Keywords:
Improperly posed problems.
;
Electronic books.
Description / Table of Contents:
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.
Type of Medium:
Online Resource
Pages:
1 online resource (319 pages)
Edition:
1st ed.
ISBN:
9780080537740
Series Statement:
Issn Series ; v.Volume 194
URL:
https://ebookcentral.proquest.com/lib/geomar/detail.action?docID=311369
DDC:
530.15/5353
Language:
English
Note:
Cover -- Non-Standard and Improperly Posed Problems -- Copyright -- Contents -- Preface -- 1. Introduction -- 1.1 Introductory Remarks -- 1.2 Notation and Some Important Inequalities -- 1.3 Examples of Non-Uniqueness and Breakdown of Continuous Dependence -- 1.4 Four Classical Improperly Posed Problems -- 1.5 Methods of Analysis -- 1.6 Existence of Solutions and Outline of the Book -- 2. Continuous Dependence on the Geometry -- 2.1 Continuous Dependence on the Initial-Time Geometry for the Heat Equation -- 2.2 Continuous Dependence on the Initial-Time Geometry for the Heat Equation on an Exterior Domain -- 2.3 Continuous Dependence on the Initial -Time Geometry for an Equation from Dynamo Theory -- 2.4 Continuous Dependence on the Initial -Time Geometry for the Navier-Stokes Equations -- 2.5 Continuous Dependence on the Initial -Time Geometry for Solutions to the Darcy and Brinkman Equations for Flow in Porous Media -- 2.6 Continuous Dependence on the Spatial Geometry -- 3. Continuous Dependence on Modeling Backward in Time -- 3.1 Singular Perturbation in Improperly Posed Problems -- 3.2 Modeling Errors for the Navier-Stokes Equations Backward in Time -- 3.3 Modeling Errors for First and Second Order Operator Equations -- 3.4 Continuous Dependence on Modeling in the Cauchy Problem for Second Order Partial Differential Equations: Some Examples -- 3.5 Modeling Errors in Micropolar Fluid Dynamics and in Magnetohydrodynamics -- 3.6 Continuous Dependence on Modeling in Porous Convection Problems -- 3.7 Modeling Errors in Theories of Heat Conduction with Finite Propagation Speed -- 4. Continuous Dependence on Modeling Forward in Time -- 4.1 Modeling Errors for the Navier-Stokes Equations Forward in Time -- 4.2 Modeling Errors in Micropolar Fluid Dynamics -- 4.3 Continuous Dependence on the Velocity for an Equation Arising from Dynamo Theory.
,
4.4 Structural Stability for Infinite Prandtl Number Thermal Convection -- 5. Non-Standard and Non-Characteristic Problems -- 5.1 The "Sideways" Problem for the Heat Equation -- 5.2 The "Sideways" Problem for a Hyperbolic Equation -- 5.3 The Cauchy Problem for the Laplace Equation and Other Elliptic Equations -- 6. Some Further Improperly Posed Problems -- 6.1 Uniqueness and Continuous Dependence for Singular Equations -- 6.2 Spatial Decay in Improperly Posed Parabolic Problems -- 6.3 Uniqueness for the Backward in Time Navier-Stokes Equations on an Unbounded Spatial Domain -- 6.4 Improperly Posed Problems for Dusty Gases -- 6.5 Linear Thermoelasticity -- 6.6 Overcoming Hölder Continuity, and Image Restoration -- Bibliography -- Index.
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