The mathematics of shock reflection-diffraction and von Neumann's conjectures:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Princeton and Oxford
Princeton University Press
2018
|
| Schriftenreihe: | Annals of mathematics studies
number 197 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | xiv, 814 Seiten Illustrationen |
| ISBN: | 9780691160542 9780691160559 |
Internformat
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| 245 | 1 | 0 | |a The mathematics of shock reflection-diffraction and von Neumann's conjectures |c Gui-Qiang G. Chen and Mikhail Feldman |
| 264 | 1 | |a Princeton and Oxford |b Princeton University Press |c 2018 | |
| 300 | |a xiv, 814 Seiten |b Illustrationen | ||
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| 490 | 1 | |a Annals of mathematics studies |v number 197 | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Shock waves |x Diffraction | |
| 650 | 4 | |a Shock waves |x Mathematics | |
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Datensatz im Suchindex
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|---|---|
| adam_text | The Mathematics of
Shock Reflection-Diffraction
and
von Neumann’s Conjectures
Gui-Qiang G Chen
Mikhail Feldman
PRINCETON UNIVERSITY PRESS
PRINCETON AND OXFORD
2018
Contents
Preface
xi
I Shock Reflection-Diffraction, Nonlinear Conservation Laws
of Mixed Type, and von Neumann’s Conjectures 1
1 Shock Reflection-Diffraction, Nonlinear Partial Differential
Equations of Mixed Type, and Free Boundary Problems 3
2 Mathematical Formulations and Main Theorems 16
2 1 The potential flow equation 16
2 2 Mathematical problems for shock reflection-diffraction 19
2 3 Weak solutions of Problem 221 and Problem 223 23
2 4 Structure of solutions: Regular reflection-diffraction
configurations 24
2 5 Existence of state (2) and continuous dependence on
the parameters 27
2 6 Von Neumann’s conjectures, Problem 261 (free boundary
problem), and main theorems 28
3 Main Steps and Related Analysis in the Proofs of the Main
Theorems 37
3 1 Normal reflection 37
3 2 Main steps and related analysis in the proof of the sonic conjecture 37
3 3 Main steps and related analysis in the proof of the detachment
conjecture 55
3 4 Appendix: The method of continuity and fixed point theorems 65
II Elliptic Theory and Related Analysis for Shock
Reflection-Diffraction 67
4 Relevant Results for Nonlinear Elliptic Equations of Second
Order 69
4 1 Notations: Holder norms and ellipticity 69
4 2 Quasilinear uniformly elliptic equations 72
VI
CONTENTS
4 3 Estimates for Lipschitz solutions of elliptic boundary value
problems 105
4 4 Comparison principle for a mixed boundary value problem in a
domain with corners 142
4 5 Mixed boundary value problems in a domain with corners for
uniforihly elliptic equations 145
4 6 Holder spaces with parabolic scaling 192
4 7 Degenerate elliptic equations 197
4 8 Uniformly elliptic equations in a curved triangle-shaped domain
with one-point Dirichlet condition 207
5 Basic Properties of the Self-Similar Potential Flow Equation 216
5 1 Some basic facts and formulas for the potential flow equation 216
5 2 Interior ellipticity principle for self-similar potential flow 222
5 3 Ellipticity principle for self-similar potential flow with slip
condition on the flat boundary 227
III Proofs of the Main Theorems for the Sonic Conjecture
and Related Analysis 229
6 Uniform States and Normal Reflection 231
6 1 Uniform states for self-similar potential flow 231
6 2 Normal reflection and its uniqueness 238
6 3 The self-similar potential flow equation in the coordinates
flattening the sonic circle of a uniform state 239
7 Local Theory and von Neumann’s Conjectures 242
7 1 Local regular reflection and state (2) 242
7 2 Local theory of shock reflection for large-angle wedges 245
7 3 The shock polar for steady potential flow and its properties 248
7 4 Local theory for shock reflection: Existence of the weak and
strong state (2) up to the detachment angle 263
7 5 Basic properties of the weak state (2) and the definition of
supersonic and subsonic wedge angles 273
7 6 Von Neumann’s sonic and detachment conjectures 279
8 Admissible Solutions and Features of Problem 261 281
