Chueshov / Lasiecka Von Karman Evolution Equations

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

E-Book, Englisch, 770 Seiten

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.

Autoren/Hrsg.

Chueshov, Igor

Lasiecka, Irena

Inhaltsverzeichnis

1;Contents;6
3;Introduction;16
3.1;0.1 Von Karman evolutions;17
3.2;0.2 Sources;1
3.3;0.3 Damping;1
3.4;0.4 Goals;20
3.5;0.5 Brief outline of the book;21
4;Part I Well-Posedness;25
4.1;1 Preliminaries;26
4.1.1;1.1 Function spaces and embedding theorems;26
4.1.1.1;1.1.1 Sobolev spaces;26
4.1.1.2;1.1.2 Besov spaces;28
4.1.1.3;1.1.3 Lizorkin and real Hardy spaces;29
4.1.1.4;1.1.4 Vector-valued spaces;32
4.1.2;1.2 Nonlinear operators and related operator equations;33
4.1.2.1;1.2.1 Monotone and pseudomonotone operators;33
4.1.2.2;1.2.2 Proper Fredholm operators;37
4.1.3;1.3 Biharmonic operator;39
4.1.3.1;1.3.1 Clamped (Dirichlet) boundary conditions;42
4.1.3.2;1.3.2 Hinged boundary conditions;44
4.1.3.3;1.3.3 Simply supported (hinged revisited) boundary conditions;46
4.1.3.4;1.3.4 Free-type boundary conditions;47
4.1.3.5;1.3.5 Mixed boundary conditions;49
4.1.4;1.4 Properties of the von Karman bracket;51
4.1.5;1.5 Stationary von Karman equations;58
4.1.5.1;1.5.1 Clamped and hinged boundary conditions;62
4.1.5.2;1.5.2 General mixed boundary conditions;65
4.1.5.3;1.5.3 Modified mixed boundary conditions;69
4.2;2 Evolutionary Equations;72
4.2.1;2.1 Overview;72
4.2.2;2.2 Accretive operators in Hilbert spaces;72
4.2.3;2.3 Abstract differential equations;74
4.2.4;2.4 Second-order abstract equations;80
4.2.4.1;2.4.1 General model;80
4.2.4.2;2.4.2 Simplified nonlinear model;90
4.2.4.3;2.4.3 Linear nonhomogeneous problem;97
4.2.4.4;2.4.4 On higher regularity of solutions;112
4.2.5;2.5 Linear plate models;115
4.2.5.1;2.5.1 Homogeneous boundary conditions;115
4.2.5.2;2.5.2 Nonhomogeneous boundary conditions. Regularity theory;129
4.3;3 Von Karman Models with Rotational Forces;142
4.3.1;3.1 Well-posedness for models with internal dissipation;143
4.3.1.1;3.1.1 Clamped boundary condition;145
4.3.1.2;3.1.2 Hinged boundary conditions;156
4.3.1.3;3.1.3 Boundary conditions of the free type;160
4.3.1.4;3.1.4 Regular solutions;166
4.3.2;3.2 Well-posedness in the case of nonlinear boundary dissipation;170
4.3.2.1;3.2.1 Clamped--hinged boundary conditions;172
4.3.2.2;3.2.2 Clamped--free boundary conditions;183
4.3.2.3;3.2.3 Regular solutions;199
4.3.3;3.3 Other models with rotational inertia;201
4.3.3.1;3.3.1 Models with delay;202
4.3.3.2;3.3.2 Models with memory;205
4.3.3.3;3.3.3 Quasi-static model;206
4.4;4 Von Karman Equations Without Rotational Inertia;208
4.4.1;4.1 Models with interior dissipation;208
4.4.1.1;4.1.1 Clamped boundary conditions;210
4.4.1.2;4.1.2 Hinged boundary conditions;215
4.4.1.3;4.1.3 Free boundary conditions;217
4.4.1.4;4.1.4 Weak solutions;224
4.4.1.5;4.1.5 Regular solutions;228
4.4.1.6;4.1.6 On a model with delay;234
4.4.2;4.2 Models with nonlinear boundary dissipation;235
4.4.2.1;4.2.1 Clamped--hinged boundary conditions;236
4.4.2.2;4.2.2 Clamped--free boundary conditions;243
4.4.3;4.3 Quasi-static model with clamped boundary condition;251
4.5;5 Thermoelastic Plates;256
4.5.1;5.1 PDE model;256
4.5.2;5.2 Abstract formulation;258
