The global stability of Minkowski space-time in harmonic gauge

Pages 1401-1477 from Volume 171 (2010), Issue 3 by Hans Lindblad, Igor Rodnianski

Abstract

We give a new proof of the global stability of Minkowski space originally established in the vacuum case by Christodoulou and Klainerman. The new approach, which relies on the classical harmonic gauge, shows that the Einstein-vacuum and the Einstein-scalar field equations with asymptotically flat initial data satisfying a global smallness condition produce global (causally geodesically complete) solutions asymptotically convergent to the Minkowski space-time.

  • [A1] Go to document S. Alinhac, "Rank 2 singular solutions for quasilinear wave equations," Internat. Math. Res. Notices, iss. 18, pp. 955-984, 2000.
    @article {A1, MRKEY = {1792284},
      AUTHOR = {Alinhac, Serge},
      TITLE = {Rank 2 singular solutions for quasilinear wave equations},
      JOURNAL = {Internat. Math. Res. Notices},
      FJOURNAL = {International Mathematics Research Notices},
      YEAR = {2000},
      NUMBER = {18},
      PAGES = {955--984},
      ISSN = {1073-7928},
      MRCLASS = {35L70 (35B40)},
      MRNUMBER = {2002e:35157},
      MRREVIEWER = {Ralph Saxton},
      DOI = {10.1155/S1073792800000507},
      ZBLNUMBER = {0971.35053},
      }
  • [A2] Go to document S. Alinhac, "The null condition for quasilinear wave equations in two space dimensions I," Invent. Math., vol. 145, iss. 3, pp. 597-618, 2001.
    @article {A2, MRKEY = {1856402},
      AUTHOR = {Alinhac, Serge},
      TITLE = {The null condition for quasilinear wave equations in two space dimensions {I}},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {145},
      YEAR = {2001},
      NUMBER = {3},
      PAGES = {597--618},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {35L70 (35A05 35L80)},
      MRNUMBER = {2002i:35127},
      MRREVIEWER = {Michael Renardy},
      DOI = {10.1007/s002220100165},
      ZBLNUMBER = {1112.35341},
      }
  • [A3] S. Alinhac, "An example of blowup at infinity for a quasilinear wave equation," Astérisque, vol. 284, pp. 1-91, 2003.
    @article {A3, MRKEY = {2003417},
      AUTHOR = {Alinhac, Serge},
      TITLE = {An example of blowup at infinity for a quasilinear wave equation},
      JOURNAL = {Astérisque},
      FJOURNAL = {Astérisque},
      VOLUME = {284},
      YEAR = {2003},
      PAGES = {1--91},
      ISSN = {0303-1179},
      MRCLASS = {35L70 (35B40 35Q60 35S50)},
      MRNUMBER = {2005a:35197},
      MRREVIEWER = {Varga Kalantarov},
      ZBLNUMBER = {1053.35097},
      }
  • [Bl] Go to document L. Blanchet, "Gravitational radiation from post-Newtonian sources and inspiralling compact binaries," Living Reviews in Relativity, vol. 5, 2002.
    @article{Bl,
      author = {Blanchet, L.},
      TITLE = {Gravitational radiation from post-Newtonian sources and inspiralling compact binaries},
      JOURNAL = {Living Reviews in Relativity},
      VOLUME = {5},
      YEAR = {2002},
      URL = {http://relativity.livingreviews.org/Articles/lrr-2006-4/},
      NOTE={http://relativity.livingreviews.org/Articles/lrr-2006-4/},
      }
  • [B-D] Go to document L. Blanchet and T. Damour, "Hereditary effects in gravitational radiation," Phys. Rev. D, vol. 46, iss. 10, pp. 4304-4319, 1992.
