Von Karman Evolution Equations (eBook)

Contents 6
Introduction 16
0.1 Von Karman evolutions 17
0.2 Sources 1
0.3 Damping 1
0.4 Goals 20
0.5 Brief outline of the book 21
Part I Well-Posedness 25
1 Preliminaries 26
1.1 Function spaces and embedding theorems 26
1.1.1 Sobolev spaces 26
1.1.2 Besov spaces 28
1.1.3 Lizorkin and real Hardy spaces 29
1.1.4 Vector-valued spaces 32
1.2 Nonlinear operators and related operator equations 33
1.2.1 Monotone and pseudomonotone operators 33
1.2.2 Proper Fredholm operators 37
1.3 Biharmonic operator 39
1.3.1 Clamped (Dirichlet) boundary conditions 42
1.3.2 Hinged boundary conditions 44
1.3.3 Simply supported (hinged revisited) boundary conditions 46
1.3.4 Free-type boundary conditions 47
1.3.5 Mixed boundary conditions 49
1.4 Properties of the von Karman bracket 51
1.5 Stationary von Karman equations 58
1.5.1 Clamped and hinged boundary conditions 62
1.5.2 General mixed boundary conditions 65
1.5.3 Modified mixed boundary conditions 69
2 Evolutionary Equations 72
2.1 Overview 72
2.2 Accretive operators in Hilbert spaces 72
2.3 Abstract differential equations 74
2.4 Second-order abstract equations 80
2.4.1 General model 80
2.4.2 Simplified nonlinear model 90
2.4.3 Linear nonhomogeneous problem 97
2.4.4 On higher regularity of solutions 112
2.5 Linear plate models 115
2.5.1 Homogeneous boundary conditions 115
2.5.2 Nonhomogeneous boundary conditions. Regularity theory 129
3 Von Karman Models with Rotational Forces 142
3.1 Well-posedness for models with internal dissipation 143
3.1.1 Clamped boundary condition 145
3.1.2 Hinged boundary conditions 156
3.1.3 Boundary conditions of the free type 160
3.1.4 Regular solutions 166
3.2 Well-posedness in the case of nonlinear boundary dissipation 170
3.2.1 Clamped--hinged boundary conditions 172
3.2.2 Clamped--free boundary conditions 183
3.2.3 Regular solutions 199
3.3 Other models with rotational inertia 201
3.3.1 Models with delay 202
3.3.2 Models with memory 205
3.3.3 Quasi-static model 206
4 Von Karman Equations Without Rotational Inertia 208
4.1 Models with interior dissipation 208
4.1.1 Clamped boundary conditions 210
4.1.2 Hinged boundary conditions 215
4.1.3 Free boundary conditions 217
4.1.4 Weak solutions 224
4.1.5 Regular solutions 228
4.1.6 On a model with delay 234
4.2 Models with nonlinear boundary dissipation 235
4.2.1 Clamped--hinged boundary conditions 236
4.2.2 Clamped--free boundary conditions 243
4.3 Quasi-static model with clamped boundary condition 251
5 Thermoelastic Plates 256
5.1 PDE model 256
5.2 Abstract formulation 258
5.3 Linear problem 259
5.3.1 Generation of strongly continuous semigroup 260
5.3.2 Analyticity of the semigroup for the model without rotational inertia 263
5.4 Generation of a nonlinear semigroup 270
5.5 Regularity of the semiflow 274
5.6 Backward uniqueness of the semiflow 277
5.7 Stationary solutions 284
6 Structural Acoustic Problems and Plates in a Potential Flow of Gas 286
6.1 Introduction 286
6.2 Structural acoustic problem 289
6.2.1 Description of the model 289
6.2.2 Basic assumption 290
6.2.3 Abstract formulation 291
6.2.4 Well-posedness 294
6.3 Coupled wave and thermoelastic plate equations 300
6.3.1 Description of the model 300
6.3.2 Abstract formulation 301
6.3.3 Well-posedness 303
6.4 Plates in a flow of gas: Description of the model 306
6.5 Plates in a flow of gas: Subsonic case 309
6.5.1 The statement of the main results 310
6.5.2 Preliminaries and abstract setting 312
6.5.3 Galerkin approximations 315
6.5.4 Strong solutions---Proof of Part I of Theorem 6.5.2 318
6.5.5 Generalized and weak solutions---Proof of Part II of Theorem 6.5.2 321
6.5.6 Stationary solutions 323
