A stability criterion for high-frequency oscillations:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Paris
Société Mathématique de France
2015
|
| Schriftenreihe: | Mémoires de la SMF
142 |
| Schlagworte: | |
| Online-Zugang: | Klappentext Inhaltsverzeichnis |
| Beschreibung: | VII, 130 Seiten |
| ISBN: | 9782856298121 |
Internformat
MARC
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| 245 | 1 | 0 | |a A stability criterion for high-frequency oscillations |c Yong Lu, Benjamin Texier |
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Datensatz im Suchindex
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|---|---|
| adam_text | We show that a simple Levi compatibility condition determines stability of
WKB solutions to semilinear hyperbolic initial-value problems issued from
highly-oscillating initial data with large amplitudes. The compatibility condi-
tion involves the hyperbolic operator, the fundamental phase associated with
the initial oscillation, and the semilinear source term; it states roughly that
hyperbolicity is preserved around resonances.
If the compatibility condition is satisfied, the solutions are defined over time
intervals independent of the wavelength, and the associated WKB solutions are
stable under a large class of initial perturbations. If the compatibility condition
is not satisfied, resonances are exponentially amplified, and arbitrarily small
initial perturbations can destabilize the WKB solutions in small time.
In the unstable case, the key observation is that resonances correspond to
weakly hyperbolic frequencies; the amplification proof then relies on a short-
time Duhamel representation formula for solutions of zeroth-order pseudo-
differential equations.
Our examples include coupled Klein-Gordon systems, and systems describ-
ing Raman and Brillouin instabilities.
CONTENTS^
1. Introduction .............................................................. 1
1.1. Background ............................................................. 2
1.2. Resonances, transparency, and WKB solutions .......................... 3
1.3. A criterion for stability ............................................ 6
1.4. On the class of initial perturbations ................................ 10
1.5. Overview of the results .............................................. 11
1.6. On related instability results ........................................ 12
1.7. Examples .............................................................. 13
1.8. Open problems ......................................................... 14
2. Assumptions and results .................................................... 17
2.1. Main result ........................................................... 19
2.2. Comments .............................................................. 21
2.3. Extensions ...................................................... 23
2.3.1. Several non֊transparent resonances ................................ 23
2.3.2. All non-transparent resonances are amplified ..................... 24
2.3.3. Improved spatial localization ..................................... 20
2.3.4. A greater deviation estimate ...................................... 26
3. Main proof.................................................................. 29
3.1. Proof of Theorem 2.7: instability .................................... 29
3.1.1. Overview of the instability proof ............................... 29
3.1.2. Preparation ....................................................... 29
3.1.3. Duhamel representation ............................................ 39
3.1.4. Lower bound ....................................................... 40
3.1.5. Existence over logarithmic times and upper bound ................ 44
3.1.6. Endgame: proof of the deviation estimate (2.16) ................... 49
3.2. Proof of Theorem 2.7: stability ...................................... 51
3.2.1. Symmetrizcr ....................................................... 52
3.2.2. Uniform bounds .................................................... 56
vi CONTENTS
4. Other proofs ............................................................ 57
4.1. Proof of Theorem 2.9 ............................................... 57
4.1.1. Coordinatization ................................................ 57
4.1.2. Normal form reduction ...................................... 59
4.1.3. Space-frequency localization .................................... 54
4.1.4. Conclusion ................................................... · - * 55
4.2. Proof of Theorem 2.11 .............................................. 56
4.2.1. Preparation ................................................... 56
4.2.2. Upper bounds .................................................. 58
4.2.3. Conclusion .................................................... 59
4.3. Proof of Theorem 2.12 ................................................ 70
4.4. Proof of Theorem 2.13 ................................................ 71
5. Examples ................................................................ 73
5.1. Raman and Brillouin instabilities .................................. 73
5.1.1. Three-wave interaction systems .................................. 74
5.1.2. Derivation of three-wave interaction systems from Euler-Maxwell ... 76
