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Long time behaviour of strong solutions to interactive fluid-plate system without rotational inertia
Semigroup well-posedness of a linearized, compressible fluid with an elastic boundary
1. | University of Nebraska-Lincoln, Lincoln, NE, USA |
2. | University of Nebraska-Lincoln, Lincoln, NE, USA, & Hacettepe University, Ankara, Turkey |
3. | University of Maryland, Baltimore County, Baltimore, MD, USA |
We address semigroup well-posedness of the fluid-structure interaction of a linearized compressible, viscous fluid and an elastic plate (in the absence of rotational inertia). Unlike existing work in the literature, we linearize the compressible Navier-Stokes equations about an arbitrary state (assuming the fluid is barotropic), and so the fluid PDE component of the interaction will generally include a nontrivial ambient flow profile $\mathbf{U}$. The appearance of this term introduces new challenges at the level of the stationary problem. In addition, the boundary of the fluid domain is unavoidably Lipschitz, and so the well-posedness argument takes into account the technical issues associated with obtaining necessary boundary trace and elliptic regularity estimates. Much of the previous work on flow-plate models was done via Galerkin-type constructions after obtaining good a priori estimates on solutions (specifically [
References:
[1] |
R. Aoyama and Y. Kagei,
Spectral properties of the semigroup for the linearized compressible Navier-Stokes equation around a parallel flow in a cylindrical domain, Advances in Differential Equations, 21 (2016), 265-300.
|
[2] |
J. P. Aubin, Applied Functional Analysis, John Wiley & Sons, New York-Chichester-Brisbane, 1979. |
[3] |
G. Avalos and F. Bucci, Exponential decay properties of a mathematical model for a certain fluid-structure interaction. In New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer International Publishing, (2014), 49-78. |
[4] |
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. |
[5] |
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. |
[6] |
G. Avalos and M. Dvorak,
A new maximality argument for a coupled fluid-structure interaction, with implications for a divergence free finite element method, Applicationes Mathematicae, 35 (2008), 259-280.
doi: 10.4064/am35-3-2. |
[7] |
G. Avalos and P. G. Geredeli, Exponential stability and supporting spectral analysis of a linearized compressible flow-structure PDE model, preprint, 2018. |
[8] |
G. Avalos and R. Triggiani,
The coupled PDE system arising in fluid-structure interaction, Part Ⅰ: Explicit semigroup generator and its spectral properties, Contemporary Mathematics, 440 (2007), 15-54.
|
[9] |
G. Avalos and R. Triggiani,
Semigroup wellposedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE of fluid-structure interactions, Discrete and Continuous Dynamical Systems, 2 (2009), 417-447.
doi: 10.3934/dcdss.2009.2.417. |
[10] |
R. L. Bisplinghoff and H. Ashley, Principles of Aeroelasticity, John Wiley and Sons, Inc., New York-London, 1962. |
[11] |
V. V. Bolotin, Nonconservative Problems of the Theory of Elastic Stability, Macmillan, 1963. |
[12] |
A. Buffa and G. Geymonat,
On traces of functions in $ W^{2, p}(Ω)$ for Lipschitz domains in R3, Comptes Rendus de l'Acad émie des Sciences-Series I-Mathematics, 332 (2001), 699-704.
doi: 10.1016/S0764-4442(01)01881-X. |
[13] |
A. Buffa, M. Costabel and D. Sheen,
On traces for $ \mathbf{H}(\text{curl}, Ω)$ in Lipschitz domains, Journal of Mathematical Analysis and Applications, 276 (2002), 845-867.
doi: 10.1016/S0022-247X(02)00455-9. |
[14] |
G. Chen,
Energy decay estimates and exact boundary-value controllability for the wave-equation in a bounded domain, Journal de Mathématiques Pures et Appliquées, 58 (1979), 249-273.
|
[15] |
A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, Texts in Applied Mathematics, 4. Springer-Verlag, New York, 1990. |
[16] |
I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999, (in Russian); English translation: 2002, Acta, Kharkov. |
[17] | |
[18] |
I. Chueshov,
Dynamics of a nonlinear elastic plate interacting with a linearized compressible viscous fluid, Nonlinear Analysis: Theory, Methods & Applications, 95 (2014), 650-665.