8 1 Definition of admissible solutions 281
8 2 Strict directional monotonicity for admissible solutions 286
8 3 Appendix: Properties of solutions of Problem 261 for large-
angle wedges 305
CONTENTS
vii
9 Uniform Estimates for Admissible Solutions 319
9 1 Bounds of the elliptic domain D and admissible solution ip in Q, 319
9 2 Regularity of admissible solutions away from rshock U rsonic U {P3} 322
9 3 Separation of rshock from rsym 339
9 4 Lower bound for the distance between rshock and rwedge 341
9 5 Uniform positive lower bound for the distance between rshock
and the sonic circle of state (1) 354
9 6 Uniform estimates of the ellipticity constant in ft rsonic 369
10 Regularity of Admissible Solutions away from the Sonic Arc 382
10 1 rshock as a graph in the radial directions with respect to state (1) 382
10 2 Boundary conditions on rshock for admissible solutions 385
10 3 Local estimates near r8hock 387
10 4 The critical angle and the distance between rshock and rwedge • 389
10 5 Regularity of admissible solutions away from rsonic 390
10 6 Regularity of the limit of admissible solutions away from rsonic • 392
11 Regularity of Admissible Solutions near the Sonic Arc 396
11 1 The equation near the sonic arc and structure of elliptic degeneracy 396
11 2 Structure of the neighborhood of rsonic in D and estimates of
) 398
11 3 Properties of the Rankine-Hugoniot condition on rshock near rsoniC 413
11 4 ^’“-estimates in the scaled Holder norms near rs0nic 421
11 5 The reflected-diffracted shock is C2,a near Pi 431
11 6 Compactness of the set of admissible solutions 434
12 Iteration Set and Solvability of the Iteration Problem 440
12 1 Statement of the existence results 440
12 2 Mapping to the iteration region 440
12 3 Definition of the iteration set 461
12 4 The equation for the iteration 469
12 5 Assigning a boundary condition on the shock for the iteration 485
12 6 Normal reflection, iteration set, and admissible solutions 504
12 7 Solvability of the iteration problem and estimates of solutions 505
12 8 Openness of the iteration set 520
13 Iteration Map, Fixed Points, and Existence of Admissible
Solutions up to the Sonic Angle 524
13 1 Iteration map 524
13 2 Continuity and compactness of the iteration map 528
13 3 Normal reflection and the iteration map for 0W = ^ 530
13 4 Fixed points of the iteration map for 0W ^ are admissible
solutions 531
13 5 Fixed points cannot lie on the boundary of the iteration set 557
CONTENTS
viii
13 6 Proof of the existence of solutions up to the sonic angle or the
critical angle 559
13 7 Proof of Theorem 2 6 2: Existence of global solutions up to the
sonic angle when u c 559
13 8 Proof of Theorem 2 6 4: Existence of global solutions when u c 562
13 9 Appendix: Extension of the functions in weighted spaces 564
14 Optimal Regularity of Solutions near the Sonic Circle 586
14 1 Regularity of solutions near the degenerate boundary for
nonlinear degenerate elliptic equations of second order 586
14 2 Optimal regularity of solutions across rsonic 599
IV Subsonic Regular Reflection-Diffraction and Global
Existence of Solutions up to the Detachment Angle 613
15 Admissible Solutions and Uniform Estimates up to the
Detachment Angle 615
15 1 Definition of admissible solutions for the supersonic and subsonic
reflections 615
15 2 Basic estimates for admissible solutions up to the detachment
angle 617
15 3 Separation of rshOCk from rsym 618
15 4 Lower bound for the distance between rshock and rwedge away
from P0 618
15 5 Uniform positive lower bound for the distance between rshock
and the sonic circle of state (1) 621
15 6 Uniform estimates of the ellipticity constant 622
15 7 Regularity of admissible solutions away from rsonic 625
16 Regularity of Admissible Solutions near the Sonic Arc
and the Reflection Point 629
16 1 Pointwise and gradient estimates near rsoniC and the reflection
point 629
16 2 The Rankine-Hugoniot condition on rshock near r8onic and the
reflection point 633
16 3 A priori estimates near rsonic in the supersonic-away-ffom-sonic
case 635
16 4 A priori estimates near rsonic in the supersonic-near-sonic case 636
16 5 A priori estimates near the reflection point in the subsonic-near-
sonic case 656
