September  2014, 13(5): 1759-1778. doi: 10.3934/cpaa.2014.13.1759

Interaction of an elastic plate with a linearized inviscid incompressible fluid

1. 

Department of Mechanics and Mathematics, Kharkov National University, 4 Svobody sq., 61077, Kharkov, Ukraine

Received  June 2013 Revised  September 2013 Published  June 2014

We prove well-posedness of energy type solutions to an interacting system consisting of the 3D linearized Euler equations and a (possibly nonlinear) elastic plate equation describing large deflections of a flexible part of the boundary. In the damped case under some conditions concerning the plate nonlinearity we prove the existence of a compact global attractor for the corresponding dynamical system and describe the situations when this attractor has a finite fractal dimension.
Citation: I. D. Chueshov. Interaction of an elastic plate with a linearized inviscid incompressible fluid. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1759-1778. doi: 10.3934/cpaa.2014.13.1759
References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992). Google Scholar

[2]

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,, \emph{J. Math. Fluid Mech.}, 7 (2005), 368. doi: 10.1007/s00021-004-0121-y. Google Scholar

[3]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, AMS, (2002). Google Scholar

[4]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, Acta, (1999). Google Scholar

[5]

I. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate,, \emph{Math. Meth. Appl. Sci.}, 34 (2011), 1801. doi: 10.1002/mma.1496. Google Scholar

[6]

I. Chueshov, Dynamics of a nonlinear elastic plate interacting with a linearized compressible viscous fluid,, \emph{Nonlinear Analysis}, 95 (2014), 650. doi: 10.1016/j.na.2013.10.018. Google Scholar

[7]

I. Chueshov and S. Kolbasin, Long-time dynamics in plate models with strong nonlinear damping,, \emph{Comm. Pure Appl. Anal.}, 11 (2012), 659. doi: 10.3934/cpaa.2012.11.659. Google Scholar

[8]

I. Chueshov and I. Lasiecka, Attractors for second order evolution equations,, J. Dynam. Diff. Eqs., 16 (2004), 469. doi: 10.1007/s10884-004-4289-x. Google Scholar

[9]

I. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with nonlinear boundary/interior damping,, \emph{J. Diff. Eqs.}, 233 (2007), 42. doi: 10.1016/j.jde.2006.09.019. Google Scholar

[10]

I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping,, Memoirs of AMS, (2008). doi: 10.1090/memo/0912. Google Scholar

[11]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Springer, (2010). doi: 10.1007/978-0-387-87712-9. Google Scholar

[12]

I. Chueshov and I. Lasiecka, On Global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity,, \emph{Commun. Partial Dif. Eqs}, 36 (2011), 67. doi: 10.1080/03605302.2010.484472. Google Scholar

[13]

I. Chueshov and I. Lasiecka, Generation of a semigroup and hidden regularity in nonlinear subsonic flow-structure interactions with absorbing boundary conditions,, \emph{J. Abstr. Differ. Equ. Appl.}, 3 (2012), 1. Google Scholar

[14]

I. Chueshov and I. Lasiecka, Well-posedness and long time behavior in nonlinear dissipative hyperbolic-like evolutions with critical exponents,, in \emph{Nonlinear Hyperbolic PDEs, (2013), 1. Google Scholar

[15]

I. Chueshov, I. Lasiecka and J. Webster, Evolution semigroups in supersonic flow-plate interactions,, \emph{J. Diff. Eqs.}, 254 (2013), 1741. doi: 10.1016/j.jde.2012.11.009. Google Scholar

[16]

I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model,, Comm. Pure Appl. Anal. \textbf{12} (2013), 12 (2013), 1635. doi: 10.3934/cpaa.2013.12.1635. Google Scholar

[17]

I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations,, \emph{J. Diff. Eqs.}, 254 (2013), 1833. doi: 10.1016/j.jde.2012.11.006. Google Scholar

[18]

I. Chueshov and I. Ryzhkova, On interaction of an elastic wall with a Poiseuille type flow,, \emph{Ukrainian Math. J.}, 65 (2013), 158. doi: 10.1007/s11253-013-0771-0. Google Scholar

[19]

I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models,, in \emph{IFIP Advances in Information and Communication Technology}, (2013), 328. Google Scholar

[20]

I. Chueshov and A. Shcherbina, Semi-weak well-posedness and attractors for 2D Schrödinger-Boussinesq equations,, \emph{Evolution Equations and Control Theory}, 1 (2012), 57. doi: 10.3934/eect.2012.1.57. Google Scholar

