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The wave maps equation
Author:
Daniel Tataru
Translated by:
Journal:
Bull. Amer. Math. Soc. 41 (2004), 185-204
MSC (2000):
Primary 35L70
Posted:
January 8, 2004
MathSciNet review:
2043751
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Abstract: The wave maps equation has become a very popular topic in recent years. The aim of these expository notes is to present a non-technical survey of the ideas and methods which have proved useful in the study of wave maps, leading up to the latest results. The remaining open problems are also stated and explained.
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Klainerman and Igor
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Sergiu
Klainerman and Sigmund
Selberg, Remark on the optimal regularity for equations of wave
maps type, Comm. Partial Differential Equations 22
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Joachim Krieger, Global regularity of wave maps from
to surfaces, Comm. Math. Phys. 238 (2003), no. 1-2, 333-366.
- 20.
O.
A. Ladyzhenskaya and V.
I. Shubov, On the unique solvability of the Cauchy problem for
equations of two-dimensional relativistic chiral fields with values in
complete Riemannian manifolds, Zap. Nauchn. Sem. Leningrad. Otdel.
Mat. Inst. Steklov. (LOMI) 110 (1981), 81–94,
242–243 (Russian). Boundary value problems of mathematical physics
and related questions in the theory of functions, 13. MR 643976
(83h:81051)
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Hans
Lindblad, A sharp counterexample to the local existence of
low-regularity solutions to nonlinear wave equations, Duke Math. J.
72 (1993), no. 2, 503–539. MR 1248683
(94h:35165), http://dx.doi.org/10.1215/S0012-7094-93-07219-5
- 22.
Stefan
Müller and Michael
Struwe, Global existence of wave maps in 1+2 dimensions with finite
energy data, Topol. Methods Nonlinear Anal. 7 (1996),
no. 2, 245–259. MR 1481698
(99b:58063)
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Andrea Nahmod, Atanas Stefanov, and Karen Uhlenbeck, On the well-posedness of the wave map problem in high dimensions, preprint.
- 24.
John
Nash, The imbedding problem for Riemannian manifolds, Ann. of
Math. (2) 63 (1956), 20–63. MR 0075639
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Jalal
Shatah, Weak solutions and development of singularities of the
𝑆𝑈(2) 𝜎-model, Comm. Pure Appl. Math.
41 (1988), no. 4, 459–469. MR 933231
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Shatah and Michael
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Mathematics, vol. 2, New York University Courant Institute of
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Jalal
Shatah and Michael
Struwe, The Cauchy problem for wave maps, Int. Math. Res. Not.
11 (2002), 555–571. MR 1890048
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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
(96c:58049), http://dx.doi.org/10.1002/cpa.3160470507
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Thomas
C. Sideris, Global existence of harmonic maps in Minkowski
space, Comm. Pure Appl. Math. 42 (1989), no. 1,
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Tao, Global regularity of wave maps. I. Small critical Sobolev norm
in high dimension, Internat. Math. Res. Notices 6
(2001), 299–328. MR 1820329
(2001m:35200), http://dx.doi.org/10.1155/S1073792801000150
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Terence
Tao, Global regularity of wave maps. II. Small energy in two
dimensions, Comm. Math. Phys. 224 (2001), no. 2,
443–544. MR 1869874
(2002h:58052), http://dx.doi.org/10.1007/PL00005588
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Daniel Tataru, Small solutions for the wave maps equation in critical Sobolev spaces, preprint.
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Daniel
Tataru, Local and global results for wave maps. I, Comm.
Partial Differential Equations 23 (1998), no. 9-10,
1781–1793. MR 1641721
(99j:58209), http://dx.doi.org/10.1080/03605309808821400
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Daniel
Tataru, On global existence and scattering for the wave maps
equation, Amer. J. Math. 123 (2001), no. 1,
37–77. MR
1827277 (2002c:58045)
- 1.
- Piotr Bizon, Tadeusz Chmaj, and Zbis
aw Tabor, Formation of singularities for equivariant -dimensional wave maps into the 2-sphere, Nonlinearity 14 (2001), no. 5, 1041-1053. MR 2003b:58043
- 2.
- H. Brezis and L. Nirenberg, Degree theory and BMO. I. Compact manifolds without boundaries, Selecta Math. (N.S.) 1 (1995), no. 2, 197-263. MR 96g:58023
- 3.
- 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. MR 2000g:58042
- 4.
- 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 94e:58030
- 5.
- Piero D'Ancona and V. Georgiev, Ill posedness results for the two dimensional wave maps equation, preprint.
- 6.
- Alexandre Freire, Stefan Müller, and Michael Struwe, Weak compactness of wave maps and harmonic maps, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), no. 6, 725-754. MR 2000a:58045
- 7.
- J. Ginibre and G. Velo, The Cauchy problem for the
and models, Ann. Physics 142 (1982), no. 2, 393-415. MR 84i:58027
- 8.
- M. L. Gromov, Isometric imbeddings and immersions, Dokl. Akad. Nauk SSSR 192 (1970), 1206-1209. MR 43:1212
- 9.
- 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 82g:58027
- 10.