    @article {B-D, MRKEY = {1189306},
      AUTHOR = {Blanchet, Luc and Damour, Thibault},
      TITLE = {Hereditary effects in gravitational radiation},
      JOURNAL = {Phys. Rev. D},
      FJOURNAL = {Physical Review. D. Third Series},
      VOLUME = {46},
      YEAR = {1992},
      NUMBER = {10},
      PAGES = {4304--4319},
      ISSN = {0556-2821},
      CODEN = {PRVDAQ},
      MRCLASS = {83C30 (83C25 83C35)},
      MRNUMBER = {94d:83021},
      MRREVIEWER = {A. D. Popova},
      DOI = {10.1103/PhysRevD.46.4304},
      }
  • [CB1] Go to document Y. Choquet-Bruhat, "Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires," Acta Math., vol. 88, pp. 141-225, 1952.
    @article {CB1, MRKEY = {0053338},
      AUTHOR = {Choquet-Bruhat, Yvonne},
      TITLE = {Théorème d'existence pour certains systèmes d'équations aux dérivées partielles non linéaires},
      JOURNAL = {Acta Math.},
      FJOURNAL = {Acta Mathematica},
      VOLUME = {88},
      YEAR = {1952},
      PAGES = {141--225},
      ISSN = {0001-5962},
      MRCLASS = {36.0X},
      MRNUMBER = {14,756g},
      MRREVIEWER = {F. John},
      DOI = {10.1007/BF02392131},
      ZBLNUMBER = {0049.19201},
      }
  • [CB2] Y. Choquet-Bruhat, "Un théorème d’instabilité pour certaines équations hyperboliques non linéaires," C. R. Acad. Sci. Paris Sér. A-B, vol. 276, p. a281-a284, 1973.
    @article {CB2, MRKEY = {0315300},
      AUTHOR = {Choquet-Bruhat, Yvonne},
      TITLE = {Un théorème d'instabilité pour certaines équations hyperboliques non linéaires},
      JOURNAL = {C. R. Acad. Sci. Paris Sér. A-B},
      VOLUME = {276},
      YEAR = {1973},
      PAGES = {A281--A284},
      MRCLASS = {35L60},
      MRNUMBER = {47 \#3849},
      MRREVIEWER = {Howard A. Levine},
      }
  • [CB3] Y. Choquet-Bruhat, "The null condition and asymptotic expansions for the Einstein equations," Ann. Phys., vol. 9, iss. 3-5, pp. 258-266, 2000.
    @article {CB3, MRKEY = {1770106},
      AUTHOR = {Choquet-Bruhat, Yvonne},
      TITLE = {The null condition and asymptotic expansions for the {E}instein equations},
      INVNOTE = {Journ{é}es Relativistes 99 (Weimar)},
      JOURNAL = {Ann. Phys.},
      FJOURNAL = {Annalen der Physik (8)},
      VOLUME = {9},
      YEAR = {2000},
      NUMBER = {3-5},
      PAGES = {258--266},
      ISSN = {0003-3804},
      MRCLASS = {83C05},
      MRNUMBER = {2001g:83009},
      MRREVIEWER = {Norbert Noutchegueme},
      DOI = {10.1002/(SICI)1521-3889(200005)9:3/5<258::AID-ANDP258>3.3.CO;2-P},
      ZBLNUMBER = {0973.83018},
      }
  • [CB-G] Go to document Y. Choquet-Bruhat and R. Geroch, "Global aspects of the Cauchy problem in general relativity," Comm. Math. Phys., vol. 14, pp. 329-335, 1969.
    @article {CB-G, MRKEY = {0250640},
      AUTHOR = {Choquet-Bruhat, Yvonne and Geroch, Robert},
      TITLE = {Global aspects of the {C}auchy problem in general relativity},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {14},
      YEAR = {1969},
      PAGES = {329--335},
      ISSN = {0010-3616},
      MRCLASS = {83.53},
      MRNUMBER = {40 \#3872},
      MRREVIEWER = {H. A. Buchdahl},
      URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1103841822},
      ZBLNUMBER = {0182.59901},
      }
  • [C1] Go to document D. Christodoulou, "Global solutions of nonlinear hyperbolic equations for small initial data," Comm. Pure Appl. Math., vol. 39, iss. 2, pp. 267-282, 1986.