[21]

G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, 2nd ed.,, Springer, (2011). doi: 10.1007/978-0-387-09620-9. Google Scholar

[22]

M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term,, \emph{Commun. Partial Dif. Eqs.}, 34 (2009), 137. doi: 10.1080/03605300802608247. Google Scholar

[23]

M. S. Howe, Acoustics of Fluid-Structure Interactions,, Cambridge University Press, (1998). doi: 10.1017/CBO9780511662898. Google Scholar

[24]

N. Kopachevskii and S. Krein, Operator Approach to Linear Problems of Hydrodynamics: Volume 1: Self-adjoint Problems for an Ideal Fluid,, Birkh\, (2001). doi: 10.1007/978-3-0348-8342-9. Google Scholar

[25]

I. Lasiecka, Mathematical Control Theory of Coupled PDE's,, CMBS-NSF Lecture Notes, (2001). doi: 10.1137/1.9780898717099. Google Scholar

[26]

J.-L. Lions and E. Magenes, Problémes aux limites non homogénes et applications, Vol. 1,, Dunod, (1968). Google Scholar

[27]

J.-L. Lions, Quelques methodes de resolution des problémes aux limites non lineair,, Dunod, (1969). Google Scholar

[28]

B. S. Massey and J. Ward-Smith, Mechanics of Fluids, 8th ed.,, Taylor & Francis, (2006). Google Scholar

[29]

A. Osses and J. Puel, Approximate controllability for a linear model of fluid structure interaction,, \emph{ESAIM: Control, 4 (1999), 497. doi: 10.1051/cocv:1999119. Google Scholar

[30]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, \emph{Annali di Matematica Pura ed Applicata}, 148 (1987), 65. doi: 10.1007/BF01762360. Google Scholar

[31]

R. Temam, Infinite-Dimensional Dynamical Dystems in Mechanics and Physics,, Springer, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar

[32]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition,, AMS Chelsea Publishing, (2001). Google Scholar

[33]

H. Triebel, Interpolation Theory, Functional Spaces and Differential Operators,, North Holland, (1978). Google Scholar

[34]

J. T. Webster, Weak and strong solutions of a nonlinear subsonic flow-structure interaction: semigroup approach,, \emph{Nonlinear Analysis}, 74 (2011), 3123. doi: 10.1016/j.na.2011.01.028. Google Scholar

show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations,, North-Holland, (1992). Google Scholar

[2]

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,, \emph{J. Math. Fluid Mech.}, 7 (2005), 368. doi: 10.1007/s00021-004-0121-y. Google Scholar

[3]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics,, AMS, (2002). Google Scholar

[4]

I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems,, Acta, (1999). Google Scholar

[5]

I. Chueshov, A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate,, \emph{Math. Meth. Appl. Sci.}, 34 (2011), 1801. doi: 10.1002/mma.1496. Google Scholar

[6]

I. Chueshov, Dynamics of a nonlinear elastic plate interacting with a linearized compressible viscous fluid,, \emph{Nonlinear Analysis}, 95 (2014), 650. doi: 10.1016/j.na.2013.10.018. Google Scholar

[7]

I. Chueshov and S. Kolbasin, Long-time dynamics in plate models with strong nonlinear damping,, \emph{Comm. Pure Appl. Anal.}, 11 (2012), 659. doi: 10.3934/cpaa.2012.11.659. Google Scholar

[8]

I. Chueshov and I. Lasiecka, Attractors for second order evolution equations,, J. Dynam. Diff. Eqs., 16 (2004), 469. doi: 10.1007/s10884-004-4289-x. Google Scholar

[9]

I. Chueshov and I. Lasiecka, Long-time dynamics of von Karman semi-flows with nonlinear boundary/interior damping,, \emph{J. Diff. Eqs.}, 233 (2007), 42. doi: 10.1016/j.jde.2006.09.019. Google Scholar

[10]

I. Chueshov and I. Lasiecka, Long-Time Behavior of Second Order Evolution Equations with Nonlinear Damping,, Memoirs of AMS, (2008). doi: 10.1090/memo/0912. Google Scholar

[11]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations,, Springer, (2010). doi: 10.1007/978-0-387-87712-9. Google Scholar

[12]

