Chueshov / Lasiecka Von Karman Evolution Equations

Well-posedness and Long Time Dynamics
1. Auflage 2010
ISBN: 978-0-387-87712-9
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 770 Seiten, eBook

Reihe: Springer Monographs in Mathematics

In the study of mathematical models that arise in the context of concrete - plications, the following two questions are of fundamental importance: (i) we- posedness of the model, including existence and uniqueness of solutions; and (ii) qualitative properties of solutions. A positive answer to the ?rst question, - ing of prime interest on purely mathematical grounds, also provides an important test of the viability of the model as a description of a given physical phenomenon. An answer or insight to the second question provides a wealth of information about the model, hence about the process it describes. Of particular interest are questions related to long-time behavior of solutions. Such an evolution property cannot be v- i?ed empirically, thus any in a-priori information about the long-time asymptotics can be used in predicting an ultimate long-time response and dynamical behavior of solutions. In recent years, this set of investigations has attracted a great deal of attention. Consequent efforts have then resulted in the creation and infusion of new methods and new tools that have been responsible for carrying out a successful an- ysis of long-time behavior of several classes of nonlinear PDEs.

Zielgruppe

Research

Autoren/Hrsg.

Chueshov, Igor

Lasiecka, Irena

Inhaltsverzeichnis

Well-Posedness.- Preliminaries.- Evolutionary Equations.- Von Karman Models with Rotational Forces.- Von Karman Equations Without Rotational Inertia.- Thermoelastic Plates.- Structural Acoustic Problems and Plates in a Potential Flow of Gas.- Long-Time Dynamics.- Attractors for Evolutionary Equations.- Long-Time Behavior of Second-Order Abstract Equations.- Plates with Internal Damping.- Plates with Boundary Damping.- Thermoelasticity.- Composite Wave–Plate Systems.- Inertial Manifolds for von Karman Plate Equations.

Leseproben

"Chapter 13 Inertial Manifolds for von Karman Plate Equations (p. 695-696)

One of the contemporary approaches to the study of long-time behavior of infinitedimensional dynamical systems is based on the concept of inertial manifolds which was introduced in [117] (see also the monographs [61, 90, 273] and the references therein and also Section 7.6 in Chapter 7). These manifolds are finite-dimensional invariant surfaces that contain global attractors and attract trajectories exponentially fast. Moreover, there is a possibility to reduce the study of limit regimes of the original infinite-dimensional system to solving similar problem for a class of ordinary differential equations. Inertial manifolds are generalizations of center-unstable manifolds and are convenient objects to capture the long-time behavior of dynamical systems.

The theory of inertial manifolds is related to the method of integral manifolds (see, e.g., [92, 139, 233]), and has been developed and widely studied for deterministic systems by many authors. All known results concerning existence of inertial manifolds require some gap condition on the spectrum of the linearized problem (see, e.g., [45, 50, 61, 90, 227, 236, 273] and the references therein). Although inertial manifolds have been mainly studied for parabolic-like equations, there are some results for damped second order in time evolution equations arising in nonlinear oscillations theory (see, e.g., [45, 50, 61, 236]).

These results rely on the approach originally developed in [236] for a one-dimensional semilinear wave equation and require the damping coefficient to be large enough. In fact, as indicated in [236], this requirement is a necessary condition in the case of hyperbolic flows. The goal of this chapter is to provide some results on existence and properties of inertial manifolds for several models of nonlinear dynamic elasticity governed by von Karman evolution equations subject to either mechanical or thermal dissipation.

The presentation below mainly follows the paper [66]. We consider three different dissipative mechanisms: viscous damping, strong structural damping (mechanical dampings), and thermal damping. Von Karman equations with viscous damping retain hyperbolic-like properties of the dynamics, whereas structural damping and thermal damping have recently been shown [216] (see also Section 5.3.2 in Chapter 5) to be related to analyticity of the semigroup generated by the linear part of the dynamics. It is thus expected that the results obtained depend heavily on the type of dissipation.

Our main results, formulated in Section 13.3, provide conditions for existence of inertial manifolds for all three models. These conditions are derived from more general results presented in Section 7.6, Theorem 7.6.3, where the main assumption is certain gap condition. Gap condition, when specialized to the concrete models considered, imposes geometric restrictions on spatial domains along with some restrictions imposed on the damping parameter. This latter constraint is essential only in the hyperbolic case. Indeed, in the hyperbolic case (viscous damping), Theorems 13.3.5 and 13.3.6 require sufficiently large values of the damping parameter. In the analytic-like case (structural damping), instead, Theorem 13.3.12 does not require large values of damping. A similar situation takes place in the thermoelastic case; see Theorem 13.3.16."