Short overview of the classical theory of attractors

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Table of contents

I Long term analysis of two fluid flows 
1 Two phase fluid flows 
1.1 Motivation—The model H
1.1.1 On the Cahn-Hilliard equation
1.1.2 On the Navier-Stokes equations
1.1.3 Coupling Cahn-Hilliard and Navier-Stokes models
1.2 Some generalisations of the model H
1.2.1 Non-newtonian fluids
1.2.2 Oono reaction term
1.2.3 Non-local models
1.3 Large-time behaviour for solutions of parabolic systems
1.3.1 Short overview of the classical theory of attractors
1.3.2 Trajectory attractors
1.3.3 Exponential attractors
1.3.4 Pullback attractors
1.3.5 Convergence to stationary states
1.4 Notation
1.4.1 General notation
1.4.2 Functional spaces
2 A LCH system 
2.1 Functional setting
2.2 Main assumptions and weak formulation
2.3 The convective Cahn-Hilliard equation
2.4 Some results on the Ladyzhenskaya model
2.5 Weak solutions and energy estimates
2.6 A weak trajectory attractor
2.7 A strong trajectory attractor
3 A 2D CHNS system 
3.1 Functional setting and main results
3.2 Exponential pullback attractors
3.3 Existence results and basic energy estimate
3.4 Higher regularity estimates
3.5 Continuous dependence
3.6 Time regularity
3.7 Proof of the main results
4 2D NSCHO 
4.1 Functional setting and main results
4.2 Existence and dissipation
4.3 Continuous dependence
4.4 Higher-order dissipative estimate
4.5 A robust family of exponential attractors
5 Nonlocal CH 
5.1 Basic tools and well-posedness
5.1.1 Function spaces
5.1.2 Weak formulation and main result
5.1.3 Evolution equations with monotone operators
5.1.4 Results on the nonlocal operator L
5.2 Subgradients
5.3 Existence of unique solutions
5.4 Long-time behaviour
5.5 Boundary conditions for variational solutions
5.5.1 Proof of Theorem 5.5.1: case n “ 1
5.5.2 Proof of Theorem 5.5.1: case n ě 2
II Strain deformation in p-n junctions  Notation used in Part II 
6 Statement of the problem 
6.1 On the physics of semiconductors
6.1.1 Energy bands
6.1.2 Doping
7 The Drift-Diffusion model and strain 
7.1 Derivation of the DD model
7.2 A brief history of strain effects on SCs’ properties
7.3 Strain effects in silicon
7.3.1 Shift of the energy bands
7.3.2 Changes in effective masses
7.3.3 Shift of the Fermi level
7.3.4 Changes for ni
7.3.5 Changes in mobilities
8 P-N junctions under strain 
8.1 P-N junctions in operation
8.2 The Shockley relation with strain effects
8.3 Maxwell stresses and reverse coupling
III High cycle fatigue in alloys  Notation used in Part III 
9 A HCF criterion 
9.1 The models at the mesoscopic scale
9.2 Separation of time scales
9.3 Lifetime estimates in HCF
9.4 A priori estimates
9.5 Identification of parameters from fatigue experiments
9.6 Results and discussion
9.7 Conclusions
Acknowledgements
Bibliography