Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors

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ISBN: 9783764388133 | 3764388137
Cover: Hardcover
Copyright: 9/1/2008

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"This book presents recent results concerning the global existence in time, the large-time behaviour, decays of solutions and the existence of global attractors for some nonlinear parabolic-hyperbolic coupled systems of evolutionary partial differential equations arising from physics, mechanics and material science, such as the compressible Navier-Stokes equations, thermo(visco)elastic systems and elastic systems. To keep the book as self-contained as possible, the first chapter introduces to the needed results and tools from functional analysis, Sobolev spaces, differential and integral inequalities in analysis, and the theory of semigroups of linear operators and of global attractors."--BOOK JACKET.

Preface	p. xi
Preliminary
Sobolev Spaces and Their Basic Properties	p. 1
Distributions	p. 2
Weak Derivatives and Sobolev Spaces	p. 4
Sobolev Inequalities, Embedding Theorems and the Trace Theorem	p. 9
Interpolation Inequalities	p. 17
The Poincare Inequality	p. 17
Some Inequalities in Analysis	p. 18
The Classical Bellman-Gronwall Inequality	p. 18
The Generalized Bellman-Gronwall Inequalities	p. 19
The Uniform Bellman-Gronwall Inequality	p. 20
The Nakao Inequalities	p. 23
Some Differential Inequalities for Nonexistence of Global Solutions	p. 25
Other Useful Inequalities	p. 26
The Young Inequalities	p. 27
The Holder Inequality	p. 28
The Minkowski Inequalities	p. 29
The Jensen Inequality	p. 30
C[subscript 0]-Semigroups of Linear Operators	p. 31
C[subscript 0]-Semigroups of Linear Operators	p. 31
Global Attractors	p. 37
Compact Semigroups (Semiflows) for Autonomous Systems	p. 39
Weakly Compact Semigroups (Semiflows) for Autonomous Systems	p. 42
Bibliographic Comments	p. 43
A One-dimensional Nonlinear Viscous and Heat-conductive Real Gas
Fixed and Thermally Insulated Boundary Conditions	p. 46
Main Results	p. 46
Uniform A Priori Estimates	p. 49
Clamped and Constant Temperature Boundary Conditions	p. 71
Exponential Stability in H[superscript 1] and H[superscript 2]	p. 78
Main Results	p. 78
Exponential Stability in H[superscript 1]	p. 80
Exponential Stability in H[superscript 2]	p. 89
Exponential Stability in H[superscript 4]	p. 97
Global Existence in H[superscript 4]	p. 100
A Nonlinear C[subscript 0]-Semigroup S(t) on H[superscript 4]	p. 111
Exponential Stability in H[superscript 4]	p. 119
Attractors in H[superscript 1] and H[superscript 2]	p. 123
An Absorbing Set in H[superscript 1]	p. 126
An Absorbing Set in H[superscript 2]	p. 132
Universal Attractor in H[superscript 4]	p. 135
Bibliographic Comments	p. 138
A One-dimensional Polytropic Viscous and Heat-conductive Gas
Initial Boundary Value Problems	p. 143
Global Existence and Asymptotic Behavior of Solutions	p. 143
Exponential Stability	p. 153
Universal Attractors	p. 154
The Cauchy Problem	p. 154
Global Existence in H[superscript 2] (R)	p. 154
Large-Time Behavior of Solutions	p. 159
Bibliographic Comments	p. 164
A Polytropic Ideal Gas in Bounded Annular Domains in R[superscript n]
Global Existence and Asymptotic Behavior in H[superscript 1] and H[superscript 2]	p. 167
Uniform A Priori Estimates in H[superscript 1]	p. 175
Uniform a priori estimates in H[superscript 2]	p. 187
Results in Eulerian Coordinates	p. 199
Exponential Stability in H[superscript 4]	p. 200
Main Results	p. 200
Global Existence in H[superscript 4]	p. 202
A Nonlinear C[subscript 0]-Semigroup S(t) on H[superscript 4]	p. 211
Exponential Stability in H[superscript 4]	p. 222
Universal Attractors	p. 227
Nonlinear Semigroups on H[superscript 2]	p. 230
Existence of an Absorbing Set in H[subscript delta superscript (1)]	p. 231
Existence of an Absorbing Set in H[subscript delta superscript (2)]	p. 236
Results of the Eulerian Coordinates	p. 241
Attractor in H[superscript 4]	p. 241
Bibliographic Comments	p. 243
A Polytropic Viscous Gas with Cylinder Symmetry in R[superscript 3]
Main Results	p. 245
Global Existence and Exponential Stability in H[superscript 1]	p. 249
Global Existence and Exponential Stability in H[superscript 2]	p. 266
Global Existence and Exponential Stability in H[superscript 4]	p. 268
Global Existence of Solutions in H[superscript 4]	p. 268
Exponential Stability in H[subscript + superscript 4]	p. 285
Bibliographic Comments	p. 290
One-dimensional Nonlinear Thermoviscoelasticity
Global Existence and Asymptotic Behavior of Solutions	p. 293
Uniform A Priori Estimates	p. 297
Exponential Stability and Maximal Attractors	p. 325
Exponential Stability in H[superscript 1] and H[superscript 2]	p. 331
Exponential Stability in H[superscript 4]	p. 332
Universal Attractors in H[superscript i] (i = 1, 2, 4)	p. 332
Existence of An Absorbing Set in H[subscript delta superscript 1]	p. 332
Existence of An Absorbing Set in H[subscript delta superscript 2]	p. 335
Existence of An Absorbing Set in H[subscript delta superscript 4]	p. 336
Bibliographic Comments	p. 336
A Nonlinear One-dimensional Thermoelastic System with a Thermal Memory
Main Results	p. 339
Global Existence and Exponential Stability	p. 342
Bibliographic Comments	p. 361
One-dimensional Thermoelastic Equations of Hyperbolic Type
Global Existence	p. 363
Global Existence and Exponential Stability	p. 365
Bibliographic Comments	p. 379
Blow-up for the Cauchy Problem in Nonlinear One-dimensional Thermoelasticity
Introduction	p. 381
Main Results - Case I	p. 382
Main Results - Case II	p. 399
Bibliographic Comments	p. 408
Large-Time Behavior of Energy in Multi-Dimensional Elasticity
Polynomial Decay of Energy	p. 409
Main Results	p. 411
Proof of Theorem 10.1.3	p. 413
Exponential Decay of Energy	p. 421
Main Results	p. 421
Proof of Theorem 10.2.3	p. 426
Bibliographic Comments	p. 433
Bibliography	p. 435
Index	p. 463
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Nonlinear Parabolic-Hyperbolic Coupled Systems and Their Attractors

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