    @article {C1, MRKEY = {820070},
      AUTHOR = {Christodoulou, Demetrios},
      TITLE = {Global solutions of nonlinear hyperbolic equations for small initial data},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {39},
      YEAR = {1986},
      NUMBER = {2},
      PAGES = {267--282},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {35L70},
      MRNUMBER = {87c:35111},
      MRREVIEWER = {R. Glassey},
      DOI = {10.1002/cpa.3160390205},
      ZBLNUMBER = {0612.35090},
      }
  • [C2] D. Christodoulou, "The global initial value problem in general relativity," in The Ninth Marcel Grossmann Meeting\/, Gurzadyan, V. G. and Jansen, R. T., Eds., River Edge, NJ: World Sci. Publ., 2002, pp. 44-54.
    @incollection{C2,
      author = {Christodoulou, Demetrios},
      TITLE={The global initial value problem in general relativity},
      BOOKTITLE={ The Ninth Marcel Grossmann Meeting\/} , VENUE={Rome 2000},
      EDITOR={V. G. Gurzadyan and R. T. Jansen},
      PUBLISHER={ World Sci. Publ.},
      ADDRESS={ River Edge, NJ},
      YEAR={2002},
      PAGES={ 44--54},
      }
  • [C-K] D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of the Minkowski Space, Princeton, NJ: Princeton Univ. Press, 1993.
    @book {C-K, MRKEY = {1316662},
      AUTHOR = {Christodoulou, Demetrios and Klainerman, Sergiu},
      TITLE = {The Global Nonlinear Stability of the {M}inkowski Space},
      SERIES = {Princeton Math. Series},
      NUMBER = {41},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {1993},
      PAGES = {x+514},
      ISBN = {0-691-08777-6},
      MRCLASS = {83C05 (35Q75 58G16 83C35)},
      MRNUMBER = {95k:83006},
      MRREVIEWER = {Alan D. Rendall},
      ZBLNUMBER = {0827.53055},
      }
  • [C-D] Go to document P. T. Chruściel and E. Delay, "Existence of non-trivial, vacuum, asymptotically simple spacetimes," Classical Quantum Gravity, vol. 19, iss. 9, p. l71-l79, 2002.
    @article {C-D, MRKEY = {1902228},
      AUTHOR = {Chru{\'s}ciel, Piotr T. and Delay, Erwann},
      TITLE = {Existence of non-trivial, vacuum, asymptotically simple spacetimes},
      JOURNAL = {Classical Quantum Gravity},
      FJOURNAL = {Classical and Quantum Gravity},
      VOLUME = {19},
      YEAR = {2002},
      NUMBER = {9},
      PAGES = {L71--L79},
      ISSN = {0264-9381},
      CODEN = {CQGRDG},
      MRCLASS = {83C30},
      MRNUMBER = {2003e:83024a},
      MRREVIEWER = {R. Beig},
      DOI = {10.1088/0264-9381/19/9/101},
      ZBLNUMBER = {1005.83009},
      }
  • [Co] Go to document J. Corvino, "Scalar curvature deformation and a gluing construction for the Einstein constraint equations," Comm. Math. Phys., vol. 214, iss. 1, pp. 137-189, 2000.
    @article {Co, MRKEY = {1794269},
      AUTHOR = {Corvino, Justin},
      TITLE = {Scalar curvature deformation and a gluing construction for the {E}instein constraint equations},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {214},
      YEAR = {2000},
      NUMBER = {1},
      PAGES = {137--189},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {53C21 (35Q75 83C05)},
      MRNUMBER = {2002b:53050},
      MRREVIEWER = {Alan D. Rendall},
      DOI = {10.1007/PL00005533},
      ZBLNUMBER = {1031.53064},
      }
  • [Fo] V. Fock, The Theory of Space, Time and Gravitation, New York: The Macmillan Co., 1964.