6.6 Plates with rotational inertia in both subsonic and supersonic gas flow cases 325
6.6.1 The statement of the main results 326
6.6.2 Flow potentials with given boundary conditions 327
6.6.3 Construction of approximate solutions 336
6.6.4 Limit transition 342
6.6.5 Reduced retarded problem 345
Part II Long-Time Dynamics 348
7 Attractors for Evolutionary Equations 349
7.1 Dissipative dynamical systems 349
7.2 Global attractors 356
7.3 Dimension of global attractors 361
7.4 Fractal exponential attractors (inertial sets) 367
7.5 Gradient systems 371
7.5.1 Geometric structure of the attractor 372
7.5.2 Rate of convergence to global attractors 375
7.6 General idea about inertial manifolds 378
7.7 Approximate inertial manifolds 380
7.7.1 The main assumptions 380
7.7.2 Construction of approximate inertial manifolds 381
7.7.3 Nonlinear Galerkin method 383
7.8 General idea about determining functionals 385
7.8.1 Concept of a set of determining functionals 385
7.8.2 Completeness defect of a set of functionals 387
7.8.3 Estimates for completeness defect in Sobolev spaces 389
7.8.4 Existence of determining functionals 391
7.9 Stabilizability estimate and its consequences 393
7.9.1 Finite dimension of global attractors 396
7.9.2 Regularity of trajectories from the attractor 398
7.9.3 Fractal exponential attractors 399
7.9.4 Determining functionals 401
8 Long-Time Behavior of Second-Order Abstract Equations 403
8.1 Main assumptions 403
8.2 Dissipativity 406
8.3 Existence of global attractors 409
8.3.1 Preliminary inequalities 409
8.3.2 Main results on asymptotic smoothness 412
8.4 Regular attractors. Rate of stabilization to equilibria 419
8.5 Stabilizability and quasi-stability estimates 422
8.5.1 Basic theorem on quasi-stability 422
8.5.2 Sufficient conditions for quasi-stability 427
8.6 Finite dimension of global attractors 435
8.7 Regularity of elements from attractors 437
8.8 On ``strong" attractors 442
8.9 Determining functionals 445
8.9.1 An approach based on stabilizability estimate 446
8.9.2 Energy approach 448
8.10 Exponential fractal attractors 455
8.11 Approximate inertial manifolds 456
9 Plates with Internal Damping 459
9.1 Existence of global attractors for von Karman model with rotational forces 459
9.1.1 Clamped boundary condition 462
9.1.2 Hinged or simply supported boundary conditions 468
9.1.3 Free boundary conditions 469
9.1.4 Mixed boundary conditions 473
9.2 Further properties of the attractor for von Karman model with rotational inertia 475
9.2.1 Regular structure of the attractor 475
9.2.2 Finite dimension 479
9.2.3 Smoothness of elements from the attractor 481
9.2.4 Strong attractors 484
9.2.5 Exponential attractor 485
9.2.6 Determining functionals 485
9.2.7 Approximate inertial manifolds 488
9.3 Attractors for other models with rotational inertia 489
9.3.1 Von Karman equations with retarded terms 489
9.3.2 Quasi-static version of von Karman equations 495
9.4 Global attractors for von Karman model without rotational inertia 500
9.4.1 Clamped boundary condition 503
9.4.2 Hinged boundary conditions 510
9.4.3 Free boundary conditions 511
9.5 Further properties of the attractor for von Karman model without rotational inertia 523
9.5.1 Regular structure of the attractor 523
9.5.2 Smoothness of elements from the attractor 526
9.5.3 Strong attractors 535
9.5.4 Exponential attractor 538
9.5.5 Upper semicontinuity of the global attractor with respect to rotational inertia 539
9.5.6 Determining functionals 541
9.6 Global attractor for quasi-static model 546
9.6.1 The existence of attractor for quasi-static problem 547
9.6.2 Upper semicontinuity of the attractor to quasi-static problem 548