- Matthias Günther, Isometric embeddings of Riemannian manifolds, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) (Tokyo), Math. Soc. Japan, 1991, pp. 1137-1143. MR 93b:53049
- 11.
- Frédéric Hélein, Régularité des applications faiblement harmoniques entre une surface et une sphère, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 9, 519-524. MR 92a:58034
- 12.
- -, Harmonic maps, conservation laws and moving frames, second ed., Cambridge Tracts in Mathematics, vol. 150, Cambridge University Press, Cambridge, 2002. MR 2003g:58024
- 13.
- James Isenberg and Steven L. Liebling, Singularity formation in
wave maps, J. Math. Phys. 43 (2002), no. 1, 678-683. MR 2002i:58034
- 14.
- Markus Keel and Terence Tao, Local and global well-posedness of wave maps on
for rough data, Internat. Math. Res. Notices (1998), no. 21, 1117-1156. MR 99k:58180
- 15.
- S. Klainerman, The null condition and global existence to nonlinear wave equations, Nonlinear systems of partial differential equations in applied mathematics, Part 1 (Santa Fe, N.M., 1984), Lectures in Appl. Math., vol. 23, Amer. Math. Soc., Providence, RI, 1986, pp. 293-326. MR 87h:35217
- 16.
- S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math. 46 (1993), no. 9, 1221-1268. MR 94h:35137
- 17.
- Sergiu Klainerman and Igor Rodnianski, On the global regularity of wave maps in the critical Sobolev norm, Internat. Math. Res. Notices (2001), no. 13, 655-677. MR 2002h:58051
- 18.
- Sergiu Klainerman and Sigmund Selberg, Remark on the optimal regularity for equations of wave maps type, Comm. Partial Differential Equations 22 (1997), no. 5-6, 901-918. MR 99c:35163
- 19.
- Joachim Krieger, Global regularity of wave maps from
to surfaces, Comm. Math. Phys. 238 (2003), no. 1-2, 333-366.
- 20.
- O. A. Ladyzhenskaya and V. I. Shubov, On the unique solvability of the Cauchy problem for equations of two-dimensional relativistic chiral fields with values in complete Riemannian manifolds, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 110 (1981), 81-94, 242-243, Boundary value problems of mathematical physics and related questions in the theory of functions, 13. MR 83h:81051
- 21.
- Hans Lindblad, A sharp counterexample to the local existence of low-regularity solutions to nonlinear wave equations, Duke Math. J. 72 (1993), no. 2, 503-539. MR 94h:35165
- 22.
- Stefan Müller and Michael Struwe, Global existence of wave maps in
dimensions with finite energy data, Topol. Methods Nonlinear Anal. 7 (1996), no. 2, 245-259. MR 99b:58063
- 23.
- Andrea Nahmod, Atanas Stefanov, and Karen Uhlenbeck, On the well-posedness of the wave map problem in high dimensions, preprint.
- 24.
- John Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20-63. MR 17:782b
- 25.
- Jalal Shatah, Weak solutions and development of singularities of the
-model, Comm. Pure Appl. Math. 41 (1988), no. 4, 459-469. MR 89f:58044
- 26.
- Jalal Shatah and Michael Struwe, Geometric wave equations, Courant Lecture Notes in Mathematics, vol. 2, New York University Courant Institute of Mathematical Sciences, New York, 1998. MR 2000i:35135
- 27.
- -, The Cauchy problem for wave maps, Int. Math. Res. Not. (2002), no. 11, 555-571. MR 2003m:58042
- 28.
- 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 96c:58049
- 29.
- Thomas C. Sideris, Global existence of harmonic maps in Minkowski space, Comm. Pure Appl. Math. 42 (1989), no. 1, 1-13. MR 89k:58069
- 30.
- Terence Tao, Global regularity of wave maps. I. Small critical Sobolev norm in high dimension, Internat. Math. Res. Notices (2001), no. 6, 299-328. MR 2001m:35200
- 31.
- -, Global regularity of wave maps. II. Small energy in two dimensions, Comm. Math. Phys. 224 (2001), no. 2, 443-544. MR 2002h:58052
- 32.
- Daniel Tataru, Small solutions for the wave maps equation in critical Sobolev spaces, preprint.
- 33.
- -, Local and global results for wave maps. I, Comm. Partial Differential Equations 23 (1998), no. 9-10, 1781-1793. MR 99j:58209
- 34.
- -, On global existence and scattering for the wave maps equation, Amer. J. Math. 123 (2001), no. 1, 37-77. MR 2002c:58045
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Additional Information
Daniel Tataru
Affiliation:
Department of Mathematics, University of California, Berkeley, California 94720
Email:
tataru@math.berkeley.edu
DOI:
http://dx.doi.org/10.1090/S0273-0979-04-01005-5
PII:
S 0273-0979(04)01005-5
Received by editor(s):
May 10, 2003
Received by editor(s) in revised form:
September 28, 2003
Posted:
January 8, 2004
Additional Notes:
Lecture presented at the AMS Special Session on Mathematical Current Events: Expository Reports in Baltimore, MD, January 17, 2003
Article copyright:
© Copyright 2004 American Mathematical Society
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