Non-standard and improperly posed problems

K. A. Ames, Brian Straughan0120567458, 9780120567454, 9780080537740

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

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Table of contents :
Front cover ……Page 1
Title ……Page 2
Series ……Page 3
Title page ……Page 4
Date-line ……Page 5
Contents ……Page 6
Preface ……Page 9
1.1 Introductory Remarks ……Page 14
1.2 Notation and Some Important Inequalities ……Page 19
1.3 Examples of Non-Uniqueness and Breakdown of Continuous Dependence ……Page 23
1.4 Four Classical Improperly Posed Problems ……Page 27
1.5 Methods of Analysis ……Page 29
1.6 Existence of Solutions and Outline of the Book ……Page 44
2.1 Continuous Dependence on the Initial-Time Geometry for the Heat Equation ……Page 47
2.2 Continuous Dependence on the Initial-Time Geometry for the Heat Equation on an Exterior Domain ……Page 55
2.3 Continuous Dependence on the Initial-Time Geometry for an Equation from Dynamo Theory ……Page 64
2.4 Continuous Dependence on the Initial-Time Geometry for the Navier-Stokes Equations ……Page 78
2.5 Continuous Dependence on the Initial-Time Geometry for Solutions to the Darcy and Brinkman Equations for Flow in Porous Media ……Page 83
2.6 Continuous Dependence on the Spatial Geometry ……Page 101
3.1 Singular Perturbation in Improperly Posed Problems ……Page 116
3.2 Modeling Errors for the Navier-Stokes Equations Backward in Time ……Page 127
3.3 Modeling Errors for First and Second Order Operator Equations ……Page 136
3.4 Continuous Dependence on Modeling in the Cauchy Problem for Second Order Partial Differential Equations: Some Examples ……Page 153
3.5 Modeling Errors in Micropolar Fluid Dynamics and in Magnetohydrodynamics ……Page 162
3.6 Continuous Dependence on Modeling in Porous Convection Problems ……Page 171
3.7 Modeling Errors in Theories of Heat Conduction with Finite Propagation Speed ……Page 185
4.1 Modeling Errors for the Navier-Stokes Equations Forward in Time ……Page 196
4.2 Modeling Errors in Micropolar Fluid Dynamics ……Page 201
4.3 Continuous Dependence on the Velocity for an Equation Arising from Dynamo Theory ……Page 208
4.4 Structural Stability for Infinite Prandtl Number Thermal Convection ……Page 214
5.1 The “Sideways” Problem for the Heat Equation ……Page 230
5.2 The “Sideways” Problem for a Hyperbolic Equation ……Page 238
5.3 The Cauchy Problem for the Laplace Equation and Other Elliptc Equations ……Page 243
6.1 Uniqueness and Continuous Dependence for Singular Equations ……Page 251
6.2 Spatial Decay in Improperly Posed Parabolic Problems ……Page 256
6.3 Uniqueness for the Backward in Time Navier-Stokes Equations on an Unbounded Spatial Domain ……Page 267
6.4 Improperly Posed Problems for Dusty Gases ……Page 272
6.5 Linear Thermoelasticity ……Page 276
6.6 Overcoming Holder Continuity, and Image Restoration ……Page 291
Bibliography ……Page 297
Index ……Page 314
MATHEMATICS IN SCIENCE AND ENGINEERING ……Page 318