Book Review
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MathSciNet review:
3090424
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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),
Z\"urich,
2012,
vi+484 pp.,
ISBN 978-3-03719-106-4
Hajer Bahouri and Patrick Gérard, High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1999), no. 1, 131–175. MR 1705001
Thierry Cazenave, Jalal Shatah, and A. Shadi 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), no. 3, 315–349 (English, with English and French summaries). MR 1622539
Yvonne Choquet-Bruhat, Global existence theorems for hyperbolic harmonic maps, Ann. Inst. H. Poincaré Phys. Théor. 46 (1987), no. 1, 97–111. MR 877997
Demetrios Christodoulou and A. Shadi Tahvildar-Zadeh, On the regularity of spherically symmetric wave maps, Comm. Pure Appl. Math. 46 (1993), no. 7, 1041–1091. MR 1223662, DOI 10.1002/cpa.3160460705
James Eells Jr. and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. MR 164306, DOI 10.2307/2373037
Chao Hao Gu, On the Cauchy problem for harmonic maps defined on two-dimensional Minkowski space, Comm. Pure Appl. Math. 33 (1980), no. 6, 727–737. MR 596432, DOI 10.1002/cpa.3160330604
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.
Carlos E. Kenig and Frank 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, DOI 10.1007/s11511-008-0031-6
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.
J. Krieger, Global regularity and singularity development for wave maps, Surveys in differential geometry. Vol. XII. Geometric flows, Surv. Differ. Geom., vol. 12, Int. Press, Somerville, MA, 2008, pp. 167–201. MR 2488946, DOI 10.4310/SDG.2007.v12.n1.a5
J. Krieger, W. Schlag, and D. Tataru, Renormalization and blow up for charge one equivariant critical wave maps, Invent. Math. 171 (2008), no. 3, 543–615. MR 2372807, DOI 10.1007/s00222-007-0089-3
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.)
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 (English, with French summary). MR 778970
I. Rodnianski, The wave map problem. Small data critical regularity, Seminaire Bourbaki, 58eme annee, 2005–2006, no. 965. *15pt
Igor Rodnianski and Jacob Sterbenz, On the formation of singularities in the critical $\textrm {O}(3)$ $\sigma$-model, Ann. of Math. (2) 172 (2010), no. 1, 187–242. MR 2680419, DOI 10.4007/annals.2010.172.187
Jalal Shatah, Weak solutions and development of singularities of the $\textrm {SU}(2)$ $\sigma$-model, Comm. Pure Appl. Math. 41 (1988), no. 4, 459–469. MR 933231, DOI 10.1002/cpa.3160410405
Jalal Shatah and Michael Struwe, Regularity results for nonlinear wave equations, Ann. of Math. (2) 138 (1993), no. 3, 503–518. MR 1247991, DOI 10.2307/2946554
J. Shatah and A. Tahvildar-Zadeh, Regularity of harmonic maps from the Minkowski space into rotationally symmetric manifolds, Comm. Pure Appl. Math. 45 (1992), no. 8, 947–971. MR 1168115, DOI 10.1002/cpa.3160450803
Jalal Shatah and A. Shadi Tahvildar-Zadeh, On the Cauchy problem for equivariant wave maps, Comm. Pure Appl. Math. 47 (1994), no. 5, 719–754. MR 1278351, DOI 10.1002/cpa.3160470507
Jacob Sterbenz and Daniel Tataru, Energy dispersed large data wave maps in $2+1$ dimensions, Comm. Math. Phys. 298 (2010), no. 1, 139–230. MR 2657817, DOI 10.1007/s00220-010-1061-4
Jacob Sterbenz and Daniel Tataru, Regularity of wave-maps in dimension $2+1$, Comm. Math. Phys. 298 (2010), no. 1, 231–264. MR 2657818, DOI 10.1007/s00220-010-1062-3
Michael Struwe, Wave maps, Nonlinear partial differential equations in geometry and physics (Knoxville, TN, 1995) Progr. Nonlinear Differential Equations Appl., vol. 29, Birkhäuser, Basel, 1997, pp. 113–153. MR 1437153
Michael Struwe, Radially symmetric wave maps from $(1+2)$-dimensional Minkowski space to the sphere, Math. Z. 242 (2002), no. 3, 407–414. MR 1985457, DOI 10.1007/s002090100345
Michael Struwe, Radially symmetric wave maps from $(1+2)$-dimensional Minkowski space to general targets, Calc. Var. Partial Differential Equations 16 (2003), no. 4, 431–437. MR 1971037, DOI 10.1007/s00526-002-0156-y