doi: 10.1016/j.na.2013.10.018. |
[19] |
I. Chueshov,
Interaction of an elastic plate with a linearized inviscid incompressible fluid, Communications on Pure & Applied Analysis, 13 (2014), 1459-1778.
doi: 10.3934/cpaa.2014.13.1759. |
[20] |
I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, New York: Springer, 2015. |
[21] |
I. Chueshov and T. Fastovska,
On interaction of circular cylindrical shells with a Poiseuille type flow, Evolution Equations & Control Theory, 5 (2016), 605-629.
doi: 10.3934/eect.2016021. |
[22] |
I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer-Verlag, 2010. |
[23] |
I. Chueshov, I. Lasiecka and J. T. Webster,
Evolution semigroups in supersonic flow-plate interactions, Journal of Differential Equations, 254 (2013), 1741-1773.
doi: 10.1016/j.jde.2012.11.009. |
[24] |
I. Chueshov, I. Lasiecka and J. T. Webster,
Attractors for delayed, nonrotational von Karman plates with applications to flow-structure interactions without any damping, Communications in Partial Differential Equations, 39 (2014), 1965-1997.
doi: 10.1080/03605302.2014.930484. |
[25] |
I. Chueshov, I. Lasiecka and J. T. Webster,
Flow-plate interactions: Well-posedness and long-time behavior, Discrete & Continuous Dynamical Systems-Series S, 7 (2014), 925-965.
doi: 10.3934/dcdss.2014.7.925. |
[26] |
I. Chueshov and I. Ryzhkova,
Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations, Journal of Differential Equations, 254 (2013), 1833-1862.
doi: 10.1016/j.jde.2012.11.006. |
[27] |
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. |
[28] |
I. Chueshov and I. Ryzhkova,
A global attractor for a fluid-plate interaction model, Communications on Pure & Applied Analysis, 12 (2013), 1635-1656.
|
[29] |
I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models, In IFIP Conference on System Modeling and Optimization, Springer Berlin Heidelberg, 391 (2013), 328-337. |
[30] |
H. B. da Veiga,
Stationary motions and incompressible limit for compressible viscous fluids, Houston Journal of Mathematics, 13 (1987), 527-544.
|
[31] |
E. Dowell,
A Modern Course in Aeroelasticity, Kluwer Academic Publishers, 2004. |
[32] |
P. G. Geredeli and J. T. Webster,
Qualitative results on the dynamics of a Berger plate with nonlinear boundary damping, Nonlinear Analysis: Real World Applications, 31 (2016), 227-256.
doi: 10.1016/j.nonrwa.2016.02.002. |
[33] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Society for Industrial and Applied Mathematics, 2011. |
[34] |
T. Kato, Perturbation Theory for Linear Operators, Band 132 Springer-Verlag New York, Inc., New York, 1966. |
[35] |
S. Kesavan, Topics in Functional Analysis and Applications, 1989. |
[36] |
I. Lasiecka and J. T. Webster,
Feedback stabilization of a fluttering panel in an inviscid subsonic potential flow, SIAM Journal on Mathematical Analysis, 48 (2016), 1848-1891.
doi: 10.1137/15M1040529. |
[37] |
P. D. Lax and R. S. Phillips,
Local boundary conditions for dissipative symmetric linear differential operators, Communications on Pure and Applied Mathematics, 13 (1960), 427-455.
doi: 10.1002/cpa.3160130307. |
[38] |
W. C. H. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge university press, 2000. |
[39] |
C. S. Morawetz, Energy identities for the wave equation, NYU Courant Institute, Math. Sci. Res. Rep. No., 1966. |
[40] |
J. Nečas, Direct Methods in the Theory of Elliptic Equations (translated by Gerard Tronel and Alois Kufner), Springer, New York, 2012. |
[41] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. |
[42] |
R. Triggiani,
Wave equation on a bounded domain with boundary dissipation: An operator approach, Journal of Mathematical Analysis and applications, 137 (1989), 438-461.