    @book {Fo, MRKEY = {0162586},
      AUTHOR = {Fock, V.},
      TITLE = {The Theory of Space, Time and Gravitation},
      PUBLISHER = {The Macmillan Co.},
      ADDRESS = {New York},
      YEAR = {1964},
      PAGES = {xii+448},
      MRCLASS = {83.00},
      MRNUMBER = {28 \#5784},
      ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
      ZBLNUMBER = {0112.43804},
      ZBLNUMBER = {0085.42301},
      }
  • [Fr] Go to document H. Friedrich, "On the existence of $n$-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure," Comm. Math. Phys., vol. 107, iss. 4, pp. 587-609, 1986.
    @article {Fr, MRKEY = {868737},
      AUTHOR = {Friedrich, Helmut},
      TITLE = {On the existence of {$n$}-geodesically complete or future complete solutions of {E}instein's field equations with smooth asymptotic structure},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {107},
      YEAR = {1986},
      NUMBER = {4},
      PAGES = {587--609},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {83C05 (53C50 58D25 83C20)},
      MRNUMBER = {88b:83006},
      MRREVIEWER = {Judith M. Arms},
      URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1104116232},
      ZBLNUMBER = {0659.53056},
      }
  • [H-E] S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, London: Cambridge Univ. Press, 1973.
    @book {H-E, MRKEY = {0424186},
      AUTHOR = {Hawking, S. W. and Ellis, G. F. R.},
      TITLE = {The Large Scale Structure of Space-Time},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {London},
      YEAR = {1973},
      PAGES = {xi+391},
      MRCLASS = {83.58},
      MRNUMBER = {54 \#12154},
      MRREVIEWER = {Michael P. Ryan, Jr.},
      ZBLNUMBER = {0265.53054},
      }
  • [H1] L. Hörmander, "The lifespan of classical solutions of nonlinear hyperbolic equations," in Pseudodifferential Operators, New York: Springer-Verlag, 1987, vol. 1256, pp. 214-280.
    @incollection {H1, MRKEY = {897781},
      AUTHOR = {H{ö}rmander, Lars},
      TITLE = {The lifespan of classical solutions of nonlinear hyperbolic equations},
      BOOKTITLE = {Pseudodifferential Operators},
      VENUE={{O}berwolfach, 1986},
      SERIES = {Lecture Notes in Math.},
      VOLUME = {1256},
      PAGES = {214--280},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1987},
      MRCLASS = {35B40 (35L70)},
      MRNUMBER = {88j:35024},
      MRREVIEWER = {Alain Bachelot},
      ZBLNUMBER = {0632.35045},
      }
  • [H2] L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, New York: Springer-Verlag, 1997, vol. 26.
    @book {H2, MRKEY = {1466700},
      AUTHOR = {H{ö}rmander, Lars},
      TITLE = {Lectures on Nonlinear Hyperbolic Differential Equations},
      SERIES = {Math. Appl. (Berlin)},
      VOLUME = {26},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1997},
      PAGES = {viii+289},
      ISBN = {3-540-62921-1},
      MRCLASS = {35Lxx (35A27 35B65 35D05 35Sxx)},
      MRNUMBER = {98e:35103},
      MRREVIEWER = {Jean-Marc Delort},
      ZBLNUMBER = {0881.35001},
      }
  • [H3] L. Hörmander, "On the fully nonlinear Cauchy problem with small data. II," in Microlocal Analysis and Nonlinear Waves, New York: Springer-Verlag, 1991, vol. 30, pp. 51-81.
