Wave Maps and Ill-posedness of their Cauchy Problem

  • Piero D’Ancona
  • Vladimir Georgiev
Part of the Operator Theory: Advances and Applications book series (OT, volume 159)


In this review article we present an introduction to the theory of wave maps, give a short overview of some recent methods used in this field and finally we prove some recent results on ill-posedness of the corresponding Cauchy problem for the wave maps in critical Sobolev spaces. The low regularity solutions for the wave map problem are studied by the aid of appropriate bilinear estimates in the spirit of ones introduced by Klainerman and Bourgain. Our approach to obtain ill-posedness in critical Sobolev norms uses suitable family of wave maps constructed via geodesic flow on the target manifold. We give two alternative proofs of the ill-posedness: the first approach is based on the application of Fourier analysis tools, while the second proof is based on the application of the classical fundamental solution representation for the free wave equation. For the case of subcritical Sobolev norms we establish non-uniqueness of the corresponding Cauchy problem.


Equivariant wave maps Hs-spaces blow-up of solution 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R. Adams, Sobolev Spaces. Academic Press, New York, 1975.Google Scholar
  2. [2]
    J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I, II, Geom. Funct. Anal. 3 (1993) 107–156, 209–262.zbMATHMathSciNetGoogle Scholar
  3. [3]
    J. Bourgain, Periodic Korteweg-de Vries equation with measures as initial data, Selecta Math. 3 (1997) 115–159.zbMATHMathSciNetGoogle Scholar
  4. [4]
    Ph. Brenner and P. Kumlin, On wave equations with supercritical nonlinearities, Arch. Math. 74 (2000) 129–147.MathSciNetGoogle Scholar
  5. [5]
    T. Cazenave, J. Shatah, and 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. 68,3 (1998), 315–349.MathSciNetGoogle Scholar
  6. [6]
    Y. Choquet-Bruhat, Global existence for non-linear σ-models, Rend. Sem. Mat. Univ. Pol. Torino, Special Issue (1988), 65–86.Google Scholar
  7. [7]
    Y. Choquet-Bruhat, Global wave maps on curved space times, Mathematical and quantum aspects of relativity and cosmology (Pythagoreon, 1998), 1–29, Lecture Notes in Phys., 537, Springer, Berlin, 2000.Google Scholar
  8. [8]
    D. Christodoulou and A. S. Tahvildar-Zadeh, On the regularity of spherically symmetric wave maps, Comm. Pure Appl. Math. 46,7 (1993), 1041–1091.MathSciNetGoogle Scholar
  9. [9]
    P. D’Ancona and V. Georgiev, On the continuity of the solution operator to the wave map system, accepted in CPAM.Google Scholar
  10. [10]
    P. D’Ancona and V. Georgiev, Low regularity solutions for the wave map equation into the 2-D sphere, accepted in Math. Zeitschrift.Google Scholar
  11. [11]
    A. Freire, S. Müller and M. Struwe, Weak compactness of wave maps and harmonic maps, Ann. Inst. Henri Poincaré 15 No.6 (1998) 425–759.Google Scholar
  12. [12]
    V. Georgiev and A. Ivanov, Concentration of local energy for two-dimensional wave maps, preprint 2003.Google Scholar
  13. [13]
    M. G. Grillakis, Classical solution for the equivariant wave maps in 1+2 dimensions, preprint, 1991.Google Scholar
  14. [14]
    M. G. Grillakis, The wave map problem, In Current developments in mathematics, 1997 (Cambridge, MA), Int. Press, Boston, MA, 1999, pp. 227–230.Google Scholar
  15. [15]
    J. Ginibre and G. Velo, The Cauchy problem for the O(N), CP(N−1), and GC(N, p) models, Ann. Physics 142 (1982), no. 2, 393–415.CrossRefMathSciNetGoogle Scholar
  16. [16]
    C. H. Gu, On the Cauchy problem for harmonic maps defined on two-dimensional Minkowski space, Comm. Pure Appl. Math. 33,6 (1980), 727–737.zbMATHMathSciNetGoogle Scholar
  17. [17]
    H. Karcher and J. C. Wood, Non-existence results and growth properties of harmonic maps and forms, J. Reine Angew. Math. 353 (1984) 165–180.MathSciNetGoogle Scholar
  18. [18]
    T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891–907.MathSciNetGoogle Scholar
  19. [19]
    S. Klainerman and M. Machedon, On the regularity properties of a model problem related to wave maps, Duke Math. J. 87 (1997), no. 3, 553–589.CrossRefMathSciNetGoogle Scholar
  20. [20]
    S. Klainerman and S. Selberg, Remark on the optimal regularity for equations of wave maps type, Comm. Partial Differential Equations 22,5–6 (1997), 901–918.MathSciNetGoogle Scholar
  21. [21]
    S. Klainerman and S. Selberg, Bilinear estimates and applications to nonlinear wave equations (English. English summary), Commun. Contemp. Math. 4 (2002), no. 2, 223–295.MathSciNetGoogle Scholar
