The wave maps equation

Tataru, Daniel

doi:10.1090/S0273-0979-04-01005-5

The wave maps equation
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Bull. Amer. Math. Soc. 41 (2004), 185-204 Request permission

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.

References

Piotr Bizoń, Tadeusz Chmaj, and Zbisław Tabor, Formation of singularities for equivariant $(2+1)$-dimensional wave maps into the 2-sphere, Nonlinearity 14 (2001), no. 5, 1041–1053. MR 1862811, DOI 10.1088/0951-7715/14/5/308
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 1354598, DOI 10.1007/BF01671566
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
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

Ill posedness results for the two dimensional wave maps equation

Alexandre Freire, Stefan Müller, and Michael Struwe, Weak compactness of wave maps and harmonic maps, Ann. Inst. H. Poincaré C Anal. Non Linéaire 15 (1998), no. 6, 725–754 (English, with English and French summaries). MR 1650966, DOI 10.1016/S0294-1449(99)80003-1
J. Ginibre and G. Velo, The Cauchy problem for the $\textrm {O}(N),\,\textbf {C}\textrm {P}(N-1),$ and $G_{\textbf {C}}(N,\,p)$ models, Ann. Physics 142 (1982), no. 2, 393–415. MR 678488, DOI 10.1016/0003-4916(82)90077-X
M. L. Gromov, Isometric imbeddings and immersions, Dokl. Akad. Nauk SSSR 192 (1970), 1206–1209 (Russian). MR 0275456
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
Matthias Günther, Isometric embeddings of Riemannian manifolds, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 1137–1143. MR 1159298
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 (French, with English summary). MR 1078114
Frédéric Hélein, Harmonic maps, conservation laws and moving frames, 2nd ed., Cambridge Tracts in Mathematics, vol. 150, Cambridge University Press, Cambridge, 2002. Translated from the 1996 French original; With a foreword by James Eells. MR 1913803, DOI 10.1017/CBO9780511543036
James Isenberg and Steven L. Liebling, Singularity formation in $2+1$ wave maps, J. Math. Phys. 43 (2002), no. 1, 678–683. MR 1872523, DOI 10.1063/1.1418717
Markus Keel and Terence Tao, Local and global well-posedness of wave maps on $\mathbf R^{1+1}$ for rough data, Internat. Math. Res. Notices 21 (1998), 1117–1156. MR 1663216, DOI 10.1155/S107379289800066X
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 837683
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 1231427, DOI 10.1002/cpa.3160460902
Sergiu Klainerman and Igor Rodnianski, On the global regularity of wave maps in the critical Sobolev norm, Internat. Math. Res. Notices 13 (2001), 655–677. MR 1843256, DOI 10.1155/S1073792801000344
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 1452172, DOI 10.1080/03605309708821288

Global regularity of wave maps from $R^{3+1}$ to surfaces

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
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, DOI 10.1215/S0012-7094-93-07219-5
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, DOI 10.12775/TMNA.1996.011

On the well-posedness of the wave map problem in high dimensions

Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
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, Geometric wave equations, Courant Lecture Notes in Mathematics, vol. 2, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1998. MR 1674843
Jalal Shatah and Michael Struwe, The Cauchy problem for wave maps, Int. Math. Res. Not. 11 (2002), 555–571. MR 1890048, DOI 10.1155/S1073792802109044
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
Thomas C. Sideris, Global existence of harmonic maps in Minkowski space, Comm. Pure Appl. Math. 42 (1989), no. 1, 1–13. MR 973742, DOI 10.1002/cpa.3160420102
Terence Tao, Global regularity of wave maps. I. Small critical Sobolev norm in high dimension, Internat. Math. Res. Notices 6 (2001), 299–328. MR 1820329, DOI 10.1155/S1073792801000150
Terence Tao, Global regularity of wave maps. II. Small energy in two dimensions, Comm. Math. Phys. 224 (2001), no. 2, 443–544. MR 1869874, DOI 10.1007/PL00005588

Small solutions for the wave maps equation in critical Sobolev spaces

Daniel Tataru, Local and global results for wave maps. I, Comm. Partial Differential Equations 23 (1998), no. 9-10, 1781–1793. MR 1641721, DOI 10.1080/03605309808821400
Daniel Tataru, On global existence and scattering for the wave maps equation, Amer. J. Math. 123 (2001), no. 1, 37–77. MR 1827277, DOI 10.1353/ajm.2001.0005