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E-grāmata: Dispersive Partial Differential Equations: Wellposedness and Applications

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(University of Illinois, Urbana-Champaign), (University of Illinois, Urbana-Champaign)
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Dispersive Partial Differential Equations: Wellposedness and ApplicationsPalielināt
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Provides a self-contained and accessible introduction to nonlinear dispersive partial differential equations (PDEs) for graduate or advanced undergraduate students in mathematics, engineering, and physical sciences. The book can be used for self-study, or for teaching a semester-long introductory graduate course in PDEs.

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.

Recenzijas

'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 the area, and yet the authors do a nice job discussing related well-posedness results and estimates on the real line as well. While certainly not meant for students without any analysis background, a thorough work-through of this book certainly brings one to the frontier of the theory of nonlinear dispersive PDEs.' Eric Stachura, MAA Reviews 'The book is a manual for beginning graduate students in the field of the general theory of nonlinear partial differential equations. The material is presented in the rigorous mathematical style, providing proofs of formal theorems, rather than less strict considerations which may be often encountered in physics literature.' Boris A. Malomed, Zentralblatt MATH

Papildus informācija

Introduces nonlinear dispersive partial differential equations in a detailed yet elementary way without compromising the depth and richness of the subject.
Preface ix
Notation xiv
1 Preliminaries and tools
1(11)
Exercises
8(4)
2 Linear dispersive equations
12(37)
2.1 Estimates on the real line
14(8)
2.2 Estimates on the torus
22(10)
2.3 The Talbot effect
32(17)
Exercises
46(3)
3 Methods for establishing wellposedness
49(62)
3.1 The energy method
50(10)
3.1.1 A priori bounds
51(1)
3.1.2 Existence and uniqueness
52(7)
3.1.3 Growth bounds for KdV with potential
59(1)
3.2 Oscillatory integral method
60(4)
3.3 Restricted norm method
64(18)
3.3.1 L2 solutions of KdV on the real line
64(6)
3.3.2 Low regularity solutions of KdV on the torus
70(10)
3.3.3 Forced and damped KdV with a potential
80(2)
3.4 Differentiation by parts on the torus: unconditional wellposedness
82(8)
3.5 Local theory for NLS on the torus
90(7)
3.5.1 L2 wellposedness of cubic NLS on the torus
91(2)
3.5.2 Hs local wellposedness of the quintic NLS on the torus
93(4)
3.6 Illposedness results
97(14)
Exercises
107(4)
4 Global dynamics of nonlinear dispersive PDEs
111(43)
4.1 Smoothing for nonlinear dispersive PDEs on the torus
112(17)
4.1.1 Cubic NLS on the torus
112(5)
4.1.2 The KdV equation on the torus
117(10)
4.1.3 Proof of Proposition 4.7
127(2)
4.2 High--low decomposition method
129(6)
4.3 The I-method for the quintic NLS equation on the torus
135(19)
Exercises
151(3)
5 Applications of smoothing estimates
154(21)
5.1 Bounds for higher order Sobolev norms
154(6)
5.2 Almost everywhere convergence to initial data
160(2)
5.3 Nonlinear Talbot effect
162(2)
5.4 Global attractors for dissipative and dispersive PDEs
164(11)
5.4.1 The global attractor is trivial for large damping
169(2)
5.4.2 Bounds on the forced KdV equation
171(1)
Exercises
172(3)
References 175(10)
Index 185
M. Burak Erdoan is a professor in the Department of Mathematics at the University of Illinois, Urbana-Champaign. Nikolaos Tzirakis is an associate professor in the Department of Mathematics at the University of Illinois, Urbana-Champaign.
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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