Fourier analysis and nonlinear partial differential equations:
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Heidelberg
Springer
[2011]
|
| Series: | Grundlehren der mathematischen Wissenschaften
343 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | xv, 523 Seiten |
| ISBN: | 9783642168291 |
| ISSN: | 0072-7830 |
Staff View
MARC
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| 245 | 1 | 0 | |a Fourier analysis and nonlinear partial differential equations |c Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin |
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Record in the Search Index
| _version_ | 1807500491145347072 |
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| adam_text |
IMAGE 1
CONTENTS
1 BASIC ANALYSIS 1
1.1 BASIC REAL ANALYSIS 1
1.1.1 HOLDER AND CONVOLUTION INEQUALITIES 1
1.1.2 THE ATOMIC DECOMPOSITION 7
1.1.3 PROOF OF REFINED YOUNG INEQUALITY 8
1.1.4 A BILINEAR INTERPOLATION THEOREM 10
1.1.5 A LINEAR INTERPOLATION RESULT 11
1.1.6 THE HARDY-LITTLEWOOD MAXIMAL FUNCTION 13
1.2 THE FOURIER TRANSFORM 16
1.2.1 FOURIER TRANSFORMS OF FUNCTIONS AND THE SCHWARTZ SPACE 16 1.2.2
TEMPERED DISTRIBUTIONS AND THE FOURIER TRANSFORM 18 1.2.3 A FEW
CALCULATIONS OF FOURIER TRANSFORMS 23
1.3 HOMOGENEOUS SOBOLEV SPACES 25
1.3.1 DEFINITION AND BASIC PROPERTIES 25
1.3.2 SOBOLEV EMBEDDING IN LEBESGUE SPACES 29
1.3.3 THE LIMIT CASE H? 36
1.3.4 THE EMBEDDING THEOREM IN HOLDER SPACES 37
1.4 NONHOMOGENEOUS SOBOLEV SPACES ON R D 38
1.4.1 DEFINITION AND BASIC PROPERTIES 38
1.4.2 EMBEDDING 44
1.4.3 A DENSITY THEOREM 47
1.4.4 HARDY INEQUALITY 48
1.5 REFERENCES AND REMARKS 49
2 LITTLEWOOD-PALEY THEORY 51
2.1 FUNCTIONS WITH COMPACTLY SUPPORTED FOURIER TRANSFORMS 51 2.1.1
BERNSTEIN-TYPE LEMMAS 52
2.1.2 THE SMOOTHING EFFECT OF HEAT FLOW 53
2.1.3 THE ACTION OF A DIFFEOMORPHISM 56
2.1.4 THE EFFECTS OF SOME NONLINEAR FUNCTIONS 58
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1007313889
DIGITALISIERT DURCH
IMAGE 2
CONTENTS
2.2 DYADIC PARTITION OF UNITY 59
2.3 HOMOGENEOUS BESOV SPACES 63
2.4 CHARACTERIZATIONS OF HOMOGENEOUS BESOV SPACES 72 2.5 BESOV SPACES,
LEBESGUE SPACES, AND REFINED INEQUALITIES 78 2.6 HOMOGENEOUS
PARADIFFERENTIAL CALCULUS 85
2.6.1 HOMOGENEOUS BONY DECOMPOSITION 85
2.6.2 ACTION OF SMOOTH FUNCTIONS 93
2.6.3 TIME-SPACE BESOV SPACES 98
2.7 NONHOMOGENEOUS BESOV SPACES 98
2.8 NONHOMOGENEOUS PARADIFFERENTIAL CALCULUS 102
2.8.1 THE BONY DECOMPOSITION 102
2.8.2 THE PARALINEARIZATION THEOREM 104
2.9 BESOV SPACES AND COMPACT EMBEDDINGS 108
2.10 COMMUTATOR ESTIMATES 110
2.11 AROUND THE SPACE 5^ )OC 116
2.12 REFERENCES AND REMARKS 120
TRANSPORT AND TRANSPORT-DIFFUSION EQUATIONS 123 3.1 ORDINARY
DIFFERENTIAL EQUATIONS 124
3.1.1 THE CAUCHY-LIPSCHITZ THEOREM REVISITED 124 3.1.2 ESTIMATES FOR THE
FLOW 129
3.1.3 A BLOW-UP CRITERION FOR ORDINARY DIFFERENTIAL EQUATIONSL31 3.2
TRANSPORT EQUATIONS: THE LIPSCHITZ CASE 132
3.2.1 A PRIORI ESTIMATES IN GENERAL BESOV SPACES 132 3.2.2 REFINED
ESTIMATES IN BESOV SPACES WITH INDEX 0 135 3.2.3 SOLVING THE TRANSPORT
EQUATION IN BESOV SPACES 136 3.2.4 APPLICATION TO A SHALLOW WATER
EQUATION 140
3.3 LOSING ESTIMATES FOR TRANSPORT EQUATIONS 147
3.3.1 LINEAR LOSS OF REGULARITY IN BESOV SPACES 147 3.3.2 THE
EXPONENTIAL LOSS 151
3.3.3 LIMITED LOSS OF REGULARITY 153
