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Long time behaviour of strong solutions to interactive fluid-plate system without rotational inertia

  • * Corresponding author: Iryna Ryzhkova-Gerasymova

    * Corresponding author: Iryna Ryzhkova-Gerasymova
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  • We study well-posedness and asymptotic dynamics of a coupled system consisting of linearized 3D Navier-Stokes equations in a bounded domain and a classical (nonlinear) full von Karman plate equations that accounts for both transversal and lateral displacements on a flexible part of the boundary. Rotational inertia of the filaments of the plate is not taken into account. Our main result shows well-posedness of strong solutions to the problem, thus the problem generates a semiflow in an appropriate phase space. We also prove uniform stability of strong solutions to homogeneous problem.

    Mathematics Subject Classification: Primary: 35B40, 35D35; Secondary: 35Q35, 35Q74.


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  •   G. Avalos  and  F. Bucci , Rational rates of uniform decay for strong solutions to a fluid-structure PDE system, Journal of Differential Equations, 258 (2015) , 4398-4423.  doi: 10.1016/j.jde.2015.01.037.
      G. Avalos  and  T. Clark , A mixed variational formulation for the wellposedness and numerical approximation of a PDE model arising in a 3-D fluid-structure interaction, Evolution Equations and Control Theory, 3 (2014) , 557-578.  doi: 10.3934/eect.2014.3.557.
      A. Chambolle , B. Desjardins , M. Esteban  and  C. Grandmont , Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, J. Math. Fluid Mech., 7 (2005) , 368-404.  doi: 10.1007/s00021-004-0121-y.
      I. D. Chueshov , Strong slutions and attractor of the von Karman equations (in Russian), Mathematics of the USSR-Sbornik, 69 (1990) , 25-36. 
      I. Chueshov , A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate, Math. Meth. Appl. Sci., 34 (2011) , 1801-1812. 
      I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc. , 195 (2008), ⅷ+183 pp.
      I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer, New York, 2010.
      I. Chueshov  and  I. Ryzhkova , A global attractor for a fluid-plate interaction model, Comm. Pure Appl. Anal., 12 (2013) , 1635-1656. 
      I. Chueshov  and  I. Ryzhkova , Unsteady interaction of a viscous fluid with an elastic plate modeled by full von Karman equations, J. Diff. Eqs, 254 (2013) , 1833-1862.  doi: 10.1016/j.jde.2012.11.006.
      G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer Berlin, New York, 1976.
      M. Grobbelaar-Van Dalsen , On a fluid-structure model in which the dynamics of the structure involves the shear stress due to the fluid, J. Math. Fluid Mech., 10 (2008) , 388-401.  doi: 10.1007/s00021-006-0236-4.
      M. Grobbelaar-Van Dalsen , A new approach to the stabilization of a fluid-structure interaction model, Appl. Anal., 88 (2009) , 1053-1065.  doi: 10.1080/00036810903114841.
      M. Grobbelaar-Van Dalsen , Strong stability for a fluid-structure model, Math. Methods Appl. Sci., 32 (2009) , 1452-1466.  doi: 10.1002/mma.1104.
      H. Koch and I. Lasiecka, Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, in Prog. Nonlinear Differ. Equ. Appl. , Basel: Birkhäuser, 50 (2002), 197-216.
      I. Lasiecka , Uniform stabilizability of a full von Karman system with nonlinear boundary feedback, SIAM J. Control Optim., 36 (1998) , 1376-1422.  doi: 10.1137/S0363012996301907.
      J. -L. Lions and E. Magenes, Problémes Aux Limites non Homogénes et Applications, Vol. 1, Dunod, Paris, 1968.
      V. I. Sedenko, Global in Time Well-posedness of Initial-boundary Value Problems for Marguerre-Vlasov Equations of Nonlinear Elastic Shells Theory, (in russian) Docthral Thesis, Rostov state university, 1995.
      V. I. Sedenko, The classical solubility of initial-boundary-value problems in the non-linear theory of oscillations of shallow shells, (in russian) Izvestiya: Mathematics, 60 (1996), 1027-1059.
      J. Simon , Compact sets in the space Lp(0, T; B), Annali di Matematica Pura ed Applicata, Ser.4, 146 (1987) , 65-96. 
      R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001.
      H. Triebel, Interpolation Theory, Functional Spaces and Differential Operators, North Holland, Amsterdam, 1978.
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