On the formation of singularities in the critical $O(3)$ $\sigma$-model

Abstract

We study the phenomena of energy concentration for the critical $O(3)$ sigma model, also known as the wave map flow from $\mathbb{R}^{2+1}$ Minkowski space into the sphere $\mathbb{S}^2$. We establish rigorously and constructively existence of a set of smooth initial data resulting in a dynamic finite time formation of singularities. The construction and analysis are done in the context of the $k$-equivariant symmetry reduction, and we restrict to maps with homotopy class $k\geqslant 4$. The concentration mechanism we uncover is essentially due to a resonant self-focusing (shrinking) of a corresponding harmonic map. We show that the phenomenon is generic (e.g. in certain Sobolev spaces) in that it persists under small perturbations of initial data, while the resulting blowup is bounded by a log-modified self-similar asymptotic.

  • [BP_sigma] A. A. Belavin and A. M. Polyakov, "Metastable states of two-dimensional isotropic ferromagnets (Russian)," JETP Lett., vol. 22, pp. 245-247, 1975.
    @article{BP_sigma,
      author={Belavin, A. A. and Polyakov, A. M.},
      TITLE={Metastable states of two-dimensional isotropic ferromagnets (Russian)},
      JOURNAL={JETP Lett.},
      VOLUME={22},
      YEAR={1975},
      PAGES={245--247},
      }
  • [B_WM] Go to document P. Bizoń, T. Chmaj, and Z. Tabor, "Formation of singularities for equivariant $(2+1)$-dimensional wave maps into the 2-sphere," Nonlinearity, vol. 14, iss. 5, pp. 1041-1053, 2001.
    @article {B_WM, MRKEY = {1862811},
      AUTHOR = {Bizo{ń},
      Piotr and Chmaj, Tadeusz and Tabor, Zbis{\l}aw},
      TITLE = {Formation of singularities for equivariant {$(2+1)$}-dimensional wave maps into the 2-sphere},
      JOURNAL = {Nonlinearity},
      FJOURNAL = {Nonlinearity},
      VOLUME = {14},
      YEAR = {2001},
      NUMBER = {5},
      PAGES = {1041--1053},
      ISSN = {0951-7715},
      CODEN = {NONLE5},
      MRCLASS = {58J45 (35L67 35L70)},
      MRNUMBER = {2003b:58043},
      DOI = {10.1088/0951-7715/14/5/308},
      ZBLNUMBER = {0988.35010},
      }
  • [BS_YM] Go to document P. Bizoń, Y. N. Ovchinnikov, and I. M. Sigal, "Collapse of an instanton," Nonlinearity, vol. 17, iss. 4, pp. 1179-1191, 2004.
    @article {BS_YM, MRKEY = {2069700},
      AUTHOR = {Bizo{ń},
      P. and Ovchinnikov, Yu. N. and Sigal, I. M.},
      TITLE = {Collapse of an instanton},
      JOURNAL = {Nonlinearity},
      FJOURNAL = {Nonlinearity},
      VOLUME = {17},
      YEAR = {2004},
      NUMBER = {4},
      PAGES = {1179--1191},
      ISSN = {0951-7715},
      CODEN = {NONLE5},
      MRCLASS = {35L10 (35A30 58E15)},
      MRNUMBER = {2005e:35145},
      DOI = {10.1088/0951-7715/17/4/003},
      ZBLNUMBER = {1059.35081},
      }
  • [B_stab] E. B. Bogomol’nyui, "The stability of classical solutions," Jadernaja Fiz., vol. 24, iss. 4, pp. 861-870, 1976.
    @article {B_stab, MRKEY = {0443719},
      AUTHOR = {Bogomol'ny{\u\i},
      E. B.},
      TITLE = {The stability of classical solutions},
      JOURNAL = {Jadernaja Fiz.},
      FJOURNAL = {Jadernaja Fiz.},
      VOLUME = {24},
      YEAR = {1976},
      NUMBER = {4},
      PAGES = {861--870},
      MRCLASS = {81.49},
      MRNUMBER = {56 \#2082},
      }
  • [BP_NLS] V. S. Buslaev and G. S. Perelman, "Scattering for the nonlinear Schrödinger equation: states that are close to a soliton," Algebra i Analiz, vol. 4, iss. 6, pp. 63-102, 1992.
