Nonlinear partial differential equations, Volume 44: Sequential and weak solutions (North-Holland Mathematics Studies)

✍ Scribed by E. E Rosinger

• Publisher: North Holland
• Year: 1980
• Tongue: English
• Leaves: 339
• Category: Library

✦ Table of Contents

Nonlinear Partial Differntial Equations: Sequential and Weak Solutions
Copyright Page
TABLE OF CONTENTS
Chapter 1. Sequential Solutions of Nonlinear PDEs
0 Introduction
1 The Problem of Stability of Sequential Solutions
2 Defining the Quotient Spaces and Algebras
3 The Notion of Sequential Solution
4 Exactness of Sequential Solutions
5 Interplay between Stability, Generality and Exactness
6 The Deficiency of the Distributional Approach Concerning Exactness
7 Resolution of Nowhere Dense Singularities
8 Preservation of Ellipticity and Hypoellipticity
9 General Nonlinear PDEs
10 Variable Transforms
11 Stronger Conditions for Partial Derivative Operators
12 Systems of Nonlinear PDEs
13 A Cesaro Characterization of the Nowhere Dense Ideal
Chapter 2. Necessary and/or Sufficient Conditions for the Existence of Sequential Solutions
0 Introduction
1 Subsequence Quasi Invariant Sequential Solutions
2 Applications to Linear and Nonlinear PDEs
3 Subsequence Invariant Sequential Solution
4 Resolvent Sets
5 Domains of Solvability
6 Remarks on Lemma 2
Chapter 3. Algebras Containing the Distribution
0 Introduction
1 Embedding the Distributions into Quotient Algebras
2 Simpler Inclusion Diagrams
3 Regularizations and Chains of Quotient Algebras Containing the Distributions
4 Existence, Characterization and Construction of Regularizations
5 Additional Chains of Quotient Algebras Containing Distributions
6 Quotient Algebras Containing Large Subspaces of Distributions
7 Quotient Algebras Possessing Special Properties
8 Definition of Positive Powers for Certain Generalized Functions
9 Limitations on Embedding Smooth Functions into Chains of Quotient Algebras
10 Special Classes of Regular Ideals
11 The Proof of Lemma 1
12 Example of Shrinking Non-δ Sequence
13 Inexistence of the Largest Regular Ideals
Chapter 4. Resolution of Singularities of Weak Solutions for Polynomial Nonlinear PDEs
0 Introduction
1 The Case of Simple Polynomial Nonlinear PDEs
2 Resolution of Singularities of Nonlinear Shock Waves
3 Resolution of Singularities of Klein-Gordon Type Nonlinear Waves
4 Resolution of Singularities of Weak Solutions for General Polynomial Nonlinear PDEs
5 Junction Conditions and Resolution of Singularities of Weak Solutions for the Equations of Magneto-hydrodynamics and General Relativity
6 Resoluble Systems of Polynomial Nonlinear PDEs
7 Computation of the Junction Conditions
8 Examples of Resoluble Systems of PDEs
Chapter 5. Stability and Exactness of Sequential and Weak Solutions for Polynomial Nonlinear PDEs
0 Introduction
1 Basic Facts and Remarks Concerning Stabilit
2 Maximal Stability
3 Interplay between Stability and Exactness
4 Remarks in the Finite Smoothness Case
5 Stability of Regular Weak Solutions
6 Stability of Solutions with Jump Discontinuities for Resoluble Systems of PDEs
Chapter 6. Characterization of the Necessary Structure of the Algebras Containing the Distributions
0 Introduction
1 Zero Filters
2 Density Characterization of Z-Filters
3 Large C°–Regular Ideals
4 Further Properties of Z-Filters
5 Construction and Characterization of a Class of Vanishing Ideals
6 Countable Reduced Products of Vector Spaces
Chapter 7. Quantum Scattering in Potentials Positive Powers of the Dirac δ Distribution
0 Introduction
1 Wave Function Solutions and Junction Relations
2 Weak Wave Function Solutions
3 Smooth Representations forthe Dirac δ Distribution and Smooth Weak Wave Function Solutions
4 Wave Function Solutions in the Algebras Containing the Distributions
5 Stability and Exactness of the Wave Function Solutions
Chapter 8. Products with Dirac δ Distribution
0 Introduction
1 A Class of Regularizations
2 Products with Dirac δ Distributions
3 Application to a Riccati Differential Equation
4 Validity of Formulas in Quantum Mechanics
5 The Existence of the Sequences of Functions in Zδ
6 Remark on Nilpotent Elements in the Quotient Algebras
Chapter 9. Linear Independent Families of Dirac δ Distributions at a Point
0 Introduction
1 Compatible Quotient Algebras and Independent Variable Transforms
2 Linear Independent Families of Dirac δ Distributions at a Point
3 Generalized Dirac δ Distributions
Chapter 10. Support and Local Properties
0 Introduction
1 Support of Elements in Quotient Algebras
2 Localization with in the Quotient Algebras
3 Equivalence between S=O and supp S=Q
4 Domains of Solvability for Polynomial Nonlinear PDEs
FINAL REMARKS
Appendix 1: Neutrix Calculus and Negligible Sequences of Functions
Appendix 2: The Embedding Impossibility Result of L.Schwartz
Appendix 3: A Nonlinear Extension of the Lax-Richtmyer Equivalence between Stability and Convergence of Difference Schemes
Appendix 4: The Cauchy-Bolzano Quotient Algebra Construction of the Real Numbers
REFERENCES