16 6 A priori estimates near the reflection point in the subsonic-away-
from-sonic case 665
CONTENTS ix
17 Existence of Global Regular Reflection-Diffraction Solutions
up to the Detachment Angle 690
17 1 Statement of the existence results 690
17 2 Mapping to the iteration region 690
17 3 Iteration set 707
17 4 Existence and estimates of solutions of the iteration problem 725
17 5 Openness of the iteration set 737
17 6 Iteration map and its properties 741
17 7 Compactness of the iteration map 745
17 8 Normal reflection and the iteration map for 0W = ^ 747
17 9 Fixed points of the iteration map for 0W § are admissible
solutions 747
17 10 Fixed points cannot lie on the boundary of the iteration set 752
17 11 Proof of the existence of solutions up to the critical angle 753
17 12 Proof of Theorem 2 6 6: Existence of global solutions up to the
detachment angle when ui c 753
17 13 Proof of Theorem 2 6 8: Existence of global solutions when
u ci 753
V Connections and Open Problems 755
18 The Full Euler Equations and the Potential Flow Equation 757
18 1 The full Euler equations 757
18 2 Mathematical formulation I: Initial-boundary value problem 761
18 3 Mathematical formulation II: Boundary value problem 762
18 4 Normal reflection 768
18 5 Local theory for regular reflection near the reflection point 769
18 6 Von Neumann’s conjectures 777
18 7 Connections with the potential flow equation 781
19 Shock Reflection-Diffraction and New Mathematical
Challenges 785
19 1 Mathematical theory for multidimensional conservation laws 785
19 2 Nonlinear partial differential equations of mixed elliptic-hyperbolic
type 788
19 3 Free boundary problems and techniques 790
19 4 Numerical methods for multidimensional conservation laws 791
Bibliography 794
Index
|
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| indexdate | 2024-07-10T08:04:07Z |
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| language | English |
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| physical | xiv, 814 Seiten Illustrationen |
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| spelling | Chen, Gui-Qiang 1963- Verfasser (DE-588)140086951 aut The mathematics of shock reflection-diffraction and von Neumann's conjectures Gui-Qiang G. Chen and Mikhail Feldman Princeton and Oxford Princeton University Press 2018 xiv, 814 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Annals of mathematics studies number 197 Includes bibliographical references and index Shock waves Diffraction Shock waves Mathematics Von Neumann algebras Feldman, Mikhail 1960- Verfasser (DE-588)1154964019 aut Annals of mathematics studies number 197 (DE-604)BV000000991 197 HEBIS Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030289901&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chen, Gui-Qiang 1963- Feldman, Mikhail 1960- The mathematics of shock reflection-diffraction and von Neumann's conjectures Annals of mathematics studies Shock waves Diffraction Shock waves Mathematics Von Neumann algebras |
| title | The mathematics of shock reflection-diffraction and von Neumann's conjectures |
| title_auth | The mathematics of shock reflection-diffraction and von Neumann's conjectures |
| title_exact_search | The mathematics of shock reflection-diffraction and von Neumann's conjectures |
| title_full | The mathematics of shock reflection-diffraction and von Neumann's conjectures Gui-Qiang G. Chen and Mikhail Feldman |
| title_fullStr | The mathematics of shock reflection-diffraction and von Neumann's conjectures Gui-Qiang G. Chen and Mikhail Feldman |
| title_full_unstemmed | The mathematics of shock reflection-diffraction and von Neumann's conjectures Gui-Qiang G. Chen and Mikhail Feldman |
| title_short | The mathematics of shock reflection-diffraction and von Neumann's conjectures |
| title_sort | the mathematics of shock reflection diffraction and von neumann s conjectures |
| topic | Shock waves Diffraction Shock waves Mathematics Von Neumann algebras |
| topic_facet | Shock waves Diffraction Shock waves Mathematics Von Neumann algebras |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=030289901&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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