    @article {BP_NLS, MRKEY = {1199635},
      AUTHOR = {Buslaev, V. S. and Perelman, G. S.},
      TITLE = {Scattering for the nonlinear {S}chrödinger equation: states that are close to a soliton},
      JOURNAL = {Algebra i Analiz},
      FJOURNAL = {Rossiĭskaya Akademiya Nauk. Algebra i Analiz},
      VOLUME = {4},
      YEAR = {1992},
      NUMBER = {6},
      PAGES = {63--102},
      ISSN = {0234-0852},
      MRCLASS = {35Q55 (35Q51 47N20)},
      MRNUMBER = {94b:35256},
      MRREVIEWER = {Meng Ru Li},
      ZBLNUMBER={0853.35112},
      }
  • [CST] Go to document T. Cazenave, J. Shatah, and S. A. 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., vol. 68, iss. 3, pp. 315-349, 1998.
    @article {CST, MRKEY = {1622539},
      AUTHOR = {Cazenave, Thierry and Shatah, Jalal and Tahvildar-Zadeh, A. Shadi},
      TITLE = {Harmonic maps of the hyperbolic space and development of singularities in wave maps and {Y}ang-{M}ills fields},
      JOURNAL = {Ann. Inst. H. Poincaré Phys. Théor.},
      FJOURNAL = {Annales de l'Institut Henri Poincaré. Physique Théorique},
      VOLUME = {68},
      YEAR = {1998},
      NUMBER = {3},
      PAGES = {315--349},
      ISSN = {0246-0211},
      CODEN = {AHPAAO},
      MRCLASS = {58J45 (35L70 35Q75 58E20)},
      MRNUMBER = {2000g:58042},
      MRREVIEWER = {David M. A. Stuart},
      URL = {http://www.numdam.org/item?id=AIHPA_1998__68_3_315_0},
      ZBLNUMBER = {0918.58074},
      }
  • [CDY_blowup] Go to document K. Chang, W. Y. Ding, and R. Ye, "Finite-time blow-up of the heat flow of harmonic maps from surfaces," J. Differential Geom., vol. 36, iss. 2, pp. 507-515, 1992.
    @article {CDY_blowup, MRKEY = {1180392},
      AUTHOR = {Chang, Kung-Ching and Ding, Wei Yue and Ye, Rugang},
      TITLE = {Finite-time blow-up of the heat flow of harmonic maps from surfaces},
      JOURNAL = {J. Differential Geom.},
      FJOURNAL = {Journal of Differential Geometry},
      VOLUME = {36},
      YEAR = {1992},
      NUMBER = {2},
      PAGES = {507--515},
      ISSN = {0022-040X},
      CODEN = {JDGEAS},
      MRCLASS = {58E20 (35K55 58G11)},
      MRNUMBER = {93h:58043},
      MRREVIEWER = {Martin Fuchs},
      URL = {http://projecteuclid.org/getRecord?id=euclid.jdg/1214448751},
      ZBLNUMBER = {0765.53026},
      }
  • [S_heat] Go to document Y. M. Chen and M. Struwe, "Existence and partial regularity results for the heat flow for harmonic maps," Math. Z., vol. 201, iss. 1, pp. 83-103, 1989.
    @article {S_heat, MRKEY = {990191},
      AUTHOR = {Chen, Yun Mei and Struwe, Michael},
      TITLE = {Existence and partial regularity results for the heat flow for harmonic maps},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {201},
      YEAR = {1989},
      NUMBER = {1},
      PAGES = {83--103},
      ISSN = {0025-5874},
      CODEN = {MAZEAX},
      MRCLASS = {58E20 (58G11)},
      MRNUMBER = {90i:58031},
      MRREVIEWER = {Helmut Kaul},
      DOI = {10.1007/BF01161997},
      ZBLNUMBER = {0652.58024},
      }
  • [CS_WM1] Go to document D. Christodoulou and S. A. Tahvildar-Zadeh, "On the regularity of spherically symmetric wave maps," Comm. Pure Appl. Math., vol. 46, iss. 7, pp. 1041-1091, 1993.
    @article {CS_WM1, MRKEY = {1223662},
      AUTHOR = {Christodoulou, Demetrios and Tahvildar-Zadeh, A. Shadi},
      TITLE = {On the regularity of spherically symmetric wave maps},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {46},
      YEAR = {1993},
      NUMBER = {7},
      PAGES = {1041--1091},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {58E20},
      MRNUMBER = {94e:58030},
      MRREVIEWER = {Caio J. C. Negreiros},
      DOI = {10.1002/cpa.3160460705},
      ZBLNUMBER = {0744.58071},
      }
  • [CS_WM2] Go to document D. Christodoulou and S. A. Tahvildar-Zadeh, "On the asymptotic behavior of spherically symmetric wave maps," Duke Math. J., vol. 71, iss. 1, pp. 31-69, 1993.
