equations and related thermoelastic systems, J. Inverse IllPosed
Probl. 9 (2001), no. 2, 103–148. Distributed
systems: optimization and economicenvironmental applications
(Ekaterinburg, 2000). MR 1843430
(2002m:74036), http://dx.doi.org/10.1515/jiip.2001.9.2.103
 [ELT.3]
M.
Eller, I.
Lasiecka, and R.
Triggiani, Simultaneous exact/approximate boundary controllability
of thermoelastic plates with variable thermal coefficient and moment
control, J. Math. Anal. Appl. 251 (2000), no. 2,
452–478. MR 1794432
(2001j:93008), http://dx.doi.org/10.1006/jmaa.2000.7015
 [EINT.1]
M.
Eller, V.
Isakov, G.
Nakamura, and D.
Tataru, Uniqueness and stability in the Cauchy problem for Maxwell
and elasticity systems, Nonlinear partial differential equations and
their applications. Collège de France Seminar, Vol. XIV (Paris,
1997/1998) Stud. Math. Appl., vol. 31, NorthHolland, Amsterdam,
2002, pp. 329–349. MR 1936000
(2004c:35399), http://dx.doi.org/10.1016/S01682024(02)800169
 [ELT.4]
M.
Eller, I.
Lasiecka, and R.
Triggiani, Exact/approximate controllability of thermoelastic
plates with variable thermal coefficients, Discrete Contin. Dynam.
Systems 7 (2001), no. 2, 283–302. MR 1808401
(2001k:93011), http://dx.doi.org/10.3934/dcds.2001.7.283
 [Fu.1]
Daisuke
Fujiwara, Concrete characterization of the domains of fractional
powers of some elliptic differential operators of the second order,
Proc. Japan Acad. 43 (1967), 82–86. MR 0216336
(35 #7170)
 [G.1]
P.
Grisvard, Caractérisation de quelques espaces
d’interpolation, Arch. Rational Mech. Anal. 25
(1967), 40–63 (French). MR 0213864
(35 #4718)
 [HP.1]
V.
Hutson and J.
S. Pym, Applications of functional analysis and operator
theory, Mathematics in Science and Engineering, vol. 146,
Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New
YorkLondon, 1980. MR 569354
(81i:46001)
 [I.1]
Victor
Isakov, On the uniqueness of the continuation for a
thermoelasticity system, SIAM J. Math. Anal. 33
(2001), no. 3, 509–522 (electronic). MR 1871407
(2002j:35052), http://dx.doi.org/10.1137/S0036141000366509
 [Ke.1]
B. Kellogg, Properties of solutions of elliptic boundary value problems, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, edited by A. K. Aziz, Academic Press, New York (1972), pp. 4781.
 [Kes.1]
S.
Kesavan, Topics in functional analysis and applications, John
Wiley & Sons, Inc., New York, 1989. MR 990018
(90m:46002)
 [KL.1]
H. Koch and I. Lasiecka, Backward uniqueness in linear thermoelasticity with time and space variable coefficients, in Functional Analysis and Evolution Equations, the Gunter Lumer Volume, Birkhäuser Verlag (2007), 389403, edited by H. Amann, W. Arendt, M. Hieber, F. Neubrander, S. Nicaise, and J. von Below.
 [Kr.1]
S.
G. Kreĭn, Linear differential equations in Banach
space, American Mathematical Society, Providence, R.I., 1971.
Translated from the Russian by J. M. Danskin; Translations of Mathematical
Monographs, Vol. 29. MR 0342804
(49 #7548)
 [LLT.1]
I.
Lasiecka, J.L.
Lions, and R.
Triggiani, Nonhomogeneous boundary value problems for second order
hyperbolic operators, J. Math. Pures Appl. (9) 65
(1986), no. 2, 149–192. MR 867669
(88c:35092)
 [LRT.1]
I.
Lasiecka, M.
Renardy, and R.
Triggiani, Backward uniqueness for thermoelastic plates with
rotational forces, Semigroup Forum 62 (2001),
no. 2, 217–242. MR 1831509
(2002d:35204), http://dx.doi.org/10.1007/s002330010035
 [LT.1]
I.