doi: 10.1016/0022-247X(89)90255-2. |
[43] |
A. Valli,
On the existence of stationary solutions to compressible Navier-Stokes equations, Annales de l'IHP Analyse non linéaire, 4 (1987), 99-113.
doi: 10.1016/S0294-1449(16)30374-2. |
[44] |
J. T. Webster,
Weak and strong solutions of a nonlinear subsonic flow-structure interaction: Semigroup approach, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 3123-3136.
doi: 10.1016/j.na.2011.01.028. |
show all references
References:
[1] |
R. Aoyama and Y. Kagei,
Spectral properties of the semigroup for the linearized compressible Navier-Stokes equation around a parallel flow in a cylindrical domain, Advances in Differential Equations, 21 (2016), 265-300.
|
[2] |
J. P. Aubin, Applied Functional Analysis, John Wiley & Sons, New York-Chichester-Brisbane, 1979. |
[3] |
G. Avalos and F. Bucci, Exponential decay properties of a mathematical model for a certain fluid-structure interaction. In New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer International Publishing, (2014), 49-78. |
[4] |
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. |
[5] |
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. |
[6] |
G. Avalos and M. Dvorak,
A new maximality argument for a coupled fluid-structure interaction, with implications for a divergence free finite element method, Applicationes Mathematicae, 35 (2008), 259-280.
doi: 10.4064/am35-3-2. |
[7] |
G. Avalos and P. G. Geredeli, Exponential stability and supporting spectral analysis of a linearized compressible flow-structure PDE model, preprint, 2018. |
[8] |
G. Avalos and R. Triggiani,
The coupled PDE system arising in fluid-structure interaction, Part Ⅰ: Explicit semigroup generator and its spectral properties, Contemporary Mathematics, 440 (2007), 15-54.
|
[9] |
G. Avalos and R. Triggiani,
Semigroup wellposedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE of fluid-structure interactions, Discrete and Continuous Dynamical Systems, 2 (2009), 417-447.
doi: 10.3934/dcdss.2009.2.417. |
[10] |
R. L. Bisplinghoff and H. Ashley, Principles of Aeroelasticity, John Wiley and Sons, Inc., New York-London, 1962. |
[11] |
V. V. Bolotin, Nonconservative Problems of the Theory of Elastic Stability, Macmillan, 1963. |
[12] |
A. Buffa and G. Geymonat,
On traces of functions in $ W^{2, p}(Ω)$ for Lipschitz domains in R3, Comptes Rendus de l'Acad émie des Sciences-Series I-Mathematics, 332 (2001), 699-704.
doi: 10.1016/S0764-4442(01)01881-X. |
[13] |
A. Buffa, M. Costabel and D. Sheen,
On traces for $ \mathbf{H}(\text{curl}, Ω)$ in Lipschitz domains, Journal of Mathematical Analysis and Applications, 276 (2002), 845-867.
doi: 10.1016/S0022-247X(02)00455-9. |
[14] |
G. Chen,
Energy decay estimates and exact boundary-value controllability for the wave-equation in a bounded domain, Journal de Mathématiques Pures et Appliquées, 58 (1979), 249-273.
|
[15] |
A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, Texts in Applied Mathematics, 4. Springer-Verlag, New York, 1990. |
[16] |
I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999, (in Russian); English translation: 2002, Acta, Kharkov. |
[17] | |
[18] |
I. Chueshov,
Dynamics of a nonlinear elastic plate interacting with a linearized compressible viscous fluid, Nonlinear Analysis: Theory, Methods & Applications, 95 (2014), 650-665.
doi: 10.1016/j.na.2013.10.018. |
[19] |
I. Chueshov,
Interaction of an elastic plate with a linearized inviscid incompressible fluid, Communications on Pure & Applied Analysis, 13 (2014), 1459-1778.
doi: 10.3934/cpaa.2014.13.1759. |
[20] |
I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, New York: Springer, 2015. |
[21] |
I. Chueshov and T. Fastovska,
On interaction of circular cylindrical shells with a Poiseuille type flow, Evolution Equations & Control Theory, 5 (2016), 605-629.
