000			03348nam a22004335a 4500
001			146-120207
003			CH-001817-3
005			20170613140806.0
006			a fot ||| 0|
007			cr nn mmmmamaa
008			120207e20120209sz fot ||| 0|eng d
020			_a9783037196069
024	7	0	_a10.4171/106 _2doi
040			_ach0018173
072		7	_aPBKJ _2bicssc
072		7	_aPBMP _2bicssc
084			_a35-xx _a53-xx _2msc
100	1		_aKrieger, Joachim, _eauthor.
245	1	0	_aConcentration Compactness for Critical Wave Maps _h[electronic resource] / _cJoachim Krieger, Wilhelm Schlag
260	3		_aZuerich, Switzerland : _bEuropean Mathematical Society Publishing House, _c2012
264		1	_aZuerich, Switzerland : _bEuropean Mathematical Society Publishing House, _c2012
300			_a1 online resource (490 pages)
336			_atext _btxt _2rdacontent
337			_acomputer _bc _2rdamedia
338			_aonline resource _bcr _2rdacarrier
347			_atext file _bPDF _2rda
490	0		_aEMS Monographs in Mathematics (EMM)
506	1		_aRestricted to subscribers: _uhttp://www.ems-ph.org/ebooks.php
520			_aWave 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 we restrict ourselves 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.
650	0	7	_aDifferential equations _2bicssc
650	0	7	_aDifferential & Riemannian geometry _2bicssc
650	0	7	_aPartial differential equations _2msc
650	0	7	_aDifferential geometry _2msc
700	1		_aKrieger, Joachim, _eauthor.
700	1		_aSchlag, Wilhelm, _eauthor.
856	4	0	_uhttps://doi.org/10.4171/106
856	4	2	_3cover image _uhttp://www.ems-ph.org/img/books/krieger_mini.jpg
942			_2EBK13820 _cEBK
999			_c50444 _d50444