Lasiecka and R.
Triggiani, Uniform stabilization of the wave equation with
Dirichlet or Neumann feedback control without geometrical conditions,
Appl. Math. Optim. 25 (1992), no. 2, 189–224.
MR
1142681 (93b:93099), http://dx.doi.org/10.1007/BF01182480
 [LT.2]
I.
Lasiecka and R.
Triggiani, Sharp regularity theory for elastic and thermoelastic
Kirchoff [Kirchhoff] equations with free boundary conditions, Rocky
Mountain J. Math. 30 (2000), no. 3, 981–1024.
MR
1797827 (2001i:35039), http://dx.doi.org/10.1216/rmjm/1021477256
 [LT.3]
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories; Vol. I: Abstract Hyperbolic Equations, Cambridge University Press, 2000, Encyclopedia of Mathematics and its Applications, 660 pp.
 [Li.1]
J.L.
Lions, Quelques méthodes de résolution des
problèmes aux limites non linéaires, Dunod;
GauthierVillars, Paris, 1969 (French). MR 0259693
(41 #4326)
 [LM.1]
J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Vol. 1, SpringerVerlag, 1972.
 [Lit.1]
Walter
Littman, Near optimal time boundary controllability for a class of
hyperbolic equations, Control problems for systems described by
partial differential equations and applications (Gainesville, Fla., 1986)
Lecture Notes in Control and Inform. Sci., vol. 97, Springer, Berlin,
1987, pp. 307–312. MR 910526
(88m:93015), http://dx.doi.org/10.1007/BFb0038763
 [Lit.2]
Walter
Littman, Remarks on global uniqueness theorems for partial
differential equations, Differential geometric methods in the control
of partial differential equations (Boulder, CO, 1999) Contemp. Math.,
vol. 268, Amer. Math. Soc., Providence, RI, 2000,
pp. 363–371. MR 1804800
(2002a:35006), http://dx.doi.org/10.1090/conm/268/04318
 [Pa.1]
A.
Pazy, Semigroups of linear operators and applications to partial
differential equations, Applied Mathematical Sciences, vol. 44,
SpringerVerlag, New York, 1983. MR 710486
(85g:47061)
 [Th.1]
Vidar
Thomée, Galerkin finite element methods for parabolic
problems, Springer Series in Computational Mathematics, vol. 25,
SpringerVerlag, Berlin, 1997. MR 1479170
(98m:65007)
 [Tr.1]
R.
Triggiani, Finite rank, relatively bounded perturbations of
semigroups generators. III.\ A sharp result on the lack of uniform
stabilization, Differential Integral Equations 3
(1990), no. 3, 503–522. MR 1047750
(91f:93091)
 [Tr.2]
R.
Triggiani, Backward uniqueness of semigroups arising in coupled
partial differential equations systems of structural acoustics, Adv.
Differential Equations 9 (2004), no. 12,
53–84. MR
2099606 (2005h:35218)
 [A.1]
 G. Avalos, The strong stability and instability of a fluidstructure semigroup, Ann. Math. & Optim. 55 (2007), 163184. MR 2305089 (2008b:93090)
 [AT.1]
 G. Avalos and R. Triggiani, The coupled PDEsystem arising in fluidstructure interaction. Part I: Explicit semigroup generator and its spectral properties, AMS Contemporary Mathematics, Fluids and Waves 440 (2007), 1555. MR 2359448
 [AT.2]
 G. Avalos and R. Triggiani, Uniform stabilization of a coupled PDE system arising in fluidstructure interaction with boundary dissipation at the interface, Discrete Contin. Dynam. Systems 22 (2008), no. 4, 817833. MR 2434971 (2009i:74025)
 [AT.3]
 G. Avalos and R. Triggiani, Wellposedness and stability analysis of a coupled StokesLamé PDE system, 2007.
 [AT.4]
 G. Avalos and R. Triggiani, Backward uniqueness of the s.c. semigroup arising in parabolichyperbolic fluidstructure interaction, J. Differential Equations 245 (2008), no. 3, 737761. MR 2422526
 [AT.5]
 G. Avalos and R. Triggiani, Uniform stabilization of the coupled StokesLamé PDE system with boundary dissipation at the interface, 2007.
