Dispersive partial differential equations: wellposedness and applications
The area of nonlinear dispersive partial differential equations (PDEs) is a fast developing field which has become exceedingly technical in recent years. With this book, the authors provide a self-contained and accessible introduction for graduate or advanced undergraduate students in mathematics, e...
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| Hauptverfasser: | , |
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Cambridge
Cambridge University Press
2016
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| Schriftenreihe: | London Mathematical Society student texts
86 |
| Schlagworte: | |
| Online-Zugang: | BSB01 FHN01 UBR01 Volltext Inhaltsverzeichnis |
| Zusammenfassung: | The area of nonlinear dispersive partial differential equations (PDEs) is a fast developing field which has become exceedingly technical in recent years. With this book, the authors provide a self-contained and accessible introduction for graduate or advanced undergraduate students in mathematics, engineering, and the physical sciences. Both classical and modern methods used in the field are described in detail, concentrating on the model cases that simplify the presentation without compromising the deep technical aspects of the theory, thus allowing students to learn the material in a short period of time. This book is appropriate both for self-study by students with a background in analysis, and for teaching a semester-long introductory graduate course in nonlinear dispersive PDEs. Copious exercises are included, and applications of the theory are also presented to connect dispersive PDEs with the more general areas of dynamical systems and mathematical physics |
| Beschreibung: | 1 Online-Ressource (xvi, 186 Seiten) |
| ISBN: | 9781316563267 |
| DOI: | 10.1017/CBO9781316563267 |
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Datensatz im Suchindex
| _version_ | 1804176880500211712 |
|---|---|
| adam_text | Titel: Dispersive partial differential equations
Autor: Erdoğan, M. Burak
Jahr: 2016
Contents
Preface page ix
Notation xiv
1 Preliminaries and tools 1
Exercises 8
2 Linear dispersive equations 12
2.1 Estimates on the real line 14
2.2 Estimates on the torus 22
2.3 The Talbot effect 32
Exercises 46
3 Methods for establishing wellposedness 49
3.1 The energy method 50
3.1.1 A priori bounds 51
3.1.2 Existence and uniqueness 52
3.1.3 Growth bounds for KdV with potential 59
3.2 Oscillatory integral method 60
3.3 Restricted norm method 64
3.3.1 1? Solutions of KdV on the real line 64
3.3.2 Low regularity Solutions of KdV on the torus 70
3.3.3 Forced and damped KdV with a potential 80
3.4 Differentiation by parts on the torus: unconditional
wellposedness 82
3.5 Local theory for NLS on the torus 90
3.5.1 L2 wellposedness of cubic NLS on the torus 91
3.5.2 Hs local wellposedness of the quintic NLS on
the torus 93
3.6 Illposedness results 97
Exercises 107
vii
viii Contents
4 Global dynamics of nonlinear dispersive PDEs 111
4.1 Smoothing for nonlinear dispersive PDEs on the torus 112
4.1.1 Cubic NLS on the torus 112
4.1.2 The KdV equation on the torus 117
4.1.3 Proof of Proposition 4.7 127
4.2 High-low decomposition method 129
4.3 The /-method for the quintic NLS equation on the torus 135
Exercises 151
5 Applications of smoothing estimates 154
5.1 Bounds for higher order Sobolev norms 154
5.2 Almost everywhere convergence to initial data 160
5.3 Nonlinear Talbot effect 162
5.4 Global attractors for dissipative and dispersive PDEs 164
5.4.1 The global attractor is trivial for large damping 169
5.4.2 Bounds on the forced KdV equation 171
Exercises 172
References 175
Index 185
|
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| author | Erdoğan, M. Burak Tzirakis, Nikolaos 1975- |
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| discipline | Mathematik |
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| spelling | Erdoğan, M. Burak Verfasser (DE-588)1120320682 aut Dispersive partial differential equations wellposedness and applications M. Burak Erdogan, University of Illinois, Urbana-Champaign, Nikolaos Tzirakis, University of Illinois, Urbana-Champaign Cambridge Cambridge University Press 2016 1 Online-Ressource (xvi, 186 Seiten) txt rdacontent c rdamedia cr rdacarrier London Mathematical Society student texts 86 The area of nonlinear dispersive partial differential equations (PDEs) is a fast developing field which has become exceedingly technical in recent years. With this book, the authors provide a self-contained and accessible introduction for graduate or advanced undergraduate students in mathematics, engineering, and the physical sciences. Both classical and modern methods used in the field are described in detail, concentrating on the model cases that simplify the presentation without compromising the deep technical aspects of the theory, thus allowing students to learn the material in a short period of time. This book is appropriate both for self-study by students with a background in analysis, and for teaching a semester-long introductory graduate course in nonlinear dispersive PDEs. Copious exercises are included, and applications of the theory are also presented to connect dispersive PDEs with the more general areas of dynamical systems and mathematical physics Differential equations, Partial / Textbooks Differential equations, Nonlinear / Textbooks Nonlinear partial differential operators Tzirakis, Nikolaos 1975- Verfasser (DE-588)1120320860 aut Erscheint auch als Druck-Ausgabe 978-1-107-14904-5 Erscheint auch als Druck-Ausgabe 978-1-316-60293-5 https://doi.org/10.1017/CBO9781316563267 Verlag URL des Erstveröffentlichers Volltext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029349201&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Erdoğan, M. Burak Tzirakis, Nikolaos 1975- Dispersive partial differential equations wellposedness and applications Differential equations, Partial / Textbooks Differential equations, Nonlinear / Textbooks Nonlinear partial differential operators |
| title | Dispersive partial differential equations wellposedness and applications |
| title_auth | Dispersive partial differential equations wellposedness and applications |
| title_exact_search | Dispersive partial differential equations wellposedness and applications |
| title_full | Dispersive partial differential equations wellposedness and applications M. Burak Erdogan, University of Illinois, Urbana-Champaign, Nikolaos Tzirakis, University of Illinois, Urbana-Champaign |
| title_fullStr | Dispersive partial differential equations wellposedness and applications M. Burak Erdogan, University of Illinois, Urbana-Champaign, Nikolaos Tzirakis, University of Illinois, Urbana-Champaign |
| title_full_unstemmed | Dispersive partial differential equations wellposedness and applications M. Burak Erdogan, University of Illinois, Urbana-Champaign, Nikolaos Tzirakis, University of Illinois, Urbana-Champaign |
| title_short | Dispersive partial differential equations |
| title_sort | dispersive partial differential equations wellposedness and applications |
| title_sub | wellposedness and applications |
| topic | Differential equations, Partial / Textbooks Differential equations, Nonlinear / Textbooks Nonlinear partial differential operators |
| topic_facet | Differential equations, Partial / Textbooks Differential equations, Nonlinear / Textbooks Nonlinear partial differential operators |
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