    @article {CS_WM2, MRKEY = {1230285},
      AUTHOR = {Christodoulou, Demetrios and Tahvildar-Zadeh, A. Shadi},
      TITLE = {On the asymptotic behavior of spherically symmetric wave maps},
      JOURNAL = {Duke Math. J.},
      FJOURNAL = {Duke Mathematical Journal},
      VOLUME = {71},
      YEAR = {1993},
      NUMBER = {1},
      PAGES = {31--69},
      ISSN = {0012-7094},
      CODEN = {DUMJAO},
      MRCLASS = {58E20 (53C25)},
      MRNUMBER = {94j:58044},
      MRREVIEWER = {Caio J. C. Negreiros},
      DOI = {10.1215/S0012-7094-93-07103-7},
      ZBLNUMBER = {0791.58105},
      }
  • [C_inst] R. Côte, "Instability of nonconstant harmonic maps for the $(1+2)$-dimensional equivariant wave map system," Internat. Math. Res. Not., iss. 57, pp. 3525-3549, 2005.
    @article {C_inst, MRKEY = {2199855},
      AUTHOR = {C{ô}te, Rapha{ë}l},
      TITLE = {Instability of nonconstant harmonic maps for the {$(1+2)$}-dimensional equivariant wave map system},
      JOURNAL = {Internat. Math. Res. Not.},
      FJOURNAL = {International Mathematics Research Notices},
      YEAR = {2005},
      NUMBER = {57},
      PAGES = {3525--3549},
      ISSN = {1073-7928},
      MRCLASS = {58E20 (35L15 35L70 35Q53 35Q55 58J45)},
      MRNUMBER = {2006k:58023},
      MRREVIEWER = {Vladimir Tulovsky},
      ZBLNUMBER = {1101.35055},
      }
  • [ES_original] Go to document J. Eells Jr. and J. H. Sampson, "Harmonic mappings of Riemannian manifolds," Amer. J. Math., vol. 86, pp. 109-160, 1964.
    @article {ES_original, MRKEY = {0164306},
      AUTHOR = {Eells, Jr., James and Sampson, J. H.},
      TITLE = {Harmonic mappings of {R}iemannian manifolds},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {86},
      YEAR = {1964},
      PAGES = {109--160},
      ISSN = {0002-9327},
      MRCLASS = {53.72 (57.50)},
      MRNUMBER = {29 \#1603},
      MRREVIEWER = {J. A. Wolf},
      DOI = {10.2307/2373037},
      ZBLNUMBER = {0122.40102},
      }
  • [G_NLS] Go to document R. T. Glassey, "On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations," J. Math. Phys., vol. 18, iss. 9, pp. 1794-1797, 1977.
    @article {G_NLS, MRKEY = {0460850},
      AUTHOR = {Glassey, R. T.},
      TITLE = {On the blowing up of solutions to the {C}auchy problem for nonlinear {S}chrödinger equations},
      JOURNAL = {J. Math. Phys.},
      FJOURNAL = {Journal of Mathematical Physics},
      VOLUME = {18},
      YEAR = {1977},
      NUMBER = {9},
      PAGES = {1794--1797},
      ISSN = {0022-2488},
      MRCLASS = {35B35 (35G25)},
      MRNUMBER = {57 \#842},
      ZBLNUMBER={0372.35009},
      MRREVIEWER = {A. A. Arsenev},
      DOI = {10.1063/1.523491},
      }
  • [SG] Go to document J. F. Grotowski and J. Shatah, "Geometric evolution equations in critical dimensions," Calc. Var. Partial Differential Equations, vol. 30, iss. 4, pp. 499-512, 2007.
    @article {SG, MRKEY = {2332425},
      AUTHOR = {Grotowski, Joseph F. and Shatah, Jalal},
      TITLE = {Geometric evolution equations in critical dimensions},
      JOURNAL = {Calc. Var. Partial Differential Equations},
      FJOURNAL = {Calculus of Variations and Partial Differential Equations},
      VOLUME = {30},
      YEAR = {2007},
      NUMBER = {4},
      PAGES = {499--512},
      ISSN = {0944-2669},
      MRCLASS = {53C44 (35K55 58E20)},
      MRNUMBER = {2008e:53122},
      MRREVIEWER = {James McCoy},
      DOI = {10.1007/s00526-007-0100-2},
      ZBLNUMBER = {1127.58003},
      }
  • [H_moving] F. Hélein, Harmonic Maps, Conservation Laws and Moving Frames, Second ed., Cambridge: Cambridge Univ. Press, 2002.
