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Author : Fedor A. Chechkin
Release : 2000
Genre :
Kind : eBook
Book Rating : /5 ( reviews)
Book Synopsis Convergence of Wave Maps by : Fedor A. Chechkin
Download or read book Convergence of Wave Maps written by Fedor A. Chechkin. This book was released on 2000. Available in PDF, EPUB and Kindle. Book excerpt:
Author : Joachim Krieger
Release : 2012
Genre : Differential equations, Hyperbolic
Kind : eBook
Book Rating : 064/5 ( reviews)
Book Synopsis Concentration Compactness for Critical Wave Maps by : Joachim Krieger
Download or read book Concentration Compactness for Critical Wave Maps written by Joachim Krieger. This book was released on 2012. Available in PDF, EPUB and Kindle. Book excerpt: Wave maps are the simplest wave equations taking their values in a Riemannian manifold $(M,g)$. Their Lagrangian is the same as for the scalar equation, the only difference being that lengths are measured with respect to the metric $g$. By Noether's theorem, symmetries of the Lagrangian imply conservation laws for wave maps, such as conservation of energy. In coordinates, wave maps are given by a system of semilinear wave equations. Over the past 20 years important methods have emerged which address the problem of local and global wellposedness of this system. Due to weak dispersive effects, wave maps defined on Minkowski spaces of low dimensions, such as $\mathbb R^{2+1}_{t,x}$, present particular technical difficulties. This class of wave maps has the additional important feature of being energy critical, which refers to the fact that the energy scales exactly like the equation. Around 2000 Daniel Tataru and Terence Tao, building on earlier work of Klainerman-Machedon, proved that smooth data of small energy lead to global smooth solutions for wave maps from 2+1 dimensions into target manifolds satisfying some natural conditions. In contrast, for large data, singularities may occur in finite time for $M =\mathbb S^2$ as target. This monograph establishes that for $\mathbb H$ as target the wave map evolution of any smooth data exists globally as a smooth function. While the authors restrict themselves to the hyperbolic plane as target the implementation of the concentration-compactness method, the most challenging piece of this exposition, yields more detailed information on the solution. This monograph will be of interest to experts in nonlinear dispersive equations, in particular to those working on geometric evolution equations.
Author : Dan-andrei Geba
Release : 2016-08-18
Genre : Mathematics
Kind : eBook
Book Rating : 929/5 ( reviews)
Book Synopsis An Introduction To The Theory Of Wave Maps And Related Geometric Problems by : Dan-andrei Geba
Download or read book An Introduction To The Theory Of Wave Maps And Related Geometric Problems written by Dan-andrei Geba. This book was released on 2016-08-18. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Author : Yuan-Jen Chiang
Release : 2013-06-18
Genre : Mathematics
Kind : eBook
Book Rating : 349/5 ( reviews)
Book Synopsis Developments of Harmonic Maps, Wave Maps and Yang-Mills Fields into Biharmonic Maps, Biwave Maps and Bi-Yang-Mills Fields by : Yuan-Jen Chiang
Download or read book Developments of Harmonic Maps, Wave Maps and Yang-Mills Fields into Biharmonic Maps, Biwave Maps and Bi-Yang-Mills Fields written by Yuan-Jen Chiang. This book was released on 2013-06-18. Available in PDF, EPUB and Kindle. Book excerpt: Harmonic maps between Riemannian manifolds were first established by James Eells and Joseph H. Sampson in 1964. Wave maps are harmonic maps on Minkowski spaces and have been studied since the 1990s. Yang-Mills fields, the critical points of Yang-Mills functionals of connections whose curvature tensors are harmonic, were explored by a few physicists in the 1950s, and biharmonic maps (generalizing harmonic maps) were introduced by Guoying Jiang in 1986. The book presents an overview of the important developments made in these fields since they first came up. Furthermore, it introduces biwave maps (generalizing wave maps) which were first studied by the author in 2009, and bi-Yang-Mills fields (generalizing Yang-Mills fields) first investigated by Toshiyuki Ichiyama, Jun-Ichi Inoguchi and Hajime Urakawa in 2008. Other topics discussed are exponential harmonic maps, exponential wave maps and exponential Yang-Mills fields.
Author : Björn Sandstede
Release : 2023-05-23
Genre : Mathematics
Kind : eBook
Book Rating : 091/5 ( reviews)
Book Synopsis Spiral Waves: Linear and Nonlinear Theory by : Björn Sandstede
Download or read book Spiral Waves: Linear and Nonlinear Theory written by Björn Sandstede. This book was released on 2023-05-23. Available in PDF, EPUB and Kindle. Book excerpt: View the abstract.