 [BS.1]
 S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, SpringerVerlag, New York (1994). MR 1278258 (95f:65001)
 [BGLT.1]
 V. Barbu, Z. Grujić, I. Lasiecka, and A. Tuffaha, Existence of the energy level solutions for a weak nonlinear fluidstructure interaction model, AMS Contemporary Mathematics 440 (2007), 5563. MR 2359449
 [CT.1]
 S. K. Chang and R. Triggiani, Spectral analysis of thermoelastic plates with rotational forces, Optimal Control: Theory, Algorithms, and Applications, Kluwer (1998), 84115. MR 1635994 (99g:35126)
 [CR.1]
 H. Cohen and S. I. Rubinow, Some mathematical topics in biology, Proc. Symp. on System Theory, Polytechnic Press, New York (1965), 321337.
 [DGHL.1]
 Q. Du, M. D. Gunzburger, L. S. Hou, and J. Lee, Analysis of a linear fluidstructure interaction problem, Discr. & Cont. Dynam. Sys. 9(3) (2003), 633650. MR 1974530 (2004c:74017)
 [ELT.1]
 M. Eller, I. Lasiecka, and R. Triggiani, Simultaneous exact/approximate boundary controllability of thermoelastic plates with variable coefficient, Marcel Dekker Lecture Notes Pure and Applied Mathematics 216 (February 2001), 109230, invited paper for the special volume entitled Shape Optimization and Optimal Designs, J. Cagnol and J. P. Zolesio (Editors). [The preliminary version is in invited paper in Semigroup of Operators and Applications, Birkhäuser (2000), 335351, A. V. Balakrishnan (editor).]
 [ELT.2]
 M. Eller, I. Lasiecka, and R. Triggiani, Unique continuation result for thermoelastic plates, Inverse and IllPosed Problems 9(2) (2001), 109148. MR 1843430 (2002m:74036)
 [ELT.3]
 M. Eller, I. Lasiecka, and R. Triggiani, Simultaneous exact/approximate boundary controllability of thermoelastic plates with variable thermal coefficient and moment control, J. Math. Anal. Appl. 251 (2000), 452478. MR 1794432 (2001j:93008)
 [EINT.1]
 M. Eller, V. Isakov, G. Nakamura, and D. Tataru, ``Uniqueness and stability in the Cauchy Problem for Maxwell's and elasticity systems,'' Nonlinear partial differential equations and their applications, Collège de France, Vol. XIV (Paris, 1997/1998), Stud. Math. Appl., 31, NorthHolland, Amsterdam (2002), pp. 329349. MR 1936000 (2004c:35399)
 [ELT.4]
 M. Eller, I. Lasiecka, and R. Triggiani, Simultaneous exact/approximate boundary controllability of thermoelastic plates with variable thermal coefficient and clamped controls, Discrete Cont. Dynam. Sys. 7(2) (2001), 283301. MR 1808401 (2001k:93011)
 [Fu.1]
 D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad. 43 (1967), 8286. MR 0216336 (35:7170)
 [G.1]
 P. Grisvard, Caracterization de quelques espaces d'interpolation, Arch. Ration. Mech. Anal. 25 (1967), 4063. MR 0213864 (35:4718)
 [HP.1]
 V. Hutson and J. S. Pym, Applications of Functional Analysis and Operator Theory, Academic Press, New York (1979). MR 569354 (81i:46001)
 [I.1]
 V. Isakov, On the uniqueness of continuation for a thermoelastic system, Math. Anal., 33 (2001), 509522. MR 1871407 (2002j:35052)
 [Ke.1]
 B. Kellogg, Properties of solutions of elliptic boundary value problems, in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, edited by A. K. Aziz, Academic Press, New York (1972), pp. 4781.