3.3.4 A FEW APPLICATIONS 155
3.4 TRANSPORT-DIFFUSION EQUATIONS 156
3.4.1 A PRIORI ESTIMATES 157
3.4.2 EXPONENTIAL DECAY 163
3.5 REFERENCES AND REMARKS 166
QUASILINEAR SYMMETRIC SYSTEMS 169
4.1 DEFINITION AND EXAMPLES 169
4.2 LINEAR SYMMETRIC SYSTEMS 172
4.2.1 THE WELL-POSEDNESS OF LINEAR SYMMETRIC SYSTEMS 172 4.2.2 FINITE
PROPAGATION SPEED 180
4.2.3 FURTHER WELL-POSEDNESS RESULTS FOR LINEAR SYMMETRIC SYSTEMS 183
4.3 THE RESOLUTION OF QUASILINEAR SYMMETRIC SYSTEMS 187
IMAGE 3
CONTENTS XIII
4.3.1 PARALINEARIZATION AND ENERGY ESTIMATES 189
4.3.2 CONVERGENCE OF THE SCHEME 190
4.3.3 COMPLETION OF THE PROOF OF EXISTENCE 191
4.3.4 UNIQUENESS AND CONTINUATION CRITERION 192
4.4 DATA WITH CRITICAL REGULARITY AND BLOW-UP CRITERIA 193
4.4.1 CRITICAL BESOV REGULARITY 193
4.4.2 A REFINED BLOW-UP CONDITION 196
4.5 CONTINUITY OF THE FLOW MAP 198
4.6 REFERENCES AND REMARKS 201
5 THE INCOMPRESSIBLE NAVIER-STOKES SYSTEM 203
5.1 BASIC FACTS CONCERNING THE NAVIER-STOKES SYSTEM 204
5.2 WELL-POSEDNESS IN SOBOLEV SPACES 209
5.2.1 A GENERAL RESULT 209
5.2.2 THE BEHAVIOR OF THE H^ 1 NORM NEAR 0 214
5.3 RESULTS RELATED TO THE STRUCTURE OF THE SYSTEM 215
5.3.1 THE PARTICULAR CASE OF DIMENSION TWO 215
5.3.2 THE CASE OF DIMENSION THREE 217
5.4 AN ELEMENTARY V APPROACH 220
5.5 THE ENDPOINT SPACE FOR PICARD'S SCHEME 227
5.6 THE USE OF THE J^-SMOOTHING EFFECT OF THE HEAT FLOW 233
5.6.1 THE CANNONE-MEYER-PLANCHON THEOREM REVISITED . . . 234 5.6.2 THE
FLOW OF THE SOLUTIONS OF THE NAVIER-STOKES SYSTEM 236 5.7 REFERENCES AND
REMARKS 242
6 ANISOTROPIE VISCOSITY 245
6.1 THE CASE OF L 2 DATA WITH ONE VERTICAL DERIVATIVE IN L 2 246
6.2 A GLOBAL EXISTENCE RESULT IN ANISOTROPIE BESOV SPACES 254 6.2.1
ANISOTROPIE LOCALIZATION IN FOURIER SPACE 254
6.2.2 THE FUNCTIONAL FRAMEWORK 256
6.2.3 STATEMENT OF THE MAIN RESULT 258
6.2.4 SOME TECHNICAL LEMMAS 261
6.3 THE PROOF OF EXISTENCE 266
6.4 THE PROOF OF UNIQUENESS 276
6.5 REFERENCES AND REMARKS 289
7 EULER SYSTEM FOR PERFECT INCOMPRESSIBLE FLUIDS 291
7.1 LOCAL WELL-POSEDNESS RESULTS FOR INVISCID FLUIDS 292
7.1.1 THE BIOT-SAVART LAW 293
7.1.2 ESTIMATES FOR THE PRESSURE 296
7.1.3 ANOTHER FORMULATION OF THE EULER SYSTEM 301
7.1.4 LOCAL EXISTENCE OF SMOOTH SOLUTIONS 302
7.1.5 UNIQUENESS 304
7.1.6 CONTINUATION CRITERIA 307
7.2 GLOBAL EXISTENCE RESULTS IN DIMENSION TWO 310
IMAGE 4
CONTENTS
7.2.1 SMOOTH SOLUTIONS 311
7.2.2 THE BORDERLINE CASE 311
7.2.3 THE YUDOVICH THEOREM 312
7.3 THE INVISCID LIMIT 313
7.3.1 REGULARITY RESULTS FOR THE NAVIER-STOKES SYSTEM 314 7.3.2 THE
SMOOTH CASE 314
7.3.3 THE ROUGH CASE 316
7.4 VISCOUS VORTEX PATCHES 318
7.4.1 RESULTS RELATED TO STRIATED REGULARITY 319
7.4.2 A STATIONARY ESTIMATE FOR THE VELOCITY FIELD 320 7.4.3 UNIFORM
ESTIMATES FOR STRIATED REGULARITY 324 7.4.4 A GLOBAL CONVERGENCE RESULT
FOR STRIATED REGULARITY. 326 7.4.5 APPLICATION TO SMOOTH VORTEX
PATCHES 330
7.5 REFERENCES AND REMARKS 331
STRICHARTZ ESTIMATES AND APPLICATIONS TO SEMILINEAR DISPERSIVE EQUATIONS
335
8.1 EXAMPLES OF DISPERSIVE ESTIMATES 336
8.1.1 THE DISPERSIVE ESTIMATE FOR THE FREE TRANSPORT EQUATION 336
8.1.2 THE DISPERSIVE ESTIMATES FOR THE SCHROEDINGER EQUATION 337 8.1.3
INTEGRAL OF OSCILLATING FUNCTIONS 339
8.1.4 DISPERSIVE ESTIMATES FOR THE WAVE EQUATION 344 8.1.5 THE 1?
