Inhalt

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  Chapter I. Introduction and Motivation
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   1. Tamed MHD Equations - Deterministic and Stochastic
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   2. Random Dynamical Systems and Random Attractors
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   3. Acknowledgements
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  Chapter II. The Deterministic Tamed MHD Equations
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  1. Introduction
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   1.1. Magnetohydrodynamics
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   1.2. Regularised Fluid Dynamical Equations
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   1.2.1. Mollifying the Nonlinearity and the Force
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   1.2.2. Leray- Model and Related Models (Clark-, LANS, …)
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   1.2.3. A Cutoff Scheme due to Yoshida and Giga
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   1.2.4. Globally Modified Navier-Stokes Equations
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   1.2.5. Regularisation by Delay
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   1.2.6. Lions' Hyperviscosity Method
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   1.2.7. Navier-Stokes-Voigt Equations
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   1.2.8. Damped Navier-Stokes Equations (or Brinkman-Forchheimer-extended Darcy Models)
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  1.3. The Tamed Equations
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   1.3.1. Physical Motivation
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   1.3.2. Mathematical Motivation
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   1.3.3. Review of Results for Tamed Navier-Stokes Equations
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   1.3.4. The Magnetic Pressure Problem
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   1.3.5. The Magnetic Field: To Regularise or Not to Regularise?
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   1.3.6. The Tamed MHD Equations
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   1.4. Results and Structure of This Chapter
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   1.5. Notation
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  2. The Case of the Whole Space
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   2.1. Auxiliary Results
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   2.2. Existence and Uniqueness of Weak Solutions
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   2.3. Existence, Uniqueness and Regularity of a Strong Solution
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   2.4. Convergence to the Untamed MHD Equations
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  Chapter III. The Stochastic Tamed MHD Equations
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   1. Introduction
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  1.1. Regularisation of Fluid Dynamical Equations
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   1.1.1. Leray- Model
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   1.1.2. Globally Modified Navier-Stokes Equations
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   1.1.3. Regularisation by Delay
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   1.1.4. Lions' Hyperviscosity Method
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   1.1.5. Navier-Stokes-Voigt Equations
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   1.1.6. Damped Navier-Stokes Equations (or Brinkman-Forchheimer-extended Darcy Models)
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   1.2. Results and Structure of This Chapter
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  2. Preliminaries
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   2.1. Notation and Assumptions
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   2.2. Estimates on the Operators A and B
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   2.3. A Tightness Criterion
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  3. Existence and Uniqueness of Strong Solutions
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   3.1. Weak and Strong Solutions
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   3.2. Pathwise Uniqueness
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   3.3. Existence of Martingale Solutions
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   3.4. Proof of Theorem 1.1
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   4. Feller Property and Existence of Invariant Measures
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  Chapter IV. Dynamical Systems and Random Attractors
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   1. Introduction
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   1.1. Literature
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   1.2. Overview
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   2. Main Framework
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   3. Strictly Stationary Solutions for Monotone SPDE
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   4. Generation of Random Dynamical Systems
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   5. Existence of a Random Attractor
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   6. Examples
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   6.1. Stochastic Burgers-type and Reaction-Diffusion Equations
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   6.2. Stochastic 2D Navier-Stokes Equation and Other Hydrodynamical Models
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   6.3. Stochastic 3D Leray- Model
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   6.4. Stochastic Power Law Fluids
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   6.5. Stochastic Ladyzhenskaya Model
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   6.6. Stochastic Cahn-Hilliard-type Equations
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   6.7. Stochastic Kuramoto-Sivashinsky Equation
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   6.8. SPDE with Monotone Coefficients
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   7. Existence and Uniqueness of Solutions to Locally Monotone PDE
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   Appendices
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  A. Lp Solutions and Integral Equations
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   A.1. A Divergence-Free Solution to the Heat Equation on the Whole Space
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   A.2. Equivalence of Weak Solutions to the MHD Equations and Solutions to the Integral Equation
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   A.3. Regularity of Solutions to the Integral Equation
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  B. A Note on Vector Calculus
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   B.1. Gradient of a Vector – Navier-Stokes Case
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   B.2. Gradient of a Vector – MHD Case
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   C. Stochastic Flows and Random Dynamical Systems
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   Bibliography