4.5.3;5.3 Linear problem;259
4.5.3.1;5.3.1 Generation of strongly continuous semigroup;260
4.5.3.2;5.3.2 Analyticity of the semigroup for the model without rotational inertia;263
4.5.4;5.4 Generation of a nonlinear semigroup;270
4.5.5;5.5 Regularity of the semiflow;274
4.5.6;5.6 Backward uniqueness of the semiflow;277
4.5.7;5.7 Stationary solutions;284
4.6;6 Structural Acoustic Problems and Plates in a Potential Flow of Gas;286
4.6.1;6.1 Introduction;286
4.6.2;6.2 Structural acoustic problem;289
4.6.2.1;6.2.1 Description of the model;289
4.6.2.2;6.2.2 Basic assumption;290
4.6.2.3;6.2.3 Abstract formulation;291
4.6.2.4;6.2.4 Well-posedness;294
4.6.3;6.3 Coupled wave and thermoelastic plate equations;300
4.6.3.1;6.3.1 Description of the model;300
4.6.3.2;6.3.2 Abstract formulation;301
4.6.3.3;6.3.3 Well-posedness;303
4.6.4;6.4 Plates in a flow of gas: Description of the model;306
4.6.5;6.5 Plates in a flow of gas: Subsonic case;309
4.6.5.1;6.5.1 The statement of the main results;310
4.6.5.2;6.5.2 Preliminaries and abstract setting;312
4.6.5.3;6.5.3 Galerkin approximations;315
4.6.5.4;6.5.4 Strong solutions---Proof of Part I of Theorem 6.5.2;318
4.6.5.5;6.5.5 Generalized and weak solutions---Proof of Part II of Theorem 6.5.2;321
4.6.5.6;6.5.6 Stationary solutions;323
4.6.6;6.6 Plates with rotational inertia in both subsonic and supersonic gas flow cases;325
4.6.6.1;6.6.1 The statement of the main results;326
4.6.6.2;6.6.2 Flow potentials with given boundary conditions;327
4.6.6.3;6.6.3 Construction of approximate solutions;336
4.6.6.4;6.6.4 Limit transition;342
4.6.6.5;6.6.5 Reduced retarded problem;345
5;Part II Long-Time Dynamics;348
5.1;7 Attractors for Evolutionary Equations;349
5.1.1;7.1 Dissipative dynamical systems;349
5.1.2;7.2 Global attractors;356
5.1.3;7.3 Dimension of global attractors;361
5.1.4;7.4 Fractal exponential attractors (inertial sets);367
5.1.5;7.5 Gradient systems;371
5.1.5.1;7.5.1 Geometric structure of the attractor;372
5.1.5.2;7.5.2 Rate of convergence to global attractors;375
5.1.6;7.6 General idea about inertial manifolds;378
5.1.7;7.7 Approximate inertial manifolds;380
5.1.7.1;7.7.1 The main assumptions;380
5.1.7.2;7.7.2 Construction of approximate inertial manifolds;381
5.1.7.3;7.7.3 Nonlinear Galerkin method;383
5.1.8;7.8 General idea about determining functionals;385
5.1.8.1;7.8.1 Concept of a set of determining functionals;385
5.1.8.2;7.8.2 Completeness defect of a set of functionals;387
5.1.8.3;7.8.3 Estimates for completeness defect in Sobolev spaces;389
5.1.8.4;7.8.4 Existence of determining functionals;391
5.1.9;7.9 Stabilizability estimate and its consequences;393
5.1.9.1;7.9.1 Finite dimension of global attractors;396
5.1.9.2;7.9.2 Regularity of trajectories from the attractor;398
5.1.9.3;7.9.3 Fractal exponential attractors;399
5.1.9.4;7.9.4 Determining functionals;401
5.2;8 Long-Time Behavior of Second-Order Abstract Equations;403
5.2.1;8.1 Main assumptions;403
5.2.2;8.2 Dissipativity;406
5.2.3;8.3 Existence of global attractors;409
5.2.3.1;8.3.1 Preliminary inequalities;409
5.2.3.2;8.3.2 Main results on asymptotic smoothness;412
5.2.4;8.4 Regular attractors. Rate of stabilization to equilibria;419
