Nonlinear Partial Differential Equations (eBook)

Sequential and weak solutions

E.E. Rosinger (Herausgeber)

eBook Download: PDF

1980 | 1. Auflage
316 Seiten
Elsevier Science (Verlag)
978-0-08-087155-4 (ISBN)

Infos

Nonlinear partial differential equations

Inhalt

Front Cover 1
Nonlinear Partial Differntial Equations: Sequential and Weak Solutions 4
Copyright Page 5
TABLE OF CONTENTS 14
Chapter 1. Sequential Solutions of Nonlinear PDEs 22
0 Introduction 22
1 The Problem of Stability of Sequential Solutions 23
2 Defining the Quotient Spaces and Algebras 26
3 The Notion of Sequential Solution 31
4 Exactness of Sequential Solutions 33
5 Interplay between Stability, Generality and Exactness 34
6 The Deficiency of the Distributional Approach Concerning Exactness 38
7 Resolution of Nowhere Dense Singularities 40
8 Preservation of Ellipticity and Hypoellipticity 45
9 General Nonlinear PDEs 46
10 Variable Transforms 48
11 Stronger Conditions for Partial Derivative Operators 51
12 Systems of Nonlinear PDEs 53
13 A Cesaro Characterization of the Nowhere Dense Ideal 55
Chapter 2. Necessary and/or Sufficient Conditions for the Existence of Sequential Solutions 58
0 Introduction 58
1 Subsequence Quasi Invariant Sequential Solutions 59
2 Applications to Linear and Nonlinear PDEs 65
3 Subsequence Invariant Sequential Solution 67
4 Resolvent Sets 72
5 Domains of Solvability 76
6 Remarks on Lemma 2 83
Chapter 3. Algebras Containing the Distribution 86
0 Introduction 86
1 Embedding the Distributions into Quotient Algebras 86
2 Simpler Inclusion Diagrams 89
3 Regularizations and Chains of Quotient Algebras Containing the Distributions 90
4 Existence, Characterization and Construction of Regularizations 96
5 Additional Chains of Quotient Algebras Containing Distributions 109
6 Quotient Algebras Containing Large Subspaces of Distributions 118
7 Quotient Algebras Possessing Special Properties 125
8 Definition of Positive Powers for Certain Generalized Functions 128
9 Limitations on Embedding Smooth Functions into Chains of Quotient Algebras 131
10 Special Classes of Regular Ideals 134
11 The Proof of Lemma 1 136
12 Example of Shrinking Non-d Sequence 138
13 Inexistence of the Largest Regular Ideals 139
Chapter 4. Resolution of Singularities of Weak Solutions for Polynomial Nonlinear PDEs 142
0 Introduction 142
1 The Case of Simple Polynomial Nonlinear PDEs 143
2 Resolution of Singularities of Nonlinear Shock Waves 152
3 Resolution of Singularities of Klein-Gordon Type Nonlinear Waves 154
4 Resolution of Singularities of Weak Solutions for General Polynomial Nonlinear PDEs 155
5 Junction Conditions and Resolution of Singularities of Weak Solutions for the Equations of Magneto-hydrodynamics and General Relativity 160
6 Resoluble Systems of Polynomial Nonlinear PDEs 173
7 Computation of the Junction Conditions 176
8 Examples of Resoluble Systems of PDEs 177
Chapter 5. Stability and Exactness of Sequential and Weak Solutions for Polynomial Nonlinear PDEs 184
0 Introduction 184
1 Basic Facts and Remarks Concerning Stabilit 184
2 Maximal Stability 186
3 Interplay between Stability and Exactness 188
4 Remarks in the Finite Smoothness Case 188
5 Stability of Regular Weak Solutions 189
6 Stability of Solutions with Jump Discontinuities for Resoluble Systems of PDEs 191
Chapter 6. Characterization of the Necessary Structure of the Algebras Containing the Distributions 194
0 Introduction 194
1 Zero Filters 195
2 Density Characterization of Z-Filters 197
3 Large C°–Regular Ideals 199
4 Further Properties of Z-Filters 203
5 Construction and Characterization of a Class of Vanishing Ideals 206
6 Countable Reduced Products of Vector Spaces 215
Chapter 7. Quantum Scattering in Potentials Positive Powers of the Dirac d Distribution 220
0 Introduction 220
1 Wave Function Solutions and Junction Relations 220
2 Weak Wave Function Solutions 222
3 Smooth Representations forthe Dirac d Distribution and Smooth Weak Wave Function Solutions 231
4 Wave Function Solutions in the Algebras Containing the Distributions 235
5 Stability and Exactness of the Wave Function Solutions 241
Chapter 8. Products with Dirac d Distribution 244
0 Introduction 244
1 A Class of Regularizations 244
2 Products with Dirac d Distributions 250
3 Application to a Riccati Differential Equation 255
4 Validity of Formulas in Quantum Mechanics 256
5 The Existence of the Sequences of Functions in Zd 263
6 Remark on Nilpotent Elements in the Quotient Algebras 270
Chapter 9. Linear Independent Families of Dirac d Distributions at a Point 272
0 Introduction 272
1 Compatible Quotient Algebras and Independent Variable Transforms 272
2 Linear Independent Families of Dirac d Distributions at a Point 276
3 Generalized Dirac d Distributions 278
Chapter 10. Support and Local Properties 286
0 Introduction 286
1 Support of Elements in Quotient Algebras 286
2 Localization with in the Quotient Algebras 291
3 Equivalence between S=O and supp S=Q 299
4 Domains of Solvability for Polynomial Nonlinear PDEs 300
FINAL REMARKS 302
Appendix 1: Neutrix Calculus and Negligible Sequences of Functions 306
Appendix 2: The Embedding Impossibility Result of L.Schwartz 310
Appendix 3: A Nonlinear Extension of the Lax-Richtmyer Equivalence between Stability and Convergence of Difference Schemes 314
Appendix 4: The Cauchy-Bolzano Quotient Algebra Construction of the Real Numbers 322
REFERENCES 326

Produkt-Details

• Erscheint lt. Verlag: 1.1.1980
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Naturwissenschaften
• Themenwelt: Technik
• ISBN-10: 0-08-087155-0 / 0080871550
• ISBN-13: 978-0-08-087155-4 / 9780080871554