Approximation of nonlinear evolution systems, Volume 164 (Mathematics in Science and Engineering)

✍ Scribed by Jerome (editor)

• Publisher: Academic Press
• Year: 1983
• Tongue: English
• Leaves: 301
• Category: Library

✦ Table of Contents

Front Cover
Approximation of Nonlinear Evolution Systems
Copyright Page
Contents
Preface
Acknowledgments
List of Symbols and Definitions
Introduction
PART I: GLOBAL WEAK SOLUTIONS
Chapter 1. Problem Formulations and Uniqueness for Dissipative Parabolic Models
1.0 Introduction
1.1 Heat Conduction with Change of Phase: Stefan Problems
1.2 Unsaturated Fluid Infiltration in Porous Media
1.3 Reaction–Diffusion Systems
1.4 Incompressible, Viscous Fluid Dynamics at Constant Temperature: Navier–Stokes Equations and Generalizations
1.5 Uniqueness of Solutions
1.6 Bibliographical Remarks
References
Chapter 2. Convergent Regularizations and Pointwise Stability of Implicit Schemes
2.0 Introduction
2.1 Regularization in the Stefan Problem
2.2 Semidiscrete Regularization and Maximum Principles in the Stefan Problem
2.3 Regularization in the Porous-Medium Equation
2.4 Nonnegative Semidiscrete Solutions of Porous-Medium Equation and Maximum Principles
2.5 Invariant Rectangles and Maximum Principles for Reaction–Diffusion Systems in Semidiscrete Form
2.6 Bibliographical Remarks
References
Chapter 3. Nonlinear Elliptic Equations and Inequalities
3.0 Introduction
3.1 General Operator Results in Banach Spaces and Ordered Spaces
3.2 Applications and Examples
3.3 Semidiscretizations Defined by Quadrature
3.4 Bibliographical Remarks
References
Chapter 4. Numerical Optimality and the Approximate Solution of Degenerate Parabolic Equations
4.0 Introduction
4.1 Representations of Sobolev-Type and Upper-Bound Estimates
4.2 Lower-Bound Estimates and N-Widths
4.3 Convergence Rates for the Continuous Galerkin Method
4.4 Convergence Rates for Semidiscrete Approximations
4.5 Bibliographical Remarks
References
Chapter 5. Existence Analysis via the Stability of Consistent Semidiscrete Approximations
5.0 Introduction
5.1 Stability in Sobolev Norms for Semidiscretizations of Degenerate Parabolic Equations
5.2 Existence of Weak Solutions for the Stefan Problem and the Porous-Medium Equation and Approximation Results
5.3 Existence for Reaction-Diffusion Systems
5.4 Existence for the Generalized Form of the Navier–Stokes Equations for Incompressible Fluids
5.5 Bibliographical Remarks
References
PART II: LOGCAL SMOOTH SOLUTIONS
Chapter 6. Linear Evolution Operators
6.0 Introduction
6.1 Semigroup Preliminaries
6.2 The Linear Evolution Equation and Evolution Operators
6.3 Peturbations of Generators and Regularity of Evolution Operators
6.4 The Inhomogeneous Problem and an Application to Linear Symmetric Hyperbolic Systems
6.5 Bibliographical Remarks
References
Chapter 7. Quasi-linear Equations of Evolution
7.0 Introduction
7.1 Perturbation of the Linear Problem and Nonlinear Preliminaries
7.2 The Quasi-linear Cauchy Problem in Banach Space
7.3 Quasi-linear Second-Order Hyperbolic Systems
7.4 The Vacuum Field Equations of General Relativity
7.5 Invariant Time Intervals for the Artificial Viscosity Method
7.6 Bibliographical Remarks
References
Subject Index