5.2.5;8.5 Stabilizability and quasi-stability estimates;422
5.2.5.1;8.5.1 Basic theorem on quasi-stability;422
5.2.5.2;8.5.2 Sufficient conditions for quasi-stability;427
5.2.6;8.6 Finite dimension of global attractors;435
5.2.7;8.7 Regularity of elements from attractors;437
5.2.8;8.8 On ``strong" attractors;442
5.2.9;8.9 Determining functionals;445
5.2.9.1;8.9.1 An approach based on stabilizability estimate;446
5.2.9.2;8.9.2 Energy approach;448
5.2.10;8.10 Exponential fractal attractors;455
5.2.11;8.11 Approximate inertial manifolds;456
5.3;9 Plates with Internal Damping;459
5.3.1;9.1 Existence of global attractors for von Karman model with rotational forces;459
5.3.1.1;9.1.1 Clamped boundary condition;462
5.3.1.2;9.1.2 Hinged or simply supported boundary conditions;468
5.3.1.3;9.1.3 Free boundary conditions;469
5.3.1.4;9.1.4 Mixed boundary conditions;473
5.3.2;9.2 Further properties of the attractor for von Karman model with rotational inertia;475
5.3.2.1;9.2.1 Regular structure of the attractor;475
5.3.2.2;9.2.2 Finite dimension;479
5.3.2.3;9.2.3 Smoothness of elements from the attractor;481
5.3.2.4;9.2.4 Strong attractors;484
5.3.2.5;9.2.5 Exponential attractor;485
5.3.2.6;9.2.6 Determining functionals;485
5.3.2.7;9.2.7 Approximate inertial manifolds;488
5.3.3;9.3 Attractors for other models with rotational inertia;489
5.3.3.1;9.3.1 Von Karman equations with retarded terms;489
5.3.3.2;9.3.2 Quasi-static version of von Karman equations;495
5.3.4;9.4 Global attractors for von Karman model without rotational inertia;500
5.3.4.1;9.4.1 Clamped boundary condition;503
5.3.4.2;9.4.2 Hinged boundary conditions;510
5.3.4.3;9.4.3 Free boundary conditions;511
5.3.5;9.5 Further properties of the attractor for von Karman model without rotational inertia;523
5.3.5.1;9.5.1 Regular structure of the attractor;523
5.3.5.2;9.5.2 Smoothness of elements from the attractor;526
5.3.5.3;9.5.3 Strong attractors;535
5.3.5.4;9.5.4 Exponential attractor;538
5.3.5.5;9.5.5 Upper semicontinuity of the global attractor with respect to rotational inertia;539
5.3.5.6;9.5.6 Determining functionals;541
5.3.6;9.6 Global attractor for quasi-static model;546
5.3.6.1;9.6.1 The existence of attractor for quasi-static problem;547
5.3.6.2;9.6.2 Upper semicontinuity of the attractor to quasi-static problem;548
5.4;10 Plates with Boundary Damping;551
5.4.1;10.1 Introduction: Overview;551
5.4.2;10.2 Global attractors for von Karman models with rotational forces and with dissipation in free boundary conditions;554
5.4.2.1;10.2.1 The model and the main result on the existence of compact attractors;554
5.4.2.2;10.2.2 Asymptotic smoothness;560
5.4.2.3;10.2.3 Proof of the main result on attractors (Theorem 10.2.11) ;568
5.4.2.4;10.2.4 Rate of convergence to the equilibria;574
5.4.2.5;10.2.5 Determining functionals;581
5.4.3;10.3 Global attractors for von Karman models with rotational forces and with dissipation in hinged boundary conditions;582
5.4.3.1;10.3.1 The model and the main result;583
5.4.3.2;10.3.2 Asymptotic smoothness;586
5.4.3.3;10.3.3 Proof of the main result on attractors (Theorem 10.3.5);592
5.4.3.4;10.3.4 Rate of convergence to equilibria;595
5.4.3.5;10.3.5 Determining functionals;595
