Weak and strong solutions of a nonlinear subsonic flowstructure interaction: Semigroup approach

Abstract We consider a non-rotational, 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. It is shown that the linearization of the model generates a strongly continuous semigroup with respect to the topology generated by “finite energy” considerations. An interesting feature of the problem is that linear perturbed flow–structure interaction is not monotone with respect to the standard norm describing the finite energy space. The main tool used in overcoming this difficulty is the construction of a suitable inner product on the finite energy space which allows the application of ω -maximal monotone operator theory. The obtained result allows us to employ suitable perturbation theory in order to discuss well-posedness of weak and strong solutions corresponding to several classes of nonlinear dynamics including the full flow–structure interaction with von Karman, Berger’s and other semilinear plate equations.

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

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

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

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

[5]  Dewey H. Hodges,et al.  Flight Dynamics of Highly Flexible Flying Wings , 2006 .

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

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

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

[9]  I. Lasiecka,et al.  Long-time Behavior of Second Order Evolution Equations With Nonlinear Damping , 2008 .

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

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

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

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

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

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

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

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

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

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