  22. [22]
    C. Kenig, G. Ponce and L. Vega, On the ill-posedness of some canonical dispersive equations, Duke Math. Journal 106 (2001) 617–632.MathSciNetGoogle Scholar
  23. [23]
    J.L. Lions and E. Magenes, Non-homogeneous boundary value problems and applications, Vol.I, Springer Verlag, Berlin 1972.Google Scholar
  24. [24]
    L. Molinet, J. C. Saut and N. Tzvetkov, Ill-posedness issues for the Benjamin-Ono and related equations., SIAM J. Math. Anal. 33 (2001) 982–988.CrossRefMathSciNetGoogle Scholar
  25. [25]
    S. Müller and M. Struwe, Global existence of wave maps in 1 + 2 dimensions with finite energy data, Topol. Methods Nonlinear Anal. 7,2 (1996), 245–259.MathSciNetGoogle Scholar
  26. [26]
    A. Nahmod, A. Stefanov and K. Uhlenbeck, On the well-posedness of the wave map problem in high dimensions, Comm. Anal. Geom. 11, Number 1, 49–83, 2003.MathSciNetGoogle Scholar
  27. [27]
    K. Nakanishi and M. Ohta, On global existence of solutions to nonlinear wave equations of wave map type, Nonlinear Anal. TMA 42, (2000), 1231–1252.CrossRefMathSciNetGoogle Scholar
  28. [28]
    T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear Partial Differential Equations, Walter de Gruyter, Berlin 1996.Google Scholar
  29. [29]
    J. Shatah, Weak solutions and development of singularities of the su(2) σ-model, Comm. Pure Appl. Math. 41,4 (1988), 459–469.zbMATHMathSciNetGoogle Scholar
  30. [30]
    J. Shatah and M. Struwe, Geometric wave equations, New York University Courant Institute of Mathematical Sciences, New York, 1998.Google Scholar
  31. [31]
    J. Shatah and M. Struwe, The Cauchy problem for wave maps, Preprint; to appear on International Math. Research Notices.Google Scholar
  32. [32]
    J. Shatah and A. Sh. Tahvildar-Zadeh, On the Cauchy problem for equivariant wave maps, Comm. Pure Appl. Math. 47 (1994), no. 5, 719–754.MathSciNetGoogle Scholar
  33. [33]
    I. Sigal, Nonlinear semi-groups, Ann. of Math. 78 No. 2 (1963) 339–364.MathSciNetGoogle Scholar
  34. [34]
    E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton University Press, Princeton, N.J., 1971, Princeton Mathematical Series, No. 32.Google Scholar
  35. [35]
    M. Struwe, Wave maps, In Nonlinear partial differential equations in geometry and physics (Knoxville, TN, 1995). Birkhäuser, Basel, 1997, pp. 113–153.Google Scholar
  36. [36]
    M. Struwe, Equivariant wave maps in two space dimensions, preprint, to appear in Comm. Pure and Appl. Math.Google Scholar
  37. [37]
    M. Struwe, Radially symmetric wave maps from 1+2-dimensional Minkowski space to the sphere, Preprint; to appear on Math. Zeitschrift.Google Scholar
  38. [38]
    M. Struwe, Radially symmetric wave maps from 1+2-dimensional Minkowski space to general targets, Preprint.Google Scholar
  39. [39]
    T. Tao, Ill-posedness for one-dimensional wave maps at the critical regularity, Amer. J. Math. 122 (2000), no. 3, 451–463.zbMATHMathSciNetGoogle Scholar
  40. [40]
    T. Tao, Global regularity of wave maps II. Small energy in two dimensions, Comm. Math. Phys. 224, (2001), 443–544.CrossRefzbMATHMathSciNetGoogle Scholar
  41. [41]
    D. Tataru, Local and global results for wave maps I, Comm. Part. Diff. Eq. 23 (1998) 1781–1793.zbMATHMathSciNetGoogle Scholar
  42. [42]
    D. Tataru, On global existence and scattering for the wave maps equation, Amer. J. Math. 123 (2001), no. 1, 37–77.zbMATHMathSciNetGoogle Scholar
  43. [43]
    M. Taylor, Partial differential equations, Vol.III, Springer Verlag, New York, 1997.Google Scholar
  44. [44]
    H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North Holland Co., Amsterdam 1978.Google Scholar
  45. [45]
    H. Triebel, Interpolation theory, function spaces, differential operators, second ed., Johann Ambrosius Barth, Heidelberg, 1995.Google Scholar
  46. [46]
    N. Tzvetkov, Remark on the local ill-posedness for KdV equation, C. R. Acad. Sci. Paris Sèr. I Math. 329 (1999) 1043–1047.zbMATHMathSciNetGoogle Scholar
  47. [47]
    Y. Zhou, Global weak solutions for (1+2)-dimensional wave maps into homogeneous spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire 16,4 (1999), 411–422.CrossRefzbMATHGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2005

Authors and Affiliations

  • Piero D’Ancona
    • 1
  • Vladimir Georgiev
    • 2
  1. 1.Dipartimento di MatematicaUniversità di Roma “La Sapienza”RomaItaly
  2. 2.Dipartimento di MatematicaUniversità di PisaPisaItaly

Personalised recommendations