    @book {H_moving, MRKEY = {1913803},
      AUTHOR = {H{é}lein, Fr{é}d{é}ric},
      TITLE = {Harmonic Maps, Conservation Laws and Moving Frames},
      SERIES = {Cambridge Tracts in Mathematics},
      NUMBER = {150},
      EDITION = {Second},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2002},
      PAGES = {xxvi+264},
      ISBN = {0-521-81160-0},
      MRCLASS = {58E20 (35A22 35J15 53C43 58E12)},
      MRNUMBER = {2003g:58024},
      MRREVIEWER = {Andreas Gastel},
      ZBLNUMBER = {1010.58010},
      }
  • [H_int] F. Hélein, Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems, Basel: Birkhäuser, 2001.
    @book {H_int, MRKEY = {1844305},
      AUTHOR = {H{é}lein, Fr{é}d{é}ric},
      TITLE = {Constant Mean Curvature Surfaces, Harmonic Maps and Integrable Systems},
      SERIES = {Lectures in Mathematics ETH Zürich},
      NOTE = {Notes taken by Roger Moser},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Basel},
      YEAR = {2001},
      PAGES = {122},
      ISBN = {3-7643-6576-5},
      MRCLASS = {53A10 (37K25 53C43 58E20)},
      MRNUMBER = {2002f:53009},
      MRREVIEWER = {Pascal S. Romon},
      ZBLNUMBER = {1158.53301},
      }
  • [IL] Go to document J. Isenberg and S. L. Liebling, "Singularity formation in $2+1$ wave maps," J. Math. Phys., vol. 43, iss. 1, pp. 678-683, 2002.
    @article {IL, MRKEY = {1872523},
      AUTHOR = {Isenberg, James and Liebling, Steven L.},
      TITLE = {Singularity formation in {$2+1$} wave maps},
      JOURNAL = {J. Math. Phys.},
      FJOURNAL = {Journal of Mathematical Physics},
      VOLUME = {43},
      YEAR = {2002},
      NUMBER = {1},
      PAGES = {678--683},
      ISSN = {0022-2488},
      CODEN = {JMAPAQ},
      MRCLASS = {58J45},
      MRNUMBER = {2002i:58034},
      MRREVIEWER = {Vladimir Tulovsky},
      DOI = {10.1063/1.1418717},
      ZBLNUMBER = {1052.58032},
      }
  • [KS] Go to document S. Klainerman and S. Selberg, "Remark on the optimal regularity for equations of wave maps type," Comm. Partial Differential Equations, vol. 22, iss. 5-6, pp. 901-918, 1997.
    @article {KS, MRKEY = {1452172},
      AUTHOR = {Klainerman, Sergiu and Selberg, Sigmund},
      TITLE = {Remark on the optimal regularity for equations of wave maps type},
      JOURNAL = {Comm. Partial Differential Equations},
      FJOURNAL = {Communications in Partial Differential Equations},
      VOLUME = {22},
      YEAR = {1997},
      NUMBER = {5-6},
      PAGES = {901--918},
      ISSN = {0360-5302},
      CODEN = {CPDIDZ},
      MRCLASS = {35L70 (58G16)},
      MRNUMBER = {99c:35163},
      MRREVIEWER = {Vladimir Tulovsky},
      DOI = {10.1080/03605309708821288},
      ZBLNUMBER = {0884.35102},
      }
  • [J_WM1] Go to document J. Krieger, "Global regularity of wave maps from $\Bbb R^{2+1}$ to $H^2$. Small energy," Comm. Math. Phys., vol. 250, iss. 3, pp. 507-580, 2004.
    @article {J_WM1, MRKEY = {2094472},
      AUTHOR = {Krieger, Joachim},
      TITLE = {Global regularity of wave maps from {$\bold R\sp {2+1}$} to {$H\sp 2$}. {S}mall energy},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {250},
      YEAR = {2004},
      NUMBER = {3},
      PAGES = {507--580},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {58J45 (35L05)},
      MRNUMBER = {2005f:58049},
      MRREVIEWER = {Terence C. Tao},
      DOI = {10.1007/s00220-004-1088-5},
      ZBLNUMBER = {1099.58010},
      }
  • [J_WM2] J. Krieger, "Stability of spherically symmetric wave maps," Mem. Amer. Math. Soc., vol. 181, iss. 853, p. viii, 2006.
