LEADER 03270nam 22004455a 450 001 9910151927803321 005 20120207234510.0 010 $a3-03719-606-8 024 70$a10.4171/106 035 $a(CKB)3710000000953880 035 $a(CH-001817-3)146-120207 035 $a(PPN)178156051 035 $a(EXLCZ)993710000000953880 100 $a20120207j20120209 fy 0 101 0 $aeng 135 $aurnn|mmmmamaa 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aConcentration Compactness for Critical Wave Maps$b[electronic resource] /$fJoachim Krieger, Wilhelm Schlag 210 3 $aZuerich, Switzerland $cEuropean Mathematical Society Publishing House$d2012 215 $a1 online resource (490 pages) 225 0 $aEMS Monographs in Mathematics (EMM) ;$x2523-5192 330 $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. 606 $aDifferential equations$2bicssc 606 $aDifferential & Riemannian geometry$2bicssc 606 $aPartial differential equations$2msc 606 $aDifferential geometry$2msc 615 07$aDifferential equations 615 07$aDifferential & Riemannian geometry 615 07$aPartial differential equations 615 07$aDifferential geometry 686 $a35-xx$a53-xx$2msc 700 $aKrieger$b Joachim$01071019 702 $aSchlag$b Wilhelm 801 0$bch0018173 906 $aBOOK 912 $a9910151927803321 996 $aConcentration Compactness for Critical Wave Maps$92565659 997 $aUNINA