doi: 10.3934/eect.2016021. |
[22] |
I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer-Verlag, 2010. |
[23] |
I. Chueshov, I. Lasiecka and J. T. Webster,
Evolution semigroups in supersonic flow-plate interactions, Journal of Differential Equations, 254 (2013), 1741-1773.
doi: 10.1016/j.jde.2012.11.009. |
[24] |
I. Chueshov, I. Lasiecka and J. T. Webster,
Attractors for delayed, nonrotational von Karman plates with applications to flow-structure interactions without any damping, Communications in Partial Differential Equations, 39 (2014), 1965-1997.
doi: 10.1080/03605302.2014.930484. |
[25] |
I. Chueshov, I. Lasiecka and J. T. Webster,
Flow-plate interactions: Well-posedness and long-time behavior, Discrete & Continuous Dynamical Systems-Series S, 7 (2014), 925-965.
doi: 10.3934/dcdss.2014.7.925. |
[26] |
I. Chueshov and I. Ryzhkova,
Unsteady interaction of a viscous fluid with an elastic shell modeled by full von Karman equations, Journal of Differential Equations, 254 (2013), 1833-1862.
doi: 10.1016/j.jde.2012.11.006. |
[27] |
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. |
[28] |
I. Chueshov and I. Ryzhkova,
A global attractor for a fluid-plate interaction model, Communications on Pure & Applied Analysis, 12 (2013), 1635-1656.
|
[29] |
I. Chueshov and I. Ryzhkova, Well-posedness and long time behavior for a class of fluid-plate interaction models, In IFIP Conference on System Modeling and Optimization, Springer Berlin Heidelberg, 391 (2013), 328-337. |
[30] |
H. B. da Veiga,
Stationary motions and incompressible limit for compressible viscous fluids, Houston Journal of Mathematics, 13 (1987), 527-544.
|
[31] |
E. Dowell,
A Modern Course in Aeroelasticity, Kluwer Academic Publishers, 2004. |
[32] |
P. G. Geredeli and J. T. Webster,
Qualitative results on the dynamics of a Berger plate with nonlinear boundary damping, Nonlinear Analysis: Real World Applications, 31 (2016), 227-256.
doi: 10.1016/j.nonrwa.2016.02.002. |
[33] |
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Society for Industrial and Applied Mathematics, 2011. |
[34] |
T. Kato, Perturbation Theory for Linear Operators, Band 132 Springer-Verlag New York, Inc., New York, 1966. |
[35] |
S. Kesavan, Topics in Functional Analysis and Applications, 1989. |
[36] |
I. Lasiecka and J. T. Webster,
Feedback stabilization of a fluttering panel in an inviscid subsonic potential flow, SIAM Journal on Mathematical Analysis, 48 (2016), 1848-1891.
doi: 10.1137/15M1040529. |
[37] |
P. D. Lax and R. S. Phillips,
Local boundary conditions for dissipative symmetric linear differential operators, Communications on Pure and Applied Mathematics, 13 (1960), 427-455.
doi: 10.1002/cpa.3160130307. |
[38] |
W. C. H. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge university press, 2000. |
[39] |
C. S. Morawetz, Energy identities for the wave equation, NYU Courant Institute, Math. Sci. Res. Rep. No., 1966. |
[40] |
J. Nečas, Direct Methods in the Theory of Elliptic Equations (translated by Gerard Tronel and Alois Kufner), Springer, New York, 2012. |
[41] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. |
[42] |
R. Triggiani,
Wave equation on a bounded domain with boundary dissipation: An operator approach, Journal of Mathematical Analysis and applications, 137 (1989), 438-461.
doi: 10.1016/0022-247X(89)90255-2. |
[43] |
A. Valli,
On the existence of stationary solutions to compressible Navier-Stokes equations, Annales de l'IHP Analyse non linéaire, 4 (1987), 99-113.
doi: 10.1016/S0294-1449(16)30374-2. |
[44] |
J. T. Webster,
Weak and strong solutions of a nonlinear subsonic flow-structure interaction: Semigroup approach, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 3123-3136.
doi: 10.1016/j.na.2011.01.028. |

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