    @article {J_WM2, MRKEY = {2214492},
      AUTHOR = {Krieger, Joachim},
      TITLE = {Stability of spherically symmetric wave maps},
      JOURNAL = {Mem. Amer. Math. Soc.},
      FJOURNAL = {Memoirs of the American Mathematical Society},
      VOLUME = {181},
      YEAR = {2006},
      NUMBER = {853},
      PAGES = {viii+80},
      ISSN = {0065-9266},
      CODEN = {MAMCAU},
      MRCLASS = {35L05 (35L70 58J45)},
      MRNUMBER = {2008j:35109},
      MRREVIEWER = {Enzo Vitillaro},
      ZBLNUMBER={05030283},
      }
  • [KS_SW] J. Krieger and W. Schlag, "On the focusing critical semi-linear wave equation," Amer. J. Math., vol. 129, iss. 3, pp. 843-913, 2007.
    @article {KS_SW, MRKEY = {2325106},
      AUTHOR = {Krieger, J. and Schlag, W.},
      TITLE = {On the focusing critical semi-linear wave equation},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {129},
      YEAR = {2007},
      NUMBER = {3},
      PAGES = {843--913},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {35L70 (35A30 35B42 35Q51)},
      MRNUMBER = {2009f:35231},
      MRREVIEWER = {Herbert Koch},
      ZBLNUMBER = {05170329},
      }
  • [Jost] J. Jost, Riemannian Geometry and Geometric Analysis, Fourth ed., New York: Springer-Verlag, 2005.
    @book {Jost, MRKEY = {2165400},
      AUTHOR = {Jost, J{ü}rgen},
      TITLE = {Riemannian Geometry and Geometric Analysis},
      SERIES = {Universitext},
      EDITION = {Fourth},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2005},
      PAGES = {xiv+566},
      ISBN = {978-3-540-25907-7; 3-540-25907-4},
      MRCLASS = {53-02 (53C20 53C21 58E05 58E10)},
      MRNUMBER = {2006c:53002},
      ZBLNUMBER = {1083.53001},
      }
  • [LPZ_stab] Go to document R. A. Leese, M. Peyrard, and W. J. Zakrzewski, "Soliton stability in the ${ O}(3) \sigma$-model in $(2+1)$ dimensions," Nonlinearity, vol. 3, iss. 2, pp. 387-412, 1990.
    @article {LPZ_stab, MRKEY = {1054581},
      AUTHOR = {Leese, Robert A. and Peyrard, Michel and Zakrzewski, Wojciech J.},
      TITLE = {Soliton stability in the {${\rm O}(3) \sigma$}-model in {$(2+1)$} dimensions},
      JOURNAL = {Nonlinearity},
      FJOURNAL = {Nonlinearity},
      VOLUME = {3},
      YEAR = {1990},
      NUMBER = {2},
      PAGES = {387--412},
      ISSN = {0951-7715},
      CODEN = {NONLE5},
      MRCLASS = {58E20 (58G99)},
      MRNUMBER = {91i:58033},
      MRREVIEWER = {Mark Temple-Raston},
      URL = {http://stacks.iop.org/0951-7715/3/387},
      ZBLNUMBER = {0721.35068},
      }
  • [LS] Go to document J. M. Linhart and L. A. Sadun, "Fast and slow blowup in the $S^2$ $\sigma$-model and the $(4+1)$-dimensional Yang-Mills model," Nonlinearity, vol. 15, iss. 2, pp. 219-238, 2002.
    @article {LS, MRKEY = {1888849},
      AUTHOR = {Linhart, Jean Marie and Sadun, Lorenzo A.},
      TITLE = {Fast and slow blowup in the {$S\sp 2$} {$\sigma$}-model and the {$(4+1)$}-dimensional {Y}ang-{M}ills model},
      JOURNAL = {Nonlinearity},
      FJOURNAL = {Nonlinearity},
      VOLUME = {15},
      YEAR = {2002},
      NUMBER = {2},
      PAGES = {219--238},
      ISSN = {0951-7715},
      CODEN = {NONLE5},
      MRCLASS = {81T13 (35B40 35L70 35Q51 35Q60 58E50)},
      MRNUMBER = {2002m:81156},
      DOI = {10.1088/0951-7715/15/2/301},
      ZBLNUMBER = {1006.81052},
      }
  • [L_NLW] Go to document H. A. Levine, "Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{tt}=-Au+\mathcal{F}(u)$," Trans. Amer. Math. Soc., vol. 192, pp. 1-21, 1974.