5.4.4;10.4 Global attractors for von Karman plates without rotational inertia and with dissipation in free boundary conditions ;596
5.4.4.1;10.4.1 The model and the main results;597
5.4.4.2;10.4.2 Asymptotic smoothness;604
5.4.4.3;10.4.3 Global attractor: Proof of Theorem 10.4.7.;611
5.4.4.4;10.4.4 Rate of stabilization: Proof of Theorem 10.4.10;616
5.4.5;10.5 Global attractors for von Karman plates without rotational inertia and with dissipation acting via hinged boundary conditions ;623
5.4.5.1;10.5.1 The model and the main results on the existence of attractors;623
5.4.5.2;10.5.2 Asymptotic smoothness;627
5.4.5.3;10.5.3 Proof of the main result (Theorem 10.5.7);632
5.5;11 Thermoelasticity;637
5.5.1;11.1 Introduction;637
5.5.2;11.2 Statements of main results;641
5.5.3;11.3 Uniform stabilizability inequality;643
5.5.4;11.4 Existence and properties of the attractor---Proof of Theorem 11.2.1 ;650
5.5.4.1;11.4.1 Existence of the attractor;650
5.5.4.2;11.4.2 Smoothness of the attractor---Proof of regularity in Theorem 11.2.1 ;651
5.5.4.3;11.4.3 Finite-dimensionality;654
5.5.4.4;11.4.4 Upper semicontinuity;658
5.5.5;11.5 Exponential rate of attraction---Proof of Theorem 11.2.2 ;660
5.6;12 Composite Wave--Plate Systems;664
5.6.1;12.1 Introduction;664
5.6.2;12.2 Structural acoustic problems;664
5.6.2.1;12.2.1 The statement of main results;666
5.6.2.2;12.2.2 Main inequality;669
5.6.2.3;12.2.3 Asymptotic smoothness;672
5.6.2.4;12.2.4 Stabilizability estimate;677
5.6.2.5;12.2.5 Additional properties of the attractor;682
5.6.2.6;12.2.6 Generalizations;683
5.6.3;12.3 Wave coupled to thermoelastic plate equation;684
5.6.3.1;12.3.1 The statement of main results;686
5.6.3.2;12.3.2 Main inequality;689
5.6.3.3;12.3.3 Asymptotic smoothness and proof of Theorem 12.3.3;692
5.6.3.4;12.3.4 Stabilizability estimate;694
5.6.3.5;12.3.5 Proofs of Theorem 12.3.5 and Theorem 12.3.7;697
5.6.4;12.4 Gas flow problems;698
5.6.4.1;12.4.1 Stabilization to a finite-dimensional set;699
5.6.4.2;12.4.2 Stabilization to equilibria;702
5.7;13 Inertial Manifolds for von Karman Plate Equations;706
5.7.1;13.1 Preliminaries;707
5.7.1.1;13.1.1 The models considered;707
5.7.1.2;13.1.2 Generation of nonlinear semigroups;709
5.7.1.3;13.1.3 Absorbing sets;711
5.7.2;13.2 Inertial manifolds for evolution equations;714
5.7.3;13.3 Inertial manifolds for second order in time evolution equation;722
5.7.3.1;13.3.1 Second-order evolutions with viscous damping;722
5.7.3.2;13.3.2 Second-order evolution equation with strong damping;727
5.7.3.3;13.3.3 Thermoelastic von Karman evolutions;731
5.8;A Jacobians and Compensated Compactness, Compactness of Vector Functions, and Sedenko's Method for Uniqueness;736
5.8.1;A.1 Jacobian regularity and compensated compactness;736
5.8.2;A.2 Compactness theorem for vector-valued functions;738
5.8.3;A.3 Logarithmic Sobolev-type inequalities and uniqueness of weak solutions by Sedenko's method;741
5.9;B Some Auxiliary Facts;752
5.9.1;B.1 Estimates for monotone functions;752
5.9.2;B.2 Concave bounds;754
5.9.3;B.3 Equation describing the convergence rates for the energy;756
5.9.4;B.4 Some convergence theorems for measurable functions;758
6;References;760
7;Index;772

Leseproben

"(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."