03167nam a22003855a 4500001001100000003001200011005001700023006001900040007001500059008004100074020001800115024002100133040001400154072001700168072001700185084002200202100003100224245011100255260008200366300003400448336002600482337002600508338003600534347002400570490005300594506006600647520178600713650003502499650004702534650004002581650003102621700003002652856003202682856006702714146-120207CH-001817-320120207234510.0a fot ||| 0|cr nn mmmmamaa120207e20120209sz fot ||| 0|eng d a978303719606970a10.4171/1062doi ach0018173 7aPBKJ2bicssc 7aPBMP2bicssc a35-xxa53-xx2msc1 aKrieger, Joachim,eauthor.10aConcentration Compactness for Critical Wave Mapsh[electronic resource] /cJoachim Krieger, Wilhelm Schlag3 aZuerich, Switzerland :bEuropean Mathematical Society Publishing House,c2012 a1 online resource (490 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier atext filebPDF2rda0 aEMS Monographs in Mathematics (EMM) ;x2523-51921 aRestricted to subscribers:uhttps://www.ems-ph.org/ebooks.php 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 Klainermanâ€“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.07aDifferential equations2bicssc07aDifferential & Riemannian geometry2bicssc07aPartial differential equations2msc07aDifferential geometry2msc1 aSchlag, Wilhelm,eauthor.40uhttps://doi.org/10.4171/106423cover imageuhttps://www.ems-ph.org/img/books/krieger_mini.jpg