    @article {L_NLW, MRKEY = {0344697},
      AUTHOR = {Levine, Howard A.},
      TITLE = {Instability and nonexistence of global solutions to nonlinear wave equations of the form {$Pu_{tt}=-Au+\mathcal{F}(u)$}},
      JOURNAL = {Trans. Amer. Math. Soc.},
      FJOURNAL = {Transactions of the American Mathematical Society},
      VOLUME = {192},
      YEAR = {1974},
      PAGES = {1--21},
      ISSN = {0002-9947},
      MRCLASS = {35L60 (47H15)},
      MRNUMBER = {49 \#9436},
      MRREVIEWER = {John M. Ball},
      DOI = {10.2307/1996814},
      ZBLNUMBER = {0288.35003},
      }
  • [Manton] N. Manton and P. Sutcliffe, Topological Solitons, Cambridge: Cambridge Univ. Press, 2004.
    @book {Manton, MRKEY = {2068924},
      AUTHOR = {Manton, Nicholas and Sutcliffe, Paul},
      TITLE = {Topological Solitons},
      SERIES = {Cambridge Monogr. Math. Phys.},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {2004},
      PAGES = {xii+493},
      ISBN = {0-521-83836-3},
      MRCLASS = {58E50 (37N20 53C07 57-02 81-02 81T10 81T13)},
      MRNUMBER = {2006d:58020},
      ZBLNUMBER = {1100.37044},
      }
  • [MM] Go to document Y. Martel and F. Merle, "Blow up in finite time and dynamics of blow up solutions for the $L^2$-critical generalized KdV equation," J. Amer. Math. Soc., vol. 15, iss. 3, pp. 617-664, 2002.
    @article {MM, MRKEY = {1896235},
      AUTHOR = {Martel, Yvan and Merle, Frank},
      TITLE = {Blow up in finite time and dynamics of blow up solutions for the {$L\sp 2$}-critical generalized {K}d{V} equation},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {15},
      YEAR = {2002},
      NUMBER = {3},
      PAGES = {617--664},
      ISSN = {0894-0347},
      MRCLASS = {35Q53 (35B40 37K40)},
      MRNUMBER = {2003c:35142},
      MRREVIEWER = {Kenji Nakanishi},
      DOI = {10.1090/S0894-0347-02-00392-2},
      ZBLNUMBER = {0996.35064},
      }
  • [MR] Go to document F. Merle and P. Raphael, "The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation," Ann. of Math., vol. 161, iss. 1, pp. 157-222, 2005.
    @article {MR, MRKEY = {2150386},
      AUTHOR = {Merle, Frank and Raphael, Pierre},
      TITLE = {The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear {S}chrödinger equation},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {161},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {157--222},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {35Q55 (35B40)},
      MRNUMBER = {2006k:35277},
      MRREVIEWER = {John Albert},
      DOI = {10.4007/annals.2005.161.157},
      }
  • [PZ_shrink] Go to document B. Piette and W. J. Zakrzewski, "Shrinking of solitons in the $(2+1)$-dimensional $S^2$ sigma model," Nonlinearity, vol. 9, iss. 4, pp. 897-910, 1996.
    @article {PZ_shrink, MRKEY = {1399478},
      AUTHOR = {Piette, B. and Zakrzewski, W. J.},
      TITLE = {Shrinking of solitons in the {$(2+1)$}-dimensional {$S\sp 2$} sigma model},
      JOURNAL = {Nonlinearity},
      FJOURNAL = {Nonlinearity},
      VOLUME = {9},
      YEAR = {1996},
      NUMBER = {4},
      PAGES = {897--910},
      ISSN = {0951-7715},
      CODEN = {NONLE5},
      MRCLASS = {81T80 (81T10)},
      MRNUMBER = {97g:81071},
      MRREVIEWER = {Bernd J. Schroers},
      DOI = {10.1088/0951-7715/9/4/005},
      ZBLNUMBER = {0895.58030},
      }
  • [Sh] Go to document J. Shatah, "Weak solutions and development of singularities of the ${ SU}(2)$ $\sigma$-model," Comm. Pure Appl. Math., vol. 41, iss. 4, pp. 459-469, 1988.
