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Long time behaviour of strong solutions to interactive fluid-plate system without rotational inertia
May  2018, 23(3): 1267-1295. doi: 10.3934/dcdsb.2018151

## 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

* Corresponding author:Justin T. Webster

In memory of Igor D. Chueshov

Received  April 2017 Revised  September 2017 Published  May 2018 Early access  February 2018

Fund Project: The research of G. Avalos was partially supported by the NSF Grants DMS-1211232 and DMS-1616425. The research of J.T. Webster was partially supported by the NSF Grant DMS-1504697.

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 [18]-the work most pertinent to ours here); in contrast, we adopt here a Lumer-Phillips approach, with a view of associating solutions of the fluid-structure dynamics with a $C_{0}$-semigroup ${{\left\{ {{e}^{\mathcal{A}t}} \right\}}_{t\ge 0}}$ on the natural finite energy space of initial data. So, given this approach, the major challenge in our work becomes establishing the maximality of the operator $\mathcal{A}$ that models the fluid-structure dynamics. In sum: our main result is semigroup well-posedness for the fully coupled fluid-structure dynamics, under the assumption that the ambient flow field $\mathbf{U}∈ \mathbf{H}^{3}(\mathcal{O})$ has zero normal component trace on the boundary (a standard assumption with respect to the literature). In the final sections we address well-posedness of the system in the presence of the von Karman plate nonlinearity, as well as the stationary problem associated to the dynamics.

Citation: George Avalos, Pelin G. Geredeli, Justin T. Webster. Semigroup well-posedness of a linearized, compressible fluid with an elastic boundary. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1267-1295. doi: 10.3934/dcdsb.2018151
##### 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] I. Chueshov, Personal communication, 2013. [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] I. Chueshov, Personal communication, 2013. [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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