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Backwards uniqueness of the $ C_{0}$-semigroup associated with a parabolic-hyperbolic Stokes-Lamé partial differential equation system

Authors: George Avalos and Roberto Triggiani
Journal: Trans. Amer. Math. Soc. 362 (2010), 3535-3561
MSC (2010): Primary 35B99, 35M30
Published electronically: February 19, 2010
MathSciNet review: 2601599
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Abstract: In this paper the ``backward-uniqueness property'' is ascertained for a two- or three-dimensional, fluid-structure interactive partial differential equation (PDE) system, for which an explicit $ C_{0}$-semigroup formulation was recently given by Avalos and Triggiani (2007) on the natural finite energy space $ \mathbf{H}$. (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 fluid-structure semigroup $ \left\{ e^{\mathcal{A}t}\right\} $, posed on the associated finite energy space $ \mathbf{H}$, the backward-uniqueness property can be stated in this way: If for given initial data $ y_{0}\in \mathbf{H}$, $ e^{\mathcal{A} T}y_{0}=0$ for some $ T>0$, then necessarily $ y_{0}=0$. The proof of this property hinges on establishing necessary PDE estimates for a certain static fluid-structure equation in order to invoke the abstract backward-uniqueness resolvent-based criterion by Lasiecka, Renardy, and Triggiani (2001). The backward-uniqueness property for the coupled Stokes-Lamé PDE is motivated by, and has positive implications to, the problem of exact controllability (in the hyperbolic state variables $ \{w,w_t\}$) and, simultaneously, approximate controllability (in the parabolic state variable $ u$) 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.

References [Enhancements On Off] (What's this?)

  • [A.1] George Avalos, The strong stability and instability of a fluid-structure semigroup, Appl. Math. Optim. 55 (2007), no. 2, 163–184. MR 2305089, 10.1007/s00245-006-0884-z
  • [A-T.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, 10.1090/conm/440/08475
  • [A-T.2] George Avalos and Roberto Triggiani, Uniform stabilization of a coupled PDE system arising in fluid-structure interaction with boundary dissipation at the interface, Discrete Contin. Dyn. Syst. 22 (2008), no. 4, 817–833. MR 2434971, 10.3934/dcds.2008.22.817
  • [A-T.3] G. Avalos and R. Triggiani, Well-posedness and stability analysis of a coupled Stokes-Lamé PDE system, 2007.
  • [A-T.4] George Avalos and Roberto Triggiani, Backward uniqueness of the s.c. semigroup arising in parabolic-hyperbolic fluid-structure interaction, J. Differential Equations 245 (2008), no. 3, 737–761. MR 2422526, 10.1016/j.jde.2007.10.036
  • [A-T.5] G. Avalos and R. Triggiani, Uniform stabilization of the coupled Stokes-Lamé PDE system with boundary dissipation at the interface, 2007.
  • [B-S.1] Susanne C. Brenner and L. Ridgway Scott, The mathematical theory of finite element methods, Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 1994. MR 1278258
  • [B-G-L-T.1] Viorel Barbu, Zoran Grujić, Irena Lasiecka, and Amjad Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, Fluids and waves, Contemp. Math., vol. 440, Amer. Math. Soc., Providence, RI, 2007, pp. 55–82. MR 2359449, 10.1090/conm/440/08476
  • [C-T.1] S. K. Chang and Roberto Triggiani, Spectral analysis of thermo-elastic plates with rotational forces, Optimal control (Gainesville, FL, 1997) Appl. Optim., vol. 15, Kluwer Acad. Publ., Dordrecht, 1998, pp. 84–115. MR 1635994, 10.1007/978-1-4757-6095-8_5
  • [C-R.1] H. Cohen and S. I. Rubinow, Some mathematical topics in biology, Proc. Symp. on System Theory, Polytechnic Press, New York (1965), 321-337.
  • [D-G-H-L.1] Q. Du, M. D. Gunzburger, L. S. Hou, and J. Lee, Analysis of a linear fluid-structure interaction problem, Discrete Contin. Dyn. Syst. 9 (2003), no. 3, 633–650. MR 1974530, 10.3934/dcds.2003.9.633
  • [E-L-T.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), 109-230, 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), 335-351, A. V. Balakrishnan (editor).]
  • [E-L-T.2] M. Eller, I. Lasiecka, and R. Triggiani, Unique continuation for over-determined Kirchoff plate equations and related thermoelastic systems, J. Inverse Ill-Posed Probl. 9 (2001), no. 2, 103–148. Distributed systems: optimization and economic-environmental applications (Ekaterinburg, 2000). MR 1843430, 10.1515/jiip.2001.9.2.103
  • [E-L-T.3] M. Eller, I. Lasiecka, and R. Triggiani, Simultaneous exact/approximate boundary controllability of thermo-elastic plates with variable thermal coefficient and moment control, J. Math. Anal. Appl. 251 (2000), no. 2, 452–478. MR 1794432, 10.1006/jmaa.2000.7015
  • [E-I-N-T.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, North-Holland, Amsterdam, 2002, pp. 329–349. MR 1936000, 10.1016/S0168-2024(02)80016-9
  • [E-L-T.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, 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
  • [G.1] P. Grisvard, Caractérisation de quelques espaces d’interpolation, Arch. Rational Mech. Anal. 25 (1967), 40–63 (French). MR 0213864
  • [H-P.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 York-London, 1980. MR 569354
  • [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, 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. 47-81.
  • [Kes.1] S. Kesavan, Topics in functional analysis and applications, John Wiley & Sons, Inc., New York, 1989. MR 990018
  • [K-L.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), 389-403, 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
  • [L-L-T.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
  • [L-R-T.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, 10.1007/s002330010035
  • [L-T.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, 10.1007/BF01182480
  • [L-T.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, 10.1216/rmjm/1021477256
  • [L-T.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; Gauthier-Villars, Paris, 1969 (French). MR 0259693
  • [L-M.1] J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. 1, Springer-Verlag, 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, 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, 10.1090/conm/268/04318
  • [Pa.1] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486
  • [Th.1] Vidar Thomée, Galerkin finite element methods for parabolic problems, Springer Series in Computational Mathematics, vol. 25, Springer-Verlag, Berlin, 1997. MR 1479170
  • [Tr.1] R. Triggiani, Finite rank, relatively bounded perturbations of semi-groups generators. III. A sharp result on the lack of uniform stabilization, Differential Integral Equations 3 (1990), no. 3, 503–522. MR 1047750
  • [Tr.2] R. Triggiani, Backward uniqueness of semigroups arising in coupled partial differential equations systems of structural acoustics, Adv. Differential Equations 9 (2004), no. 1-2, 53–84. MR 2099606

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Additional Information

George Avalos
Affiliation: Department of Mathematics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588

Roberto Triggiani
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903

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 DMS-0606776.
The research of the second author was partially supported by the NSF grant DMS-0104305.
Article copyright: © Copyright 2010 American Mathematical Society