    @article {Sh, MRKEY = {933231},
      AUTHOR = {Shatah, Jalal},
      TITLE = {Weak solutions and development of singularities of the {${\rm SU}(2)$} {$\sigma$}-model},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {41},
      YEAR = {1988},
      NUMBER = {4},
      PAGES = {459--469},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {58E20},
      MRNUMBER = {89f:58044},
      MRREVIEWER = {Allan M. Din},
      DOI = {10.1002/cpa.3160410405},
      ZBLNUMBER = {0686.35081},
      }
  • [ST_WM1] Go to document J. Shatah and A. Tahvildar-Zadeh, "Regularity of harmonic maps from the Minkowski space into rotationally symmetric manifolds," Comm. Pure Appl. Math., vol. 45, iss. 8, pp. 947-971, 1992.
    @article {ST_WM1, MRKEY = {1168115},
      AUTHOR = {Shatah, J. and Tahvildar-Zadeh, A.},
      TITLE = {Regularity of harmonic maps from the {M}inkowski space into rotationally symmetric manifolds},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {45},
      YEAR = {1992},
      NUMBER = {8},
      PAGES = {947--971},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {58E20},
      MRNUMBER = {93c:58056},
      MRREVIEWER = {Martin Fuchs},
      DOI = {10.1002/cpa.3160450803},
      ZBLNUMBER = {0769.58015},
      }
  • [ST_WM2] Go to document J. Shatah and S. A. Tahvildar-Zadeh, "On the Cauchy problem for equivariant wave maps," Comm. Pure Appl. Math., vol. 47, iss. 5, pp. 719-754, 1994.
    @article {ST_WM2, MRKEY = {1278351},
      AUTHOR = {Shatah, Jalal and Tahvildar-Zadeh, A. Shadi},
      TITLE = {On the {C}auchy problem for equivariant wave maps},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {47},
      YEAR = {1994},
      NUMBER = {5},
      PAGES = {719--754},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {58E20 (35L99)},
      MRNUMBER = {96c:58049},
      MRREVIEWER = {Yuan-Jen Chiang},
      DOI = {10.1002/cpa.3160470507},
      ZBLNUMBER = {0811.58059},
      }
  • [S_WM1] Go to document M. Struwe, "Radially symmetric wave maps from $(1+2)$-dimensional Minkowski space to the sphere," Math. Z., vol. 242, iss. 3, pp. 407-414, 2002.
    @article {S_WM1, MRKEY = {1985457},
      AUTHOR = {Struwe, Michael},
      TITLE = {Radially symmetric wave maps from {$(1+2)$}-dimensional {M}inkowski space to the sphere},
      JOURNAL = {Math. Z.},
      FJOURNAL = {Mathematische Zeitschrift},
      VOLUME = {242},
      YEAR = {2002},
      NUMBER = {3},
      PAGES = {407--414},
      ISSN = {0025-5874},
      CODEN = {MAZEAX},
      MRCLASS = {58J45 (58E20)},
      MRNUMBER = {2004d:58040},
      MRREVIEWER = {Andreas Gastel},
      DOI = {10.1007/s002090100345},
      ZBLNUMBER = {1052.58034},
      }
  • [S_WM2] Go to document M. Struwe, "Radially symmetric wave maps from $(1+2)$-dimensional Minkowski space to general targets," Calc. Var. Partial Differential Equations, vol. 16, iss. 4, pp. 431-437, 2003.
    @article {S_WM2, MRKEY = {1971037},
      AUTHOR = {Struwe, Michael},
      TITLE = {Radially symmetric wave maps from {$(1+2)$}-dimensional {M}inkowski space to general targets},
      JOURNAL = {Calc. Var. Partial Differential Equations},
      FJOURNAL = {Calculus of Variations and Partial Differential Equations},
      VOLUME = {16},
      YEAR = {2003},
      NUMBER = {4},
      PAGES = {431--437},
      ISSN = {0944-2669},
      MRCLASS = {58J45 (58E20)},
      MRNUMBER = {2004j:58033},
      DOI = {10.1007/s00526-002-0156-y},
      ZBLNUMBER = {1039.58033},
      }
  • [S_WM3] Go to document M. Struwe, "Equivariant wave maps in two space dimensions," Comm. Pure Appl. Math., vol. 56, iss. 7, pp. 815-823, 2003.
    @article {S_WM3, MRKEY = {1990477},
      AUTHOR = {Struwe, Michael},
      TITLE = {Equivariant wave maps in two space dimensions},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {56},
      YEAR = {2003},
      NUMBER = {7},
      PAGES = {815--823},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {58J45 (58E20)},
      MRNUMBER = {2004c:58061},
      MRREVIEWER = {Terence C. Tao},
      DOI = {10.1002/cpa.10074},
      ZBLNUMBER = {1033.53019},
      }
  • [Tao_WM1] Go to document T. Tao, "Global regularity of wave maps. II. Small energy in two dimensions," Comm. Math. Phys., vol. 224, iss. 2, pp. 443-544, 2001.
