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E-raamat: Non-Standard and Improperly Posed Problems

(Georgia Institute of Technology), (University of Glasgow)
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Non-Standard and Improperly Posed ProblemsSuurem pilt
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Problems for partial differential equations are defined as improperly posed if they fail to have a global or unique solution, or if the solution does not depend continuously on the data. Ames (mathematics, U. of Alabama, Huntsville) and Straughan (mathematics, U. of Glasgow) analyze such problems in relation to continuous dependence on: the geometry, modeling backward in time, and modeling forward in time. Other non-standard problems considered include the "sideways" problem for a hyperbolic equation, the classical Cauchy problem for the Laplace equation, and linear thermoplasticity. Annotation c. by Book News, Inc., Portland, Or.

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

Muu info

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
Preface viii
1. Introduction
1(33)
1.1 Introductory Remarks
1(5)
1.2 Notation and Some Important Inequalities
6(4)
1.3 Examples of Non-Uniqueness and Breakdown of Continuous Dependence
10(4)
1.4 Four Classical Improperly Posed Problems
14(2)
1.5 Methods of Analysis
16(15)
1.6 Existence of Solutions and Outline of the Book
31(3)
2. Continuous Dependence on the Geometry
34(69)
2.1 Continuous Dependence on the Initial-Time Geometry for the Heat Equation
34(8)
2.2 Continuous Dependence on the Initial-Time Geometry for the Heat Equation on an Exterior Domain
42(9)
2.3 Continuous Dependence on the Initial-Time Geometry for an Equation from Dynamo Theory
51(14)
2.4 Continuous Dependence on the Initial-Time Geometry for the Navier-Stokes Equations
65(5)
2.5 Continuous Dependence on the Initial-Time Geometry for Solutions to the Darcy and Brinkman Equations for Flow in Porous Media
70(18)
2.6 Continuous Dependence on the Spatial Geometry
88(15)
3. Continuous Dependence on Modeling Backward in Time
103(80)
3.1 Singular Perturbation in Improperly Posed Problems
103(11)
3.2 Modeling Errors for the Navier-Stokes Equations Backward in Time
114(9)
3.3 Modeling Errors for First and Second Order Operator Equations
123(17)
3.4 Continuous Dependence on Modeling in the Cauchy Problem for Second Order Partial Differential Equations: Some Examples
140(9)
3.5 Modeling Errors in Micropolar Fluid Dynamics and in Magnetohydrodynamics
149(9)
3.6 Continuous Dependence on Modeling in Porous Convection Problems
158(14)
3.7 Modeling Errors in Theories of Heat Conduction with Finite Propagation Speed
172(11)
4. Continuous Dependence on Modeling Forward in Time
183
4.1 Modeling Errors for the Navier-Stokes Equations Forward in Time
183(5)
4.2 Modeling Errors in Micropolar Fluid Dynamics
188(7)
4.3 Continuous Dependence on the Velocity for an Equation Arising from Dynamo Theory
195(6)
4.4 Structural Stability for Infinite Prandtl Number Thermal Convection
201
5. Non-Standard and Non-Characteristic Problems
217(21)
5.1 The "Sideways" Problem for the Heat Equation
217(8)
5.2 The "Sideways" Problem for a Hyperbolic Equation
225(5)
5.3 The Cauchy Problem for the Laplace Equation and Other Elliptc Equations
230(8)
6. Some Further Improperly Posed Problems
238(46)
6.1 Uniqueness and Continuous Dependence for Singular Equations
238(5)
6.2 Spatial Decay in Improperly Posed Parabolic Problems
243(11)
6.3 Uniqueness for the Backward in Time Navier-Stokes Equations on an Unbounded Spatial Domain
254(5)
6.4 Improperly Posed Problems for Dusty Gases
259(4)
6.5 Linear Thermoelasticity
263(5)
6.6 Overcoming Holder Continuity, and Image Restoration
278(6)
Bibliography 284(17)
Index 301
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