BOUNDEDNESS OF SOME FOURIER INTEGRAL OPERATORS346 8.2 BILINEAR METHODS
349
8.2.1 THE DUALITY METHOD AND THE TT* ARGUMENT 350 8.2.2 STRICHARTZ
ESTIMATES: THE CASE Q 2 351
8.2.3 STRICHARTZ ESTIMATES: THE ENDPOINT CASE Q = 2 352 8.2.4
APPLICATION TO THE CUBIC SEMILINEAR SCHROEDINGER EQUATION 355
8.3 STRICHARTZ ESTIMATES FOR THE WAVE EQUATION 359
8.3.1 THE BASIC STRICHARTZ ESTIMATE 359
8.3.2 THE REFINED STRICHARTZ ESTIMATE 362
8.4 THE QUINTIC WAVE EQUATION IN R 3 368
8.5 THE CUBIC WAVE EQUATION IN R 3 370
8.5.1 SOLUTIONS IN H 1 370
8.5.2 LOCAL AND GLOBAL WELL-POSEDNESS FOR ROUGH DATA 372 8.5.3 THE
NONLINEAR INTERPOLATION METHOD 374
8.6 APPLICATION TO A CLASS OF SEMILINEAR WAVE EQUATIONS 381 8.7
REFERENCES AND REMARKS 386
SMOOTHING EFFECT IN QUASILINEAR WAVE EQUATIONS 389 9.1 A WELL-POSEDNESS
RESULT BASED ON AN ENERGY METHOD 391 9.2 THE MAIN STATEMENT AND THE
STRATEGY OF ITS PROOF 401 9.3 REFINED PARALINEARIZATION OF THE WAVE
EQUATION 403
IMAGE 5
CONTENTS XV
9.4 REDUCTION TO A MICROLOCAL STRICHARTZ ESTIMATE 406
9.5 MICROLOCAL STRICHARTZ ESTIMATES 413
9.5.1 A RATHER GENERAL STATEMENT 413
9.5.2 GEOMETRICAL OPTICS 414
9.5.3 THE SOLUTION OF THE EIKONAL EQUATION 415
9.5.4 THE TRANSPORT EQUATION 419
9.5.5 THE APPROXIMATION THEOREM 421
9.5.6 THE PROOF OF THEOREM 9.16 423
9.6 REFERENCES AND REMARKS 427
10 THE COMPRESSIBLE NAVIER-STOKES SYSTEM 429
10.1 ABOUT THE MODEL 429
10.1.1 GENERAL OVERVIEW 430
10.1.2 THE BAROTROPIC NAVIER-STOKES EQUATIONS 432 10.2 LOCAL THEORY FOR
DATA WITH CRITICAL REGULARITY 433 10.2.1 SCALING INVARIANCE AND
STATEMENT OF THE MAIN RESULT .433 10.2.2 A PRIORI ESTIMATES 435
10.2.3 EXISTENCE OF A LOCAL SOLUTION 440
10.2.4 UNIQUENESS 445
10.2.5 A CONTINUATION CRITERION 450
10.3 LOCAL THEORY FOR DATA BOUNDED AWAY FROM THE VACUUM 451 10.3.1 A
PRIORI ESTIMATES FOR THE LINEARIZED MOMENTUM EQUATION 451
10.3.2 EXISTENCE OF A LOCAL SOLUTION 457
10.3.3 UNIQUENESS 460
10.3.4 A CONTINUATION CRITERION 462
10.4 GLOBAL EXISTENCE FOR SMALL DATA 462
10.4.1 STATEMENT OF THE RESULTS 463
10.4.2 A SPECTRAL ANALYSIS OF THE LINEARIZED EQUATION 464 10.4.3 A
PRIORI ESTIMATES FOR THE LINEARIZED EQUATION 466 10.4.4 PROOF OF GLOBAL
EXISTENCE 473
10.5 THE INCOMPRESSIBLE LIMIT 475
10.5.1 MAIN RESULTS 475
10.5.2 THE CASE OF SMALL DATA WITH CRITICAL REGULARITY 477 10.5.3 THE
CASE OF LARGE DATA WITH MORE REGULARITY 483 10.6 REFERENCES AND REMARKS
492
REFERENCES 497
LIST OF NOTATIONS 523
INDEX 527 |
| any_adam_object | 1 |
| author | Bahouri, Hajer Chemin, Jean-Yves 1959- Danchin, Raphaël 1971- |
| author_GND | (DE-588)143364219 (DE-588)143491873 (DE-588)143364286 |
| author_facet | Bahouri, Hajer Chemin, Jean-Yves 1959- Danchin, Raphaël 1971- |
| author_role | aut aut aut |
| author_sort | Bahouri, Hajer |
| author_variant | h b hb j y c jyc r d rd |
| building | Verbundindex |
| bvnumber | BV037187905 |
| classification_rvk | SK 450 SK 540 |
| ctrlnum | (OCoLC)706956659 (DE-599)DNB1007313889 |
| dewey-full | 515.353 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.353 |
| dewey-search | 515.353 |
| dewey-sort | 3515.353 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV037187905 |
| illustrated | Not Illustrated |
| indexdate | 2024-08-16T00:06:35Z |
| institution | BVB |
| isbn | 9783642168291 |
| issn | 0072-7830 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-021102390 |
