An Introduction to the Theory of Wave Maps and Related Geometric Problems

Nonfiction, Science & Nature, Mathematics, Linear Programming, Mathematical Analysis
Cover of the book An Introduction to the Theory of Wave Maps and Related Geometric Problems by Dan-Andrei Geba, Manoussos G Grillakis, World Scientific Publishing Company
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Author: Dan-Andrei Geba, Manoussos G Grillakis ISBN: 9789814713924
Publisher: World Scientific Publishing Company Publication: August 18, 2016
Imprint: WSPC Language: English
Author: Dan-Andrei Geba, Manoussos G Grillakis
ISBN: 9789814713924
Publisher: World Scientific Publishing Company
Publication: August 18, 2016
Imprint: WSPC
Language: English

The wave maps system is one of the most beautiful and challenging nonlinear hyperbolic systems, which has captured the attention of mathematicians for more than thirty years now. In the study of its various issues, such as the well-posedness theory, the formation of singularities, and the stability of the solitons, in order to obtain optimal results, one has to use intricate tools coming not only from analysis, but also from geometry and topology. Moreover, the wave maps system is nothing other than the Euler–Lagrange system for the nonlinear sigma model, which is one of the fundamental problems in classical field theory.

One of the goals of our book is to give an up-to-date and almost self-contained overview of the main regularity results proved for wave maps.

Another one is to introduce, to a wide mathematical audience, physically motivated generalizations of the wave maps system (e.g., the Skyrme model), which are extremely interesting and difficult in their own right.

Contents:

  • Introduction
  • Analytical Tools
  • Local and Small Data Global Well-Posedness Theory for Wave Maps
  • Large Data Regularity and Scattering for 2+1-Dimensional Wave Maps
  • General Well-Posedness Issues for Skyrme and Skyrme-Like Models
  • Equivariant Results
  • The Question of Collapse for Wave Maps

Readership: Advanced graduate students in mathematics, professional mathematicians and theoretical physicists.

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The wave maps system is one of the most beautiful and challenging nonlinear hyperbolic systems, which has captured the attention of mathematicians for more than thirty years now. In the study of its various issues, such as the well-posedness theory, the formation of singularities, and the stability of the solitons, in order to obtain optimal results, one has to use intricate tools coming not only from analysis, but also from geometry and topology. Moreover, the wave maps system is nothing other than the Euler–Lagrange system for the nonlinear sigma model, which is one of the fundamental problems in classical field theory.

One of the goals of our book is to give an up-to-date and almost self-contained overview of the main regularity results proved for wave maps.

Another one is to introduce, to a wide mathematical audience, physically motivated generalizations of the wave maps system (e.g., the Skyrme model), which are extremely interesting and difficult in their own right.

Contents:

Readership: Advanced graduate students in mathematics, professional mathematicians and theoretical physicists.

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