5.1.3. Raman ......................................................... 85
5.1.4. Brillouin ..................................................... 86
5.2. Coupled Klein-G or don systems with equal masses ................... 87
5.2.1. Verification of Assumption 2.1: smooth spectral decomposition.. 90
5.2.2. Verification of Assumption 2.2: WKB expansion ................... 90
5.2.3. Verification of Assumption 2.8: resonances and transparency ..... 91
5.2.4. Stability index ................................................. 93
5.3. Coupled Klein-Gordon systems with different masses ................. 94
5.3.1. Verification of Assumption 2.2: WKB expansion ................... 94
5.3.2. Verification of Assumption 2.8: resonances and transparency ..... 96
5.3.3. Stability index ................................................. 97
6. Appendix .................................................................. 99
6.1. Symbols and operators .............................................. 99
6.1.1. Estimates for Fourier multipliers ............................... 99
6.1.2. Estimates for pseudo-differential operators .....................109
6.1.3. Product laws in weighted Sobolev spaces .........................103
6.2. An integral representation formula .................................103
6.3. Bounds for the symbolic flow .......................................108
5*3.1. The autonomous case .............................................108
6.3.2. The general case ................................................Ill
6.4. A Gronwall Lemma......................................................112
6.O. On regularity of the spectral decomposition ........................113
6.6. On existence of WKB approximate solutions ............................114
6.7. On structure of the resonant set ....................................121
6.7.1. Euler-Maxwell .................................................. 122
6.7.2. Maxwell-Landau-Lifschitz ........................................122
6.8. Notation index ...................................................... P23
MÉMOIRES DE LA SMF J 42
CONTENTS
vii
6.9. Parameter list ...............................................124
Bibliography ........................................................127
|
| any_adam_object | 1 |
| author | Lu, Yong Texier, Benjamin 1976- |
| author_GND | (DE-588)1077268556 (DE-588)1054016429 |
| author_facet | Lu, Yong Texier, Benjamin 1976- |
| author_role | aut aut |
| author_sort | Lu, Yong |
| author_variant | y l yl b t bt |
| building | Verbundindex |
| bvnumber | BV042919139 |
| classification_rvk | SI 690 |
| ctrlnum | (OCoLC)923491305 (DE-599)OBVAC12646826 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV042919139 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:12:49Z |
| institution | BVB |
| isbn | 9782856298121 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-028346672 |
| oclc_num | 923491305 |
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| physical | VII, 130 Seiten |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| publisher | Société Mathématique de France |
| record_format | marc |
| series | Mémoires de la SMF |
| series2 | Mémoires de la SMF |
| spelling | Lu, Yong Verfasser (DE-588)1077268556 aut A stability criterion for high-frequency oscillations Yong Lu, Benjamin Texier Paris Société Mathématique de France 2015 VII, 130 Seiten txt rdacontent n rdamedia nc rdacarrier Mémoires de la SMF 142 Hochfrequenz (DE-588)4160130-0 gnd rswk-swf Schwingung (DE-588)4053999-4 gnd rswk-swf Hochfrequenz (DE-588)4160130-0 s Schwingung (DE-588)4053999-4 s DE-604 Texier, Benjamin 1976- Verfasser (DE-588)1054016429 aut Mémoires de la SMF 142 (DE-604)BV000000921 142 Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028346672&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028346672&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lu, Yong Texier, Benjamin 1976- A stability criterion for high-frequency oscillations Mémoires de la SMF Hochfrequenz (DE-588)4160130-0 gnd Schwingung (DE-588)4053999-4 gnd |
| subject_GND | (DE-588)4160130-0 (DE-588)4053999-4 |
| title | A stability criterion for high-frequency oscillations |
| title_auth | A stability criterion for high-frequency oscillations |
| title_exact_search | A stability criterion for high-frequency oscillations |
| title_full | A stability criterion for high-frequency oscillations Yong Lu, Benjamin Texier |
| title_fullStr | A stability criterion for high-frequency oscillations Yong Lu, Benjamin Texier |
| title_full_unstemmed | A stability criterion for high-frequency oscillations Yong Lu, Benjamin Texier |
| title_short | A stability criterion for high-frequency oscillations |
| title_sort | a stability criterion for high frequency oscillations |
| topic | Hochfrequenz (DE-588)4160130-0 gnd Schwingung (DE-588)4053999-4 gnd |
| topic_facet | Hochfrequenz Schwingung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028346672&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028346672&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000921 |
| work_keys_str_mv | AT luyong astabilitycriterionforhighfrequencyoscillations AT texierbenjamin astabilitycriterionforhighfrequencyoscillations |