Michael Struwe, Equivariant wave maps in two space dimensions, Comm. Pure Appl. Math. 56 (2003), no. 7, 815–823. Dedicated to the memory of Jürgen K. Moser. MR 1990477, DOI 10.1002/cpa.10074
Terence Tao, Geometric renormalization of large energy wave maps, Journées “Équations aux Dérivées Partielles”, École Polytech., Palaiseau, 2004, pp. Exp. No. XI, 32. MR 2135366
Terence Tao, Nonlinear dispersive equations, CBMS Regional Conference Series in Mathematics, vol. 106, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. Local and global analysis. MR 2233925, DOI 10.1090/cbms/106
T. Tao, Global regularity of wave maps III. Large energy from $R^{1+2}$ to hyperbolic spaces., preprint.
T. Tao, Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class, preprint.
T. Tao, Global regularity of wave maps V. Large data local wellposedness in the energy class, preprint.
T. Tao, Global regularity of wave maps VI. Abstract theory of minimal-energy blowup solutions, preprint.
T. Tao, Global regularity of wave maps VII. Control of delocalised or dispersed solutions, preprint.
Daniel Tataru, The wave maps equation, Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 2, 185–204. MR 2043751, DOI 10.1090/S0273-0979-04-01005-5
References
- 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)
- 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)
- Y. Choquet-Bruhat, Global existence theorems for hyperbolic harmonic maps, Ann. Inst. H. Poincare Phys. Theor. 46 (1987), 97–111. MR 877997 (88b:58037)
- 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)
- J. Eells, H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. MR 0164306 (29:1603)
- 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)
- 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.
- 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)
- 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.
- J. Krieger, Global regularity and singularity development for wave maps., preprint. MR 2488946 (2010h:58044)
- J. Krieger, W. Schlag, D. Tataru, Renormalization and blow up for charge one equivariant critical wave maps, preprint. MR 2372807 (2009b:58061)
- 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.)
- 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)
- I. Rodnianski, The wave map problem. Small data critical regularity, Seminaire Bourbaki, 58eme annee, 2005–2006, no. 965. *15pt
- I. Rodnianski, J. Sterbenz, On the formation of singularities in the critical $O(3)$ $\sigma$-models, preprint. MR 2680419 (2011i:58023)
- 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)
- J. Shatah, M. Struwe, Regularity results for non-linear wave equations, Ann. of Math. 138 (1993) 503–518. MR 1247991 (95f:35164)
- 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)
- 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)
- J. Sterbenz, D. Tataru, Energy dispersed large data wave maps in $2+1$ dimensions, preprint. MR 2657817 (2011g:58045)
- J. Sterbenz, D. Tataru, Regularity of Wave-Maps in dimension $2+1$, preprint. MR 2657818 (2011h:58026)
- 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)
- 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)
- 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)
- M. Struwe, Equivariant wave maps in two dimensions, Comm. Pure Appl. Math. 56 (2003), 815–823. MR 1990477 (2004c:58061)
- 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)
- 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)
- T. Tao, Global regularity of wave maps III. Large energy from $R^{1+2}$ to hyperbolic spaces., preprint.
- T. Tao, Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class, preprint.
- T. Tao, Global regularity of wave maps V. Large data local wellposedness in the energy class, preprint.
- T. Tao, Global regularity of wave maps VI. Abstract theory of minimal-energy blowup solutions, preprint.
- T. Tao, Global regularity of wave maps VII. Control of delocalised or dispersed solutions, preprint.
- 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
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.