I. Chueshov and I. Lasiecka, On Global attractor for 2D Kirchhoff-Boussinesq model with supercritical nonlinearity,, \emph{Commun. Partial Dif. Eqs}, 36 (2011), 67. doi: 10.1080/03605302.2010.484472. Google Scholar

[13]

I. Chueshov and I. Lasiecka, Generation of a semigroup and hidden regularity in nonlinear subsonic flow-structure interactions with absorbing boundary conditions,, \emph{J. Abstr. Differ. Equ. Appl.}, 3 (2012), 1. Google Scholar

[14]

I. Chueshov and I. Lasiecka, Well-posedness and long time behavior in nonlinear dissipative hyperbolic-like evolutions with critical exponents,, in \emph{Nonlinear Hyperbolic PDEs, (2013), 1. Google Scholar

[15]

I. Chueshov, I. Lasiecka and J. Webster, Evolution semigroups in supersonic flow-plate interactions,, \emph{J. Diff. Eqs.}, 254 (2013), 1741. doi: 10.1016/j.jde.2012.11.009. Google Scholar

[16]

I. Chueshov and I. Ryzhkova, A global attractor for a fluid-plate interaction model,, Comm. Pure Appl. Anal. \textbf{12} (2013), 12 (2013), 1635. doi: 10.3934/cpaa.2013.12.1635. Google Scholar

[17]

I. Chueshov and I. Ryzhkova, Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations,, \emph{J. Diff. Eqs.}, 254 (2013), 1833. doi: 10.1016/j.jde.2012.11.006. Google Scholar

[18]

I. Chueshov and I. Ryzhkova, On interaction of an elastic wall with a Poiseuille type flow,, \emph{Ukrainian Math. J.}, 65 (2013), 158. doi: 10.1007/s11253-013-0771-0. Google Scholar

[19]

I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models,, in \emph{IFIP Advances in Information and Communication Technology}, (2013), 328. Google Scholar

[20]

I. Chueshov and A. Shcherbina, Semi-weak well-posedness and attractors for 2D Schrödinger-Boussinesq equations,, \emph{Evolution Equations and Control Theory}, 1 (2012), 57. doi: 10.3934/eect.2012.1.57. Google Scholar

[21]

G. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems, 2nd ed.,, Springer, (2011). doi: 10.1007/978-0-387-09620-9. Google Scholar

[22]

M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term,, \emph{Commun. Partial Dif. Eqs.}, 34 (2009), 137. doi: 10.1080/03605300802608247. Google Scholar

[23]

M. S. Howe, Acoustics of Fluid-Structure Interactions,, Cambridge University Press, (1998). doi: 10.1017/CBO9780511662898. Google Scholar

[24]

N. Kopachevskii and S. Krein, Operator Approach to Linear Problems of Hydrodynamics: Volume 1: Self-adjoint Problems for an Ideal Fluid,, Birkh\, (2001). doi: 10.1007/978-3-0348-8342-9. Google Scholar

[25]

I. Lasiecka, Mathematical Control Theory of Coupled PDE's,, CMBS-NSF Lecture Notes, (2001). doi: 10.1137/1.9780898717099. Google Scholar

[26]

J.-L. Lions and E. Magenes, Problémes aux limites non homogénes et applications, Vol. 1,, Dunod, (1968). Google Scholar

[27]

J.-L. Lions, Quelques methodes de resolution des problémes aux limites non lineair,, Dunod, (1969). Google Scholar

[28]

B. S. Massey and J. Ward-Smith, Mechanics of Fluids, 8th ed.,, Taylor & Francis, (2006). Google Scholar

[29]

A. Osses and J. Puel, Approximate controllability for a linear model of fluid structure interaction,, \emph{ESAIM: Control, 4 (1999), 497. doi: 10.1051/cocv:1999119. Google Scholar

[30]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, \emph{Annali di Matematica Pura ed Applicata}, 148 (1987), 65. doi: 10.1007/BF01762360. Google Scholar

[31]

R. Temam, Infinite-Dimensional Dynamical Dystems in Mechanics and Physics,, Springer, (1988). doi: 10.1007/978-1-4684-0313-8. Google Scholar

[32]

R. Temam, Navier-Stokes Equations: Theory and Numerical Analysis, Reprint of the 1984 edition,, AMS Chelsea Publishing, (2001). Google Scholar

[33]

H. Triebel, Interpolation Theory, Functional Spaces and Differential Operators,, North Holland, (1978). Google Scholar

[34]

