Qualitative estimates for partial differential equations: an introduction
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
CRC Press
1996
|
| Series: | Library of engineering mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 368 S. graph. Darst. |
| ISBN: | 0849385121 |
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MARC
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| 100 | 1 | |a Flavin, James N. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Qualitative estimates for partial differential equations |b an introduction |c James N. Flavin ; Salvatore Rionero |
| 264 | 1 | |a Boca Raton [u.a.] |b CRC Press |c 1996 | |
| 300 | |a 368 S. |b graph. Darst. | ||
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Record in the Search Index
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|---|---|
| adam_text | CONTENTS
PREFACE vii
LIST OF FREQUENTLY USED SYMBOLS xi
PART I: METHODOLOGY
1. PRELIMINARIES AND FUNDAMENTAL ISSUES
1.1 Introduction 1
1.2 Evolution equation and choice of the state space 1
1.3 Initial and boundary value problems 3
1.4 A first requirement of the model 3
1.5 Generalized solutions 4
1.6 Dynamical system generated by the evolution equation 7
1.7 Continuous dependence 9
1.8 Ill posed problems. Counterexamples to uniqueness 10
1.9 Liapunov stability 17
1.10 Topology dependent stability 20
1.11 Errors in formulating the model 23
1.12 Mathematical modelling. Reaction diffusion equations 28
1.13 Exercises 34
2. ESTIMATES BASED ON FIRST ORDER INEQUALITD3S
1: LIAPUNOV DIRECT METHOD
2.1 Introduction 45
2.2 General estimates based on first order inequalities 46
2.3 Liapunov functions 47
2.4 Direct method 48
2.5 The norm as Liapunov function 51
Contents
2.6 Asymptotic behaviour in time of the state vector 51
2.7 Liapunov functions for the reaction diffusion equations 54
2.8 Exercises 64
3. ESTIMATES BASED ON FIRST ORDER INEQUALITIES
2: VOLUME INTEGRAL METHOD
3.1 The volume integral method 75
3.1.1 Introduction 75
3.1.2 Description of method 76
3.1.3 A Neumann problem 77
3.1.4 Phragmen Lindelof (and other) estimates 81
3.1.5 Regions with curved boundaries 83
3.1.6 An initial boundary value problem 85
3.2 Miscellanea 87
3.2.1 Introduction 87
3.2.2 A simple lemma 88
3.2.3 Linear diffusion equations 90
3.2.4 Nonlinear diffusion equations: nonexistence etc. 94
3.3 Exercises 98
3.4 Appendix 104
4. ESTIMATES BASED ON SECOND ORDER INEQUALI¬
TIES
4.1 Introduction 107
4.2 Convexity/concavity; two examples of concavity 108
4.2.1 Nonlinear Schrodinger equation 110
4.2.2 Longitudinal motion of a nonlinear elastic rod 113
4.3 Logarithmic convexity and related concepts 115
4.3.1 Introduction and basics 115
4.3.2 Cauchy (ill posed) problem 117
4.4 The method of cross sections and generalized convexity 122
4.4.1 General Principles 122
4.4.2 Rectangular region: Dirichlet problem 126
4.4.3 Lemmas relevant to regions with convex (concave) boundaries 130
4.4.4 Non rectilinear cross sections: an example 137
4.4.5 Time dependent problem 138
4.4.6 Closure 140
4.5 Exercises 141
Contents
5. IDENTITIES HOLDING ALONG SOLUTIONS TO PART¬
IAL DIFFERENTIAL EQUATIONS : LAGRANGE IDEN¬
TITY METHOD
5.1 Introduction 147
5.2 Lagrange identity method for two P.D.E. 148
i 5.3 Operator equations 151
5.4 Application to backwards heat equation: continuous dependence 156
5.5 Equipartition of energy 158
5.6 A second identity for wave like equations 161
5.7 Exercises 163
5.8 Appendix 167
6. WEIGHTED INTEGRAL METHODS
6.1 Introduction 169
6.2 Weight function approach to uniqueness on bounded domains 169
6.3 Weight function approach to backwards uniqueness on bounded
domains 171
6.4 Weight function approach to uniqueness on unbounded domains 173
6.5 Weight function approach to Cauchy problem for the Korteweg
De Vries Burgers equation 175
6.6 Weighted Poincare inequalities 179
6.7 Method of weighted cross sectional integrals 183
6.8 Exercises 186
PART H: APPLICATIONS
7. ELLD7TIC EQUATIONS
7.1 Introduction 195
7.2 Laplace s equation 196
7.3 Displacement type problems in linear elasticity 202
7.3.1 Introduction 202
7.3.2 Volume integral approach 203
7.3.3 Cross sectional method 207
7.3.4 Weighted cross sectional method 212
7.3.5 Non null conditions on lateral boundary 216
7.4 Traction type problems in linear elasticity 219
7.4.1 Two dimensional stress systems: Saint Venant issue 219
7.4.2 Knowles analysis ( Volume integral method) 219
7.4.3 Cross sectional method 223
7.4.4 Analogous issues in three dimensions 226
7.5 Semilinear equation 229
7.6 Quasilinear P.D.E. 235
Contents
7.7 Exercises 239
8. PARABOLIC EQUATIONS
8.1 Introduction 245
8.2 Boundedness and global existence of solutions to a parabolic
equation in gradient form 246
8.3 Stability of the magnetic field of Couette Poiseuille flows in ani
sotropic magneto hydrodynamics 248
8.4 Asymptotic behaviour of reaction diffusion systems 251
8.4.1 Introduction 251
8.4.2 Asymptotic homogenization 251
