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Dispersive partial differential equations : wellposedness and applications

Author: M Burak Erdogan; Nikolaos Tzirakis
Publisher: Cambridge, United Kingdom : Cambridge University Press, 2016.
Series: London Mathematical Society student texts, 86.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:

Introduces nonlinear dispersive partial differential equations in a detailed yet elementary way without compromising the depth and richness of the subject.

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Genre/Form: Textbooks
Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: M Burak Erdogan; Nikolaos Tzirakis
ISBN: 9781107149045 1107149045 9781316602935 1316602931
OCLC Number: 945356442
Description: xvi, 186 pages ; 24 cm.
Contents: 1 PRELIMINARIES AND TOOLS p.1 --
Exercises, p.8 --
2 LINEAR DISPERSIVE EQUATIONS p.12 --
2.1 Estimates on the real line, p.14 --
2.2 Estimates on the torus, p.22 --
2.3 The Talbot effect, p.32 --
Exercises, p.46 --
3. METHODS FOR ESTABLISHING WELLPOSEDNESS p.49 --
3.1 The energy method, p.50 --
3.1.1 A priori boudns, p.51 --
3.1.2 Existence and uniqueness, p.52 --
3.1.3 Growth bounds for KdV with potetial, p.59 --
3.2 Oscillatory integral method, p.60 --
3.3 Restricted norm meyhod, p.64 --
3.3.1 L2 solutions of KdV on the real line, p.64 --
3.3.2 Low regularity solutions of KdV on the torus, p.70 --
3.3.3 Forced and damped KdV with a potential, p.59 --
3.4 Differentiation by parts on the torus: unconditional wellposedness, p.82 --
3.5 Local theory for NLS on the torus, p.90 --
3.5.1 L2 wellpoesdness of cubic NLS on the torus, p.91 --
3.5.2 Hs local wellposedness of the quintic NLS on the torus, p.93 --
3.6 Illposedness results, p.97 --
Exercises, p.107 --
4 GLOBAL DYNAMICS OF NONLINEAR DISPERSIVE PDES p.111 --
4.1 Smoothing for nonlinear dispersive PDEs on the torus, p.112 --
4.1.1 Cubic NLS on the torus, p.112 --
4.1.2 The KdVequations on the torus, p.117 --
4.1.3 Proof of Proposition 4.7, p.127 --
4.2 High-low decomposition method, p.129 --
4.3 The I-method for the quintic NLS equation on the torus,p.135 --
Exercises, p.151 --
5 APPLICATIONS OF SMOOTHING ESTIMATES p.154 --
5.1 Bounds for higher order Sobolev norms, p.154 --
5.2 Almost everywhere convergence to initial data, p.160 --
5.3 Nonlinear Talbot effect, p.162 --
5.4 Global attractors for dissipative and dispersive PDEs, p.164 --
5.4.1 The global attractor is trivial for large damping, p.169 --
5.4.2 Bounds on the forced KdV equation, p.171 --
Exercises, p.172 --
References, p.175 --
Index, p.185.
Series Title: London Mathematical Society student texts, 86.
Responsibility: M. Burak Erdogan, University of Illinois, Urbana-Champaign, Nikolaos Tzirakis, University of Illinois, Urbana-Champaign.

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'The exercises in each chapter, while not at all trivial, tremendously enhance one's understanding of the material. The focus on periodic boundary conditions sets this book apart from related ones in Read more...

 
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