J. T. Webster, Weak and strong solutions of a nonlinear subsonic flow-structure interaction: semigroup approach,, \emph{Nonlinear Analysis}, 74 (2011), 3123. doi: 10.1016/j.na.2011.01.028. Google Scholar

[1]

Zhaohi Huo, Yueling Jia, Qiaoxin Li. Global well-posedness for the 3D Zakharov-Kuznetsov equation in energy space $H^1$. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 1797-1851. doi: 10.3934/dcdss.2016075

[2]

I. D. Chueshov, Iryna Ryzhkova. A global attractor for a fluid--plate interaction model. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1635-1656. doi: 10.3934/cpaa.2013.12.1635

[3]

Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D incompressible anisotropic magnetohydrodynamics equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5801-5815. doi: 10.3934/dcds.2016055

[4]

Xiaoping Zhai, Yongsheng Li, Wei Yan. Global well-posedness for the 3-D incompressible MHD equations in the critical Besov spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1865-1884. doi: 10.3934/cpaa.2015.14.1865

[5]

Ning Ju. The global attractor for the solutions to the 3D viscous primitive equations. Discrete & Continuous Dynamical Systems - A, 2007, 17 (1) : 159-179. doi: 10.3934/dcds.2007.17.159

[6]

Ning Ju. The finite dimensional global attractor for the 3D viscous Primitive Equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7001-7020. doi: 10.3934/dcds.2016104

[7]

Oualid Kafi, Nader El Khatib, Jorge Tiago, Adélia Sequeira. Numerical simulations of a 3D fluid-structure interaction model for blood flow in an atherosclerotic artery. Mathematical Biosciences & Engineering, 2017, 14 (1) : 179-193. doi: 10.3934/mbe.2017012

[8]

Chao Deng, Xiaohua Yao. Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 437-459. doi: 10.3934/dcds.2014.34.437

[9]

Qiao Liu, Ting Zhang, Jihong Zhao. Well-posedness for the 3D incompressible nematic liquid crystal system in the critical $L^p$ framework. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 371-402. doi: 10.3934/dcds.2016.36.371

[10]

Hartmut Pecher. Low regularity well-posedness for the 3D Klein - Gordon - Schrödinger system. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1081-1096. doi: 10.3934/cpaa.2012.11.1081

[11]

George Avalos, Pelin G. Geredeli, Justin T. Webster. Semigroup well-posedness of a linearized, compressible fluid with an elastic boundary. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1267-1295. doi: 10.3934/dcdsb.2018151

[12]

George Avalos, Roberto Triggiani. Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 417-447. doi: 10.3934/dcdss.2009.2.417

[13]

M. Bulíček, F. Ettwein, P. Kaplický, Dalibor Pražák. The dimension of the attractor for the 3D flow of a non-Newtonian fluid. Communications on Pure & Applied Analysis, 2009, 8 (5) : 1503-1520. doi: 10.3934/cpaa.2009.8.1503

[14]

Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D Navier-Stokes-Maxwell system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5817-5835. doi: 10.3934/dcds.2016056

[15]

Saoussen Sokrani. On the global well-posedness of 3-D Boussinesq system with partial viscosity and axisymmetric data. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1613-1650. doi: 10.3934/dcds.2019072

[16]

Elaine Cozzi, James P. Kelliher. Well-posedness of the 2D Euler equations when velocity grows at infinity. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2361-2392. doi: 10.3934/dcds.2019100

[17]

Daniela De Silva, Nataša Pavlović, Gigliola Staffilani, Nikolaos Tzirakis. Global well-posedness for a periodic nonlinear Schrödinger equation in 1D and 2D. Discrete & Continuous Dynamical Systems - A, 2007, 19 (1) : 37-65. doi: 10.3934/dcds.2007.19.37

[18]

Yong Yang, Bingsheng Zhang. On the Kolmogorov entropy of the weak global attractor of 3D Navier-Stokes equations:Ⅰ. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2339-2350. doi: 10.3934/dcdsb.2017101

[19]

T. Tachim Medjo. Non-autonomous 3D primitive equations with oscillating external force and its global attractor. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 265-291. doi: 10.3934/dcds.2012.32.265

[20]

Boling Guo, Guoli Zhou. Finite dimensionality of global attractor for the solutions to 3D viscous primitive equations of large-scale moist atmosphere. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4305-4327. doi: 10.3934/dcdsb.2018160

2018 Impact Factor: 0.925

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]