 [Kes.1]
 S. Kesavan, Topics in Functional Analysis and Applications, John Wiley & Sons, New York (1989). MR 990018 (90m:46002)
 [KL.1]
 H. Koch and I. Lasiecka, Backward uniqueness in linear thermoelasticity with time and space variable coefficients, in Functional Analysis and Evolution Equations, the Gunter Lumer Volume, Birkhäuser Verlag (2007), 389403, edited by H. Amann, W. Arendt, M. Hieber, F. Neubrander, S. Nicaise, and J. von Below.
 [Kr.1]
 S. G. Krein, Linear Differential Equations in Banach Space, American Mathematical Society, Providence Rhode Island (1971). MR 0342804 (49:7548)
 [LLT.1]
 I. Lasiecka, J. L. Lions, and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators (with I. Lasiecka and J. L. Lions), J. Math. Pures Appl. 65 (1986), 149192. MR 867669 (88c:35092)
 [LRT.1]
 I. Lasiecka, M. Renardy, and R. Triggiani, Backward uniqueness for thermoelastic plates, Semigroup Forum 62 (2001), 217242. MR 1831509 (2002d:35204)
 [LT.1]
 I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equation with Dirichlet or Neumann feedback control without geometrical conditions, Appl. Math. Optimiz. 25 (1992), 189224. (The preliminary version is in SpringerVerlag Lecture Notes LNCIS 147, 62108, J. P. Zolesio (editor).) MR 1142681 (93b:93099)
 [LT.2]
 I. Lasiecka and R. Triggiani, A sharp trace regularity result of Kirchof and thermoelastic plate equations with free boundary conditions, Rocky Mount. J. Math. 30(3) (2000), 9811023. MR 1797827 (2001i:35039)
 [LT.3]
 I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories; Vol. I: Abstract Hyperbolic Equations, Cambridge University Press, 2000, Encyclopedia of Mathematics and its Applications, 660 pp.
 [Li.1]
 J. L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires, DumodGauthierVillars, 1969. MR 0259693 (41:4326)
 [LM.1]
 J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Vol. 1, SpringerVerlag, 1972.
 [Lit.1]
 W. Littman, Near optimal time boundary controllability for a class of hyperbolic equations, Lecture Notes in Control Sciences 178, SpringerVerlag 1987, 272284. MR 910526 (88m:93015)
 [Lit.2]
 W. Littman, Remarks on global uniqueness theorems for partial differential equations, AMS Contemporary Mathematics 268, Amer. Math. Soc., 2000, 363371. MR 1804800 (2002a:35006)
 [Pa.1]
 A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, SpringerVerlag, 1983. MR 710486 (85g:47061)
 [Th.1]
 V. Thomee, Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Mathematics, vol. 1054, Springer 1984. The expanded version reprinted as a Springer Verlag book, 1997. MR 1479170 (98m:65007)
 [Tr.1]
 R. Triggiani, Finite rank, relatively bounded perturbations of semigroup generators, PART III: A sharp result on the lack of uniform stabilization, Diff. Int. Eqns. 3 (1990), 503522. (Also, preliminary version in Proceedings INRIA Conference, Paris, France (June 1988), SpringerVerlag Lecture Notes. MR 1047750 (91f:93091)
 [Tr.2]
 R. Triggiani, Backward uniqueness of semigroups arising in coupled PDE systems of structural acoustics, Advances in Diff. Eqns., Vol. 9(12) (Jan.Feb. 2004), 5384. Preliminary announcement in Semigroups of Operators: Theory and Applications (2002), 285300. C. Kubrusly, N. Levan, and M. da Silveira (editors). MR 2099606 (2005h:35218)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2010):
35B99,
35M30
Retrieve articles in all journals
with MSC (2010):
35B99,
35M30
Additional Information
George Avalos
Affiliation:
Department of Mathematics, University of NebraskaLincoln, Lincoln, Nebraska 68588
Roberto Triggiani
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903
DOI:
http://dx.doi.org/10.1090/S0002994710048518
PII:
S 00029947(10)048518
Received by editor(s):
February 15, 2008
Published electronically:
February 19, 2010
Additional Notes:
The research of the first author was partially supported by the NSF grant DMS0606776.
The research of the second author was partially supported by the NSF grant DMS0104305.
Article copyright:
© Copyright 2010
American Mathematical Society