8.4.3 Instability of the steady nonconstant solutions of reaction diffusion
equations with homogeneous Neumann boundary conditions 253
8.4.4 Travelling wave solutions of reaction diffusion equations 254
8.5 Stability of viscous flows in bounded regions 258
8.5.1 Nonlinear energy stability of steady viscous incompressible fluid
motions in bounded regions: Serrin s theorem 258
8.5.2 Connection between nonlinear energy stability and linear stability
of steady flows 261
8.5.3 Nonlinear energy stability of unsteady flows 263
8.6 Stability of viscous flows in unbounded regions 264
8.6.1 Introduction 264
8.6.2 A mixed boundary value problem: free boundary like conditions 266
8.6.3 Energy relation for the exterior of a starshaped bounded domain 265
8.6.4 Stability criteria 269
8.6.5 A non linear stability condition for a spatial sink 271
8.7 The onset of natural convection: Benard problem 272
8.7.1 Introduction 272
8.7.2 Perturbation equations to a nonconvecting stationary solution 273
8.7.3 Linear Liapunov stability 274
8.8 Exercises 280
9. HYPERBOLIC, RELATED EQUATIONS, AND OTHERS
9.1 Introduction 289
9.2 Monge Ampere equations 289
9.2.1 Introduction 289
9.2.2 Problems of Dirichlet type 293
9.2.3 Problems of initial boundary value type 296
9.2.4 Other equations of Monge Ampere type 304
9.3 Liapunov functions for some hyperbolic P.D.E.s 306
Contents
9.4 A hyperbolic heat conduction equation 309
9.5 Nonlinear stability of non viscous incompressible fluid motions:
Arnold s theorem 311
9.6 A qualitative analysis of a nonlinear differential equation of the
particle transport theory 315
9.6.1 Introduction 315
9.6.2 Global existence and uniqueness 316
9.6.3 Stability in the Li norm 318
9.6.4 Unconditional stability 320
9.7 I.B.V.P. in exterior regions 322
9.8 Exercises 329
A. APPENDIX: FUNDAMENTAL INEQUALITIES 335
A.I Cauchy inequality or arithmetic geometric inequality 335
A.2 Holder integral inequality 335
A.3 Sobolev inequality 336
A.4 Poincare inequality 337
A.5 Inequalities in one and two dimensions 338
A.6 A point wise estimate for functions p £ C1(Si) 341
A.7 A comparison theorem 342
REFERENCES 343
AUTHOR INDEX 357
SUBJECT INDEX 363
|
| any_adam_object | 1 |
| author | Flavin, James N. Rionero, Salvatore |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T18:02:29Z |
| institution | BVB |
| isbn | 0849385121 |
| language | English |
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| physical | 368 S. graph. Darst. |
| publishDate | 1996 |
| publishDateSearch | 1996 |
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| publisher | CRC Press |
| record_format | marc |
| series2 | Library of engineering mathematics |
| spelling | Flavin, James N. Verfasser aut Qualitative estimates for partial differential equations an introduction James N. Flavin ; Salvatore Rionero Boca Raton [u.a.] CRC Press 1996 368 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Library of engineering mathematics Équations aux dérivées partielles - Solutions numériques Differential equations, Partial Numerical solutions Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Rionero, Salvatore Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007368100&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Flavin, James N. Rionero, Salvatore Qualitative estimates for partial differential equations an introduction Équations aux dérivées partielles - Solutions numériques Differential equations, Partial Numerical solutions Partielle Differentialgleichung (DE-588)4044779-0 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4128130-5 |
| title | Qualitative estimates for partial differential equations an introduction |
| title_auth | Qualitative estimates for partial differential equations an introduction |
| title_exact_search | Qualitative estimates for partial differential equations an introduction |
| title_full | Qualitative estimates for partial differential equations an introduction James N. Flavin ; Salvatore Rionero |
| title_fullStr | Qualitative estimates for partial differential equations an introduction James N. Flavin ; Salvatore Rionero |
| title_full_unstemmed | Qualitative estimates for partial differential equations an introduction James N. Flavin ; Salvatore Rionero |
| title_short | Qualitative estimates for partial differential equations |
| title_sort | qualitative estimates for partial differential equations an introduction |
| title_sub | an introduction |
| topic | Équations aux dérivées partielles - Solutions numériques Differential equations, Partial Numerical solutions Partielle Differentialgleichung (DE-588)4044779-0 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Équations aux dérivées partielles - Solutions numériques Differential equations, Partial Numerical solutions Partielle Differentialgleichung Numerisches Verfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007368100&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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