Non-Standard and Improperly Posed Problems
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Table of contents
- Contentsv
- Prefaceviii
- Chapter 1. Introduction1
- 1.1 Introductory Remarks1
- 1.2 Notation and Some Important Inequalities6
- 1.3 Examples of Non-Uniqueness and Breakdown of Continuous Dependence10
- 1.4 Four Classical Improperly Posed Problems14
- 1.5 Methods of Analysis16
- 1.6 Existence of Solutions and Outline of the Book31
- Chapter 2. Continuous Dependence on the Geometry34
- 2.1 Continuous Dependence on the Initial-Time Geometry for the Heat Equation34
- 2.2 Continuous Dependence on the Initial-Time Geometry for the Heat Equation on an Exterior Domain42
- 2.3 Continuous Dependence on the Initial-Time Geometry for an Equation from Dynamo Theory51
- 2.4 Continuous Dependence on the Initial-Time Geometry for the Navier-Stokes Equations65
- 2.5 Continuous Dependence on the Initial-Time Geometry for Solutions to the Darcy and Brinkman Equat70
- 2.6 Continuous Dependence on the Spatial Geometry88
- Chapter 3. Continuous Dependence on Modeling Backward in Time103
- 3.1 Singular Perturbation in Improperly Posed Problems103
- 3.2 Modeling Errors for the Navier-Stokes Equations Backward in Time114
- 3.3 Modeling Errors for First and Second Order Operator Equations123
- 3.4 Continuous Dependence on Modeling in the Cauchy Problem for Second Order Partial Differential Eq140
- 3.5 Modeling Errors in Micropolar Fluid Dynamics and in Magnetohydrodynamics149
- 3.6 Continuous Dependence on Modeling in Porous Convection Problems158
- 3.7 Modeling Errors in Theories of Heat Conduction with Finite Propagation Speed172
- Chapter 4. Continuous Dependence on Modeling Forward in Time183
- 4.1 Modeling Errors for the Navier-Stokes Equations Forward in Time183
- 4.2 Modeling Errors in Micropolar Fluid Dynamics188
- 4.3 Continuous Dependence on the Velocity for an Equation Arising from Dynamo Theory195
- 4.4 Structural Stability for Infinite Prandtl Number Thermal Convection201
- Chapter 5. Non-Standard and Non-Characteristic Problems217
- 5.1 The "Sideways" Problem for the Heat Equation217
- 5.2 The "Sideways" Problem for a Hyperbolic Equation225
- 5.3 The Cauchy Problem for the Laplace Equation and Other Elliptc Equations230
- Chapter 6. Some Further Improperly Posed Problems238
- 6.1 Uniqueness and Continuous Dependence for Singular Equations238
- 6.2 Spatial Decay in Improperly Posed Parabolic Problems243
- 6.3 Uniqueness for the Backward in Time Navier-Stokes Equations on an Unbounded Spatial Domain254
- 6.4 Improperly Posed Problems for Dusty Gases259
- 6.5 Linear Thermoelasticity263
- 6.6 Overcoming Hölder Continuity, and Image Restoration278
- Bibliography284
- Index301
Written by two international experts in the field, this book is the first unified survey of the advances made in the last 15 years on key non-standard and improperly posed problems for partial differential equations.This reference for mathematicians, scientists, and engineers provides an overview of the methodology typically used to study improperly posed problems. It focuses on structural stability--the continuous dependence of solutions on the initial conditions and the modeling equations--and on problems for which data are only prescribed on part of the boundary.
The book addresses continuous dependence on initial-time and spatial geometry and on modeling backward and forward in time. It covers non-standard or non-characteristic problems, such as the sideways problem for a heat or hyberbolic equation and the Cauchy problem for the Laplace equation and other elliptic equations. The text also presents other relevant improperly posed problems, including the uniqueness and continuous dependence for singular equations, the spatial decay in improperly posed parabolicproblems, the uniqueness for the backward in time Navier-Stokes equations on an unbounded domain, the improperly posed problems for dusty gases, the linear thermoelasticity, and the overcoming Holder continuity and image restoration.
Key Features
* Provides the first unified survey of the advances made in the last 15 years in the field
* Includes an up-to-date compendium of the mathematical literature on these topics
The book addresses continuous dependence on initial-time and spatial geometry and on modeling backward and forward in time. It covers non-standard or non-characteristic problems, such as the sideways problem for a heat or hyberbolic equation and the Cauchy problem for the Laplace equation and other elliptic equations. The text also presents other relevant improperly posed problems, including the uniqueness and continuous dependence for singular equations, the spatial decay in improperly posed parabolicproblems, the uniqueness for the backward in time Navier-Stokes equations on an unbounded domain, the improperly posed problems for dusty gases, the linear thermoelasticity, and the overcoming Holder continuity and image restoration.
Key Features
* Provides the first unified survey of the advances made in the last 15 years in the field
* Includes an up-to-date compendium of the mathematical literature on these topics