Generation of bounded semigroups in nonlinear subsonic flow–structure interactions with boundary dissipation

We consider a subsonic flow–structure interaction describing the flow of gas above a flexible plate. A perturbed wave equation describes the flow, and a second-order nonlinear plate equation describes the plate's displacement. We consider the model that accounts for rotational inertia in the plate, parametrized by γ  ≥ 0. It is known that the presence of γ  >  0 has strong effect on regularity properties of the plate, which then allows one to establish well-posedness of finite energy solutions for the entire structure. In this paper, it is shown that semigroup well-posedness of the model is not only preserved for all γ  ≥ 0 but that the corresponding nonlinear semigroups Sγ(t) converge to S0(t) when γ   0. The above result holds also in the presence of nonlinear boundary damping. In addition, we provide a discussion of the regularity of strong solutions. Copyright © 2011 John Wiley & Sons, Ltd.

[1]  E. Dowell.  Nonlinear oscillations of a fluttering plate. II. , 1966 .

[2]  I. Lasiecka.  Mathematical control theory of couple PDEs , 2002 .

[3]  Irena Lasiecka,et al.  Sharp regularity theory for second order hyperbolic equations of Neumann type , 1990 .

[4]  A. V. Balakrishnan,et al.  Nonlinear Aeroelasticity: Continuum Theory, Flutter/Divergence Speed, and Plate Wing Model , 2006 .

[5]  Non-linear oscillations of a plate in a flow of gas , 1996 .

[6]  Justin T. Webster,et al.  Weak and strong solutions of a nonlinear subsonic flowstructure interaction: Semigroup approach , 2011 .

[7]  S. Miyatake.  Mixed problem for hyperbolic equation of second order , 1973 .

[8]  Igor Chueshov,et al.  Von Karman Evolution Equations , 2010 .

[9]  R. Showalter.  Monotone operators in Banach space and nonlinear partial differential equations , 1996 .

[10]  Irena Lasiecka,et al.  Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary , 1989 .

[11]  Igor Chueshov,et al.  The problem on interaction of von Karman plate with subsonic flow of gas , 1999 .

[12]  Robert H. Scanlan,et al.  A Modern Course in Aeroelasticity , 1981, Solid Mechanics and Its Applications.

[13]  R. Triggiani,et al.  Trace regularity of the solutions of the wave equation with homogeneous Neumann boundary conditions and data supported away from the boundary , 1989 .

[14]  I. Lasiecka,et al.  ON THE ATTRACTOR FOR A SEMILINEAR WAVE EQUATION WITH CRITICAL EXPONENT AND NONLINEAR BOUNDARY DISSIPATION , 2002 .

[15]  F. Bucci,et al.  Finite-dimensional attractor for a composite system of wave/plate equations with localized damping , 2009, 0912.5464.

[16]  K. W. Iliff,et al.  Continuum Aeroelastic Model for Inviscid Subsonic Bending-Torsion Wing Flutter , 2007 .

[17]  Iryna Ryzhkova,et al.  Dynamics of a thermoelastic von Kármán plate in a subsonic gas flow , 2007 .

[18]  Irena Lasiecka,et al.  Global existence, uniqueness and regularity of solutions to a von Kármán system with nonlinear boundary dissipation , 1996, Differential and Integral Equations.

[19]  Irina Ryzhkova,et al.  Stabilization of von Kármán plate in the presence of thermal effects in a subsonic potential flow of gas , 2004 .

[20]  J. Lions,et al.  Non homogeneous boundary value problems for second order hyperbolic operators , 1986 .

[21]  R. Temam.  Infinite Dimensional Dynamical Systems in Mechanics and Physics Springer Verlag , 1993 .

[22]  K. Zaman,et al.  Nonlinear oscillations of a fluttering plate. , 1966 .

[23]  Y. Egorov,et al.  Mixed Problems for Hyperbolic Equations , 1994 .

[24]  P. G. Ciarlet,et al.  Les équations de Von Kármán , 1980 .

[25]  Melvyn S. Berger,et al.  On von kármán's equations and the buckling of a thin elastic plate, I the clamped plate , 1967 .

[26]  J. Goldstein.  Semigroups of Linear Operators and Applications , 1985 .

[27]  Irena Lasiecka,et al.  Asymptotic Behavior with Respect to Thickness of Boundary Stabilizing Feedback for the Kirchoff Plate , 1994 .

[28]  Paul C. Fife,et al.  On von Karman's equations and the buckling of a thin elastic plate , 1966 .

[29]  V. Barbu.  Nonlinear Semigroups and di erential equations in Banach spaces , 1976 .