# American Institute of Mathematical Sciences

May 2018, 23(3): 1243-1265. doi: 10.3934/dcdsb.2018150

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

 1 V.N.Karazin Karkiv national university, Svobody sq. 4, Kharkiv, 61077, Ukraine

* Corresponding author: Iryna Ryzhkova-Gerasymova

Received  March 2017 Revised  July 2017 Published  February 2018

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.

Citation: Iryna Ryzhkova-Gerasymova. Long time behaviour of strong solutions to interactive fluid-plate system without rotational inertia. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1243-1265. doi: 10.3934/dcdsb.2018150
##### References:
 [1] 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. [2] 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. [3] 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. [4] I. D. Chueshov, Strong slutions and attractor of the von Karman equations (in Russian), Mathematics of the USSR-Sbornik, 69 (1990), 25-36. [5] 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. [6] 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. [7] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer, New York, 2010. [8] I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model, Comm. Pure Appl. Anal., 12 (2013), 1635-1656. [9] 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. [10] G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer Berlin, New York, 1976. [11] 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. [12] 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. [13] M. Grobbelaar-Van Dalsen, Strong stability for a fluid-structure model, Math. Methods Appl. Sci., 32 (2009), 1452-1466. doi: 10.1002/mma.1104. [14] 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. [15] 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. [16] J. -L. Lions and E. Magenes, Problémes Aux Limites non Homogénes et Applications, Vol. 1, Dunod, Paris, 1968. [17] 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. [18] 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. [19] J. Simon, Compact sets in the space Lp(0, T; B), Annali di Matematica Pura ed Applicata, Ser.4, 146 (1987), 65-96. [20] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. [21] H. Triebel, Interpolation Theory, Functional Spaces and Differential Operators, North Holland, Amsterdam, 1978.

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##### References:
 [1] 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. [2] 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. [3] 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. [4] I. D. Chueshov, Strong slutions and attractor of the von Karman equations (in Russian), Mathematics of the USSR-Sbornik, 69 (1990), 25-36. [5] 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. [6] 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. [7] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer, New York, 2010. [8] I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model, Comm. Pure Appl. Anal., 12 (2013), 1635-1656. [9] 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. [10] G. Duvaut and J. L. Lions, Inequalities in Mechanics and Physics, Springer Berlin, New York, 1976. [11] 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. [12] 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. [13] M. Grobbelaar-Van Dalsen, Strong stability for a fluid-structure model, Math. Methods Appl. Sci., 32 (2009), 1452-1466. doi: 10.1002/mma.1104. [14] 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. [15] 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. [16] J. -L. Lions and E. Magenes, Problémes Aux Limites non Homogénes et Applications, Vol. 1, Dunod, Paris, 1968. [17] 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. [18] 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. [19] J. Simon, Compact sets in the space Lp(0, T; B), Annali di Matematica Pura ed Applicata, Ser.4, 146 (1987), 65-96. [20] R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, 2001. [21] H. Triebel, Interpolation Theory, Functional Spaces and Differential Operators, North Holland, Amsterdam, 1978.
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