10 Plates with Boundary Damping 551
10.1 Introduction: Overview 551
10.2 Global attractors for von Karman models with rotational forces and with dissipation in free boundary conditions 554
10.2.1 The model and the main result on the existence of compact attractors 554
10.2.2 Asymptotic smoothness 560
10.2.3 Proof of the main result on attractors (Theorem 10.2.11) 568
10.2.4 Rate of convergence to the equilibria 574
10.2.5 Determining functionals 581
10.3 Global attractors for von Karman models with rotational forces and with dissipation in hinged boundary conditions 582
10.3.1 The model and the main result 583
10.3.2 Asymptotic smoothness 586
10.3.3 Proof of the main result on attractors (Theorem 10.3.5) 592
10.3.4 Rate of convergence to equilibria 595
10.3.5 Determining functionals 595
10.4 Global attractors for von Karman plates without rotational inertia and with dissipation in free boundary conditions 596
10.4.1 The model and the main results 597
10.4.2 Asymptotic smoothness 604
10.4.3 Global attractor: Proof of Theorem 10.4.7. 611
10.4.4 Rate of stabilization: Proof of Theorem 10.4.10 616
10.5 Global attractors for von Karman plates without rotational inertia and with dissipation acting via hinged boundary conditions 623
10.5.1 The model and the main results on the existence of attractors 623
10.5.2 Asymptotic smoothness 627
10.5.3 Proof of the main result (Theorem 10.5.7) 632
11 Thermoelasticity 637
11.1 Introduction 637
11.2 Statements of main results 641
11.3 Uniform stabilizability inequality 643
11.4 Existence and properties of the attractor---Proof of Theorem 11.2.1 650
11.4.1 Existence of the attractor 650
11.4.2 Smoothness of the attractor---Proof of regularity in Theorem 11.2.1 651
11.4.3 Finite-dimensionality 654
11.4.4 Upper semicontinuity 658
11.5 Exponential rate of attraction---Proof of Theorem 11.2.2 660
12 Composite Wave--Plate Systems 664
12.1 Introduction 664
12.2 Structural acoustic problems 664
12.2.1 The statement of main results 666
12.2.2 Main inequality 669
12.2.3 Asymptotic smoothness 672
12.2.4 Stabilizability estimate 677
12.2.5 Additional properties of the attractor 682
12.2.6 Generalizations 683
12.3 Wave coupled to thermoelastic plate equation 684
12.3.1 The statement of main results 686
12.3.2 Main inequality 689
12.3.3 Asymptotic smoothness and proof of Theorem 12.3.3 692
12.3.4 Stabilizability estimate 694
12.3.5 Proofs of Theorem 12.3.5 and Theorem 12.3.7 697
12.4 Gas flow problems 698
12.4.1 Stabilization to a finite-dimensional set 699
12.4.2 Stabilization to equilibria 702
13 Inertial Manifolds for von Karman Plate Equations 706
13.1 Preliminaries 707
13.1.1 The models considered 707
13.1.2 Generation of nonlinear semigroups 709
13.1.3 Absorbing sets 711
13.2 Inertial manifolds for evolution equations 714
13.3 Inertial manifolds for second order in time evolution equation 722
13.3.1 Second-order evolutions with viscous damping 722
13.3.2 Second-order evolution equation with strong damping 727
13.3.3 Thermoelastic von Karman evolutions 731
A Jacobians and Compensated Compactness, Compactness of Vector Functions, and Sedenko's Method for Uniqueness 736
A.1 Jacobian regularity and compensated compactness 736
A.2 Compactness theorem for vector-valued functions 738
A.3 Logarithmic Sobolev-type inequalities and uniqueness of weak solutions by Sedenko's method 741
B Some Auxiliary Facts 752
B.1 Estimates for monotone functions 752
B.2 Concave bounds 754
B.3 Equation describing the convergence rates for the energy 756
B.4 Some convergence theorems for measurable functions 758
References 760
Index 772

"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."