    @article {Tao_WM1, MRKEY = {1869874},
      AUTHOR = {Tao, Terence},
      TITLE = {Global regularity of wave maps. {II}. {S}mall energy in two dimensions},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {224},
      YEAR = {2001},
      NUMBER = {2},
      PAGES = {443--544},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {58J45 (35B60 35B65 35L15 58J47)},
      MRNUMBER = {2002h:58052},
      MRREVIEWER = {Joachim Krieger},
      DOI = {10.1007/PL00005588},
      ZBLNUMBER = {1020.35046},
      }
  • [Tat_WM1] Go to document D. Tataru, "On global existence and scattering for the wave maps equation," Amer. J. Math., vol. 123, iss. 1, pp. 37-77, 2001.
    @article {Tat_WM1, MRKEY = {1827277},
      AUTHOR = {Tataru, Daniel},
      TITLE = {On global existence and scattering for the wave maps equation},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {123},
      YEAR = {2001},
      NUMBER = {1},
      PAGES = {37--77},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {58J45 (35L70 35P25)},
      MRNUMBER = {2002c:58045},
      MRREVIEWER = {Kenji Nakanishi},
      URL = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v123/123.1tataru.pdf},
      ZBLNUMBER = {0979.35100},
      }
  • [Tat_WM2] Go to document D. Tataru, "Rough solutions for the wave maps equation," Amer. J. Math., vol. 127, iss. 2, pp. 293-377, 2005.
    @article {Tat_WM2, MRKEY = {2130618},
      AUTHOR = {Tataru, Daniel},
      TITLE = {Rough solutions for the wave maps equation},
      JOURNAL = {Amer. J. Math.},
      FJOURNAL = {American Journal of Mathematics},
      VOLUME = {127},
      YEAR = {2005},
      NUMBER = {2},
      PAGES = {293--377},
      ISSN = {0002-9327},
      CODEN = {AJMAAN},
      MRCLASS = {58J45 (35B30 35L15)},
      MRNUMBER = {2006a:58034},
      MRREVIEWER = {Terence C. Tao},
      URL = {http://muse.jhu.edu/journals/american_journal_of_mathematics/v127/127.2tataru.pdf},
      ZBLNUMBER={02164530},
      }
  • [W_mod] Go to document M. I. Weinstein, "Modulational stability of ground states of nonlinear Schrödinger equations," SIAM J. Math. Anal., vol. 16, iss. 3, pp. 472-491, 1985.
    @article {W_mod, MRKEY = {783974},
      AUTHOR = {Weinstein, Michael I.},
      TITLE = {Modulational stability of ground states of nonlinear {S}chrödinger equations},
      JOURNAL = {{\rm SIAM} J. Math. Anal.},
      FJOURNAL = {SIAM Journal on Mathematical Analysis},
      VOLUME = {16},
      YEAR = {1985},
      NUMBER = {3},
      PAGES = {472--491},
      ISSN = {0036-1410},
      CODEN = {SJMAAH},
      MRCLASS = {35Q20 (78A45 82A45)},
      MRNUMBER = {86i:35130},
      MRREVIEWER = {Woodford W. Zachary},
      DOI = {10.1137/0516034},
      ZBLNUMBER={0583.35028},
      }
  • [W_Ls] Go to document M. I. Weinstein, "Lyapunov stability of ground states of nonlinear dispersive evolution equations," Comm. Pure Appl. Math., vol. 39, iss. 1, pp. 51-67, 1986.
    @article {W_Ls, MRKEY = {820338},
      AUTHOR = {Weinstein, Michael I.},
      TITLE = {Lyapunov stability of ground states of nonlinear dispersive evolution equations},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {39},
      YEAR = {1986},
      NUMBER = {1},
      PAGES = {51--67},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {35B35 (35Q20)},
      MRNUMBER = {87f:35023},
      MRREVIEWER = {Roman Stankiewicz},
      DOI = {10.1002/cpa.3160390103},
      ZBLNUMBER = {0594.35005},
      }

Authors

Igor Rodnianski

Department of Mathematics
Fine Hall, Washington Road
Princeton University
Princeton, NJ 08544-1000
United States

Jacob Sterbenz

University of California at San Diego
Department of Mathematics
9500 Gilman Drive, Dept. 0112
La Jolla, 92093-0112
United States