Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

Book Review

The AMS does not provide abstracts of book reviews. You may download the entire review from the links below.


Full text of review: PDF   This review is available free of charge.
Book Information:

Authors: Joachim Krieger and Wilhelm Schlag
Title: Concentration compactness for critical wave maps
Additional book information: EMS Monographs in Modern Mathematics, European Mathematical Society (EMS), Zürich, 2012, vi+484 pp., ISBN 978-3-03719-106-4

References [Enhancements On Off] (What's this?)

  • 1. H. Bahouri, P. Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1999), no. 1, 131-175. MR 1705001 (2000i:35123)
  • 2. T. Cazenave, J. Shatah, A.S. Tahvildar-Zadeh, Harmonic maps of the hyperbolic space and development of singularities in wave maps and Yang-Mills fields, Ann. Inst. H. Poincaré Phys. Théor.68 (1998), 315-349. MR 1622539 (2000g:58042)
  • 3. Y. Choquet-Bruhat, Global existence theorems for hyperbolic harmonic maps, Ann. Inst. H. Poincare Phys. Theor. 46 (1987), 97-111. MR 877997 (88b:58037)
  • 4. D. Christodoulou, A. Tahvildar-Zadeh, On the regularity of spherically symmetric wave maps, Comm. Pure Appl. Math, 46 (1993), 1041-1091. MR 1223662 (94e:58030)
  • 5. J. Eells, H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160. MR 0164306 (29:1603)
  • 6. C. Gu, On the Cauchy problem for harmonic maps defined on two-dimensional Minkowski space, Comm. Pure Appl. Math., 33,(1980), 727-737. MR 596432 (82g:58027)
  • 7. C. Kenig, Global well-posedness and scattering for the energy critical focusing nonlinear Schrödinger and wave equations. Lectures given at ``Analyse des équations aux dérivées partielles,'' Evian-les-Bains, July 2007.
  • 8. C. Kenig, F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math. 201 (2008), no. 2, 147-212. MR 2461508 (2011a:35344)
  • 9. R. Killip, M. Visan, Nonlinear Schrödinger equations at critical regularity, To appear in proceedings of the 2008 Clay summer school, ``Evolution Equations'' held at the ETH, Zürich.
  • 10. J. Krieger, Global regularity and singularity development for wave maps., preprint. MR 2488946 (2010h:58044)
  • 11. J. Krieger, W. Schlag, D. Tataru, Renormalization and blow up for charge one equivariant critical wave maps, preprint. MR 2372807 (2009b:58061)
  • 12. O.A. Ladyzhenskaya, V.I. Shubov, Unique solvability of the Cauchy problem for the equations of the two dimensional chiral fields, taking values in complete Riemann manifolds, J. Soviet Math., 25 (1984), 855-864. (English Trans. of 1981 Article.)
  • 13. P.-L. Lions, The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), no. 2, 109-145. MR 778970 (87e:49035a)
  • 14. I. Rodnianski, The wave map problem. Small data critical regularity, Seminaire Bourbaki, 58eme annee, 2005-2006, no. 965. 15pt
  • 15. I. Rodnianski, J. Sterbenz, On the formation of singularities in the critical $ O(3)$ $ \sigma $-models, preprint. MR 2680419 (2011i:58023)
  • 16. J. Shatah, Weak solutions and development of singularities of the $ SU(2)$ $ \sigma $-model. Comm. Pure Appl. Math., 41 (1988), 459-469. MR 933231 (89f:58044)
  • 17. J. Shatah, M. Struwe, Regularity results for non-linear wave equations, Ann. of Math. 138 (1993) 503-518. MR 1247991 (95f:35164)
  • 18. J. Shatah, A. Tavildar-Zadeh, Regularity of harmonic maps from the Minkowski space into rotationally symmetric manifolds., Comm. Pure Appl. Math. 45 (1992), 947-971. MR 1168115 (93c:58056)
  • 19. J. Shatah, A. Tavildar-Zadeh, On the Cauchy problem for equivariant wave maps, Comm. Pure Appl. Math., 47 (1994), 719 - 753. MR 1278351 (96c:58049)
  • 20. J. Sterbenz, D. Tataru, Energy dispersed large data wave maps in $ 2+1$ dimensions, preprint. MR 2657817 (2011g:58045)
  • 21. J. Sterbenz, D. Tataru, Regularity of Wave-Maps in dimension $ 2+1$, preprint. MR 2657818 (2011h:58026)
  • 22. M. Struwe, Wave Maps, in Nonlinear Partial Differential Equations in Geometry and Physics, Prog. in Nonlin. Diff. Eq. and their Applic., 29, (1997), Birkhäuser, 113-150. MR 1437153 (98e:58061)
  • 23. M. Struwe, Radially symmetric wave maps from the $ (1+2)$-dimensional Minkowski space to a sphere, Math Z. 242 (2002), 407-414. MR 1985457 (2004d:58040)
  • 24. M. Struwe, Radially symmetric wave maps from $ (1+2)$-dimensional Minkowski space to general targets, Calc. Var. 16 (2003), 431-437. MR 1971037 (2004j:58033)
  • 25. M. Struwe, Equivariant wave maps in two dimensions, Comm. Pure Appl. Math. 56 (2003), 815-823. MR 1990477 (2004c:58061)
  • 26. T. Tao, Geometric renormalization of large energy wave maps, Journees ``Equations aux derives partielles'', Forges les Eaux, 7-11 June 2004, XI 1-32. MR 2135366 (2006i:58044)
  • 27. T. Tao, Nonlinear dispersive equations. Local and global analysis. CBMS Regional Conference Series in Mathematics, 106. American Mathematical Society, Providence, RI, 2006. MR 2233925 (2008i:35211)
  • 28. T. Tao, Global regularity of wave maps III. Large energy from $ R^{1+2}$ to hyperbolic spaces., preprint.
  • 29. T. Tao, Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class, preprint.
  • 30. T. Tao, Global regularity of wave maps V. Large data local wellposedness in the energy class, preprint.
  • 31. T. Tao, Global regularity of wave maps VI. Abstract theory of minimal-energy blowup solutions, preprint.
  • 32. T. Tao, Global regularity of wave maps VII. Control of delocalised or dispersed solutions, preprint.
  • 33. D. Tataru, The wave maps equation, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 2, 185-204. MR 2043751 (2005h:35245)

Review Information:

Reviewer: Terence Tao
Affiliation: Department of Mathematics, UCLA, Los Angeles, California 90095-1555
Email: tao@math.ucla.edu
Journal: Bull. Amer. Math. Soc. 50 (2013), 655-662
MSC (2010): Primary 53C44, 58E20, 35Q75
DOI: https://doi.org/10.1090/S0273-0979-2012-01395-2
Published electronically: October 16, 2012
Review copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
American Mathematical Society