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Semigroup well-posedness of a linearized, compressible fluid with an elastic boundary

  • * Corresponding author:Justin T. Webster

    * Corresponding author:Justin T. Webster

In memory of Igor D. Chueshov

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
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  • 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.

    Mathematics Subject Classification: Primary: 34A12, 74F10; Secondary: 35Q35, 76N10.


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  • Figure 1.  The Fluid-Structure Geometry

  •   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. 
      J. P. Aubin, Applied Functional Analysis, John Wiley & Sons, New York-Chichester-Brisbane, 1979.
      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.
      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.
      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.
      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.
      G. Avalos and P. G. Geredeli, Exponential stability and supporting spectral analysis of a linearized compressible flow-structure PDE model, preprint, 2018.
      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. 
      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.
      R. L. Bisplinghoff and H. Ashley, Principles of Aeroelasticity, John Wiley and Sons, Inc., New York-London, 1962.
      V. V. Bolotin, Nonconservative Problems of the Theory of Elastic Stability, Macmillan, 1963.
      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.
      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.
      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. 
      A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, Texts in Applied Mathematics, 4. Springer-Verlag, New York, 1990.
      I. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, Kharkov, 1999, (in Russian); English translation: 2002, Acta, Kharkov.
      I. Chueshov, Personal communication, 2013.
      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.
      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.
      I. Chueshov, Dynamics of Quasi-Stable Dissipative Systems, New York: Springer, 2015.
      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.
      I. Chueshov and I. Lasiecka, Von Karman Evolution Equations, Springer-Verlag, 2010.
      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.
      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.
      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.
      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.
      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.
      I. Chueshov  and  I. Ryzhkova , A global attractor for a fluid-plate interaction model, Communications on Pure & Applied Analysis, 12 (2013) , 1635-1656. 
      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.
      H. B. da Veiga , Stationary motions and incompressible limit for compressible viscous fluids, Houston Journal of Mathematics, 13 (1987) , 527-544. 
      E. Dowell, A Modern Course in Aeroelasticity, Kluwer Academic Publishers, 2004.
      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.
      P. Grisvard, Elliptic Problems in Nonsmooth Domains, Society for Industrial and Applied Mathematics, 2011.
      T. Kato, Perturbation Theory for Linear Operators, Band 132 Springer-Verlag New York, Inc., New York, 1966.
      S. Kesavan, Topics in Functional Analysis and Applications, 1989.
      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.
      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.
      W. C. H. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge university press, 2000.
      C. S. Morawetz, Energy identities for the wave equation, NYU Courant Institute, Math. Sci. Res. Rep. No., 1966.
      J. Nečas, Direct Methods in the Theory of Elliptic Equations (translated by Gerard Tronel and Alois Kufner), Springer, New York, 2012.
      A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983.
      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.
      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.
      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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