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Long-time behaviour of a radially symmetric fluid-shell interaction system

  • * Corresponding author: Tamara Fastovska

    * Corresponding author: Tamara Fastovska
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  • We study the long-time dynamics of a coupled system consisting of the 2D Navier-Stokes equations and full von Karman elasticity equations. We show that this problem generates an evolution semigroup $S_t$ possessing a compact finite-dimensional global attractor.

    Mathematics Subject Classification: Primary: 74F10, 35B41; Secondary: 35Q30, 74K25.


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  •   A. V. Babin and  M. I. VishikAttractors of Evolution Equations, North-Holland, Amsterdam, 1992. 
      V. Barbu, Z. Grujić, I. Lasiecka and A. Tuffaha, Existence of the energy-level weak solutions for a nonlinear fluid-structure interaction model, Fluids and Waves, Contemp. Math., AMS, Providence, RI, 440 (2007), 55-82.
      A. Boutet de Monvel  and  I. Chueshov , Uniqueness theorem for weak solutions of von Karman evolution equations, J. Math. Anal. Appl., 221 (1998) , 419-429.  doi: 10.1006/jmaa.1997.5681.
      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. ChueshovDynamics of Quasi-Stable Dissipative Systems, Springer-Verlag, Cham, 2015. 
      I. ChueshovIntroduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999. 
      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  T. Fastovska , On interaction of circular cylindrical shells with a Poiseuille type flow, Evolution Equations and Control Theory, 5 (2016) , 605-629.  doi: 10.3934/eect.2016021.
      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. 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 shell modeled by full von Karman equations, J. Diff. Eqs., 254 (2013) , 1833-1862.  doi: 10.1016/j.jde.2012.11.006.
      I. Chueshov  and  I. Ryzhkova , On the interaction of an elastic wall with a Poiseuille-type flow, Ukrainian Mathematical Journal, 65 (2013) , 158-177.  doi: 10.1007/s11253-013-0771-0.
      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) , 633-650.  doi: 10.3934/dcds.2003.9.633.
      G. GaldiAn Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, 2 edition, Springer-Verlag, New York, 2011. 
      G. Galdi , C. Simader  and  H. Sohr , A class of solutions to stationary Stokes and Navier-Stokes equations with boundary data in $W^{-1/q, q}$, Math. Annalen, 331 (2005) , 41-74.  doi: 10.1007/s00208-004-0573-7.
      C. Grandmont , Existence of weak solutions for the unsteady interaction of a viscous fluid with an elastic plate, SIAM J. Math. Anal., 40 (2008) , 716-737.  doi: 10.1137/070699196.
      M. Guidorzi , M. Padula  and  P. I. Plotnikov , Hopf solutions to a fluid-elastic interaction model, MMAS, 18 (2008) , 215-269.  doi: 10.1142/S0218202508002668.
      H. Koch  and  I. Lasiecka , Hadamard well-posedness of weak solutions in nonlinear dynamic elasticity-full von Karman systems, Prog. Nonlinear Differ. Equ. Appl, 50 (2002) , 197-216. 
      O. LadyzhenskayaMathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York, 1969. 
      J. -L. Lions and E. Magenes, Problémes aux Limites non Homogénes et Applications, Vol. 1, Dunod, Paris, 1968.
      J. Simon , Compact sets in the space $L^p(0, T; B)$, Ann. Mat. Pura Appl., 146 (1987) , 65-96. 
      R. TemamInfinite-Dimensional Dynamical Dystems in Mechanics and Physics, Springer-Verlag, New York, 1988. 
      R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001.
      H. TriebelInterpolation Theory, Functional Spaces and Differential Operators, North Holland, Amsterdam, 1978. 
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