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| physical | xv, 523 Seiten |
| publishDate | 2011 |
| publishDateSearch | 2011 |
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| publisher | Springer |
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| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spelling | Bahouri, Hajer Verfasser (DE-588)143364219 aut Fourier analysis and nonlinear partial differential equations Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin Berlin ; Heidelberg Springer [2011] © 2011 xv, 523 Seiten txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 343 0072-7830 Harmonische Analyse (DE-588)4023453-8 gnd rswk-swf Littlewood-Paley-Theorem (DE-588)4352642-1 gnd rswk-swf Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd rswk-swf Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 s Harmonische Analyse (DE-588)4023453-8 s Littlewood-Paley-Theorem (DE-588)4352642-1 s DE-604 Chemin, Jean-Yves 1959- Verfasser (DE-588)143491873 aut Danchin, Raphaël 1971- Verfasser (DE-588)143364286 aut Erscheint auch als Online-Ausgabe 978-3-642-16830-7 Grundlehren der mathematischen Wissenschaften 343 (DE-604)BV000000395 343 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3544047&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021102390&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bahouri, Hajer Chemin, Jean-Yves 1959- Danchin, Raphaël 1971- Fourier analysis and nonlinear partial differential equations Grundlehren der mathematischen Wissenschaften Harmonische Analyse (DE-588)4023453-8 gnd Littlewood-Paley-Theorem (DE-588)4352642-1 gnd Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd |
| subject_GND | (DE-588)4023453-8 (DE-588)4352642-1 (DE-588)4128900-6 |
| title | Fourier analysis and nonlinear partial differential equations |
| title_auth | Fourier analysis and nonlinear partial differential equations |
| title_exact_search | Fourier analysis and nonlinear partial differential equations |
| title_full | Fourier analysis and nonlinear partial differential equations Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin |
| title_fullStr | Fourier analysis and nonlinear partial differential equations Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin |
| title_full_unstemmed | Fourier analysis and nonlinear partial differential equations Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin |
| title_short | Fourier analysis and nonlinear partial differential equations |
| title_sort | fourier analysis and nonlinear partial differential equations |
| topic | Harmonische Analyse (DE-588)4023453-8 gnd Littlewood-Paley-Theorem (DE-588)4352642-1 gnd Nichtlineare partielle Differentialgleichung (DE-588)4128900-6 gnd |
| topic_facet | Harmonische Analyse Littlewood-Paley-Theorem Nichtlineare partielle Differentialgleichung |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3544047&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=021102390&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT bahourihajer fourieranalysisandnonlinearpartialdifferentialequations AT cheminjeanyves fourieranalysisandnonlinearpartialdifferentialequations AT danchinraphael fourieranalysisandnonlinearpartialdifferentialequations |