    @incollection {H3, MRKEY = {1120284},
      AUTHOR = {H{ö}rmander, Lars},
      TITLE = {On the fully nonlinear {C}auchy problem with small data. {II}},
      BOOKTITLE = {Microlocal Analysis and Nonlinear Waves },
      VENUE={{M}inneapolis, {MN},
      1988--1989},
      SERIES = {IMA Vol. Math. Appl.},
      VOLUME = {30},
      PAGES = {51--81},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1991},
      MRCLASS = {35L70 (35A05)},
      MRNUMBER = {94c:35127},
      ZBLNUMBER = {0783.35036},
      }
  • [J1] Go to document F. John, "Blow-up for quasilinear wave equations in three space dimensions," Comm. Pure Appl. Math., vol. 34, iss. 1, pp. 29-51, 1981.
    @article {J1, MRKEY = {600571},
      AUTHOR = {John, Fritz},
      TITLE = {Blow-up for quasilinear wave equations in three space dimensions},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {34},
      YEAR = {1981},
      NUMBER = {1},
      PAGES = {29--51},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {35L67},
      MRNUMBER = {83d:35096},
      MRREVIEWER = {Ronald DiPerna},
      DOI = {10.1002/cpa.3160340103},
      }
  • [J2] F. John, "Blow-up of radial solutions of $u_{tt}=c^2(u_t)\Delta u$ in three space dimensions," Mat. Apl. Comput., vol. 4, iss. 1, pp. 3-18, 1985.
    @article {J2, MRKEY = {808321},
      AUTHOR = {John, Fritz},
      TITLE = {Blow-up of radial solutions of {$u\sb {tt}=c\sp 2(u\sb t)\Delta u$} in three space dimensions},
      JOURNAL = {Mat. Apl. Comput.},
      FJOURNAL = {Matemática Aplicada e Computacional},
      VOLUME = {4},
      YEAR = {1985},
      NUMBER = {1},
      PAGES = {3--18},
      ISSN = {0101-8205},
      MRCLASS = {35L70 (35L67)},
      MRNUMBER = {87c:35114},
      MRREVIEWER = {R. Glassey},
      ZBLNUMBER = {0597.35082},
      }
  • [J3] Go to document F. John, "Blow-up of solutions of nonlinear wave equations in three space dimensions," Manuscripta Math., vol. 28, iss. 1-3, pp. 235-268, 1979.
    @article {J3, MRKEY = {535704},
      AUTHOR = {John, Fritz},
      TITLE = {Blow-up of solutions of nonlinear wave equations in three space dimensions},
      JOURNAL = {Manuscripta Math.},
      FJOURNAL = {Manuscripta Mathematica},
      VOLUME = {28},
      YEAR = {1979},
      NUMBER = {1-3},
      PAGES = {235--268},
      ISSN = {0025-2611},
      CODEN = {MSMHB2},
      MRCLASS = {35L20},
      MRNUMBER = {80i:35114},
      MRREVIEWER = {V. M. Babich},
      DOI = {10.1007/BF01647974},
      ZBLNUMBER = {0406.35042},
      }
  • [J-K] Go to document F. John and S. Klainerman, "Almost global existence to nonlinear wave equations in three space dimensions," Comm. Pure Appl. Math., vol. 37, iss. 4, pp. 443-455, 1984.
    @article {J-K, MRKEY = {745325},
      AUTHOR = {John, F. and Klainerman, S.},
      TITLE = {Almost global existence to nonlinear wave equations in three space dimensions},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {37},
      YEAR = {1984},
      NUMBER = {4},
      PAGES = {443--455},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {35L70},
      MRNUMBER = {85k:35147},
      MRREVIEWER = {R. Glassey},
      DOI = {10.1002/cpa.3160370403},
      ZBLNUMBER = {0599.35104},
      }
  • [K1] S. Klainerman, "Long time behaviour of solutions to nonlinear wave equations," in Proceedings of the International Congress of Mathematicians, Warsaw, 1984, pp. 1209-1215.
    @inproceedings {K1, MRKEY = {804771},
      AUTHOR = {Klainerman, Sergiu},
      TITLE = {Long time behaviour of solutions to nonlinear wave equations},
      BOOKTITLE = {Proceedings of the {I}nternational {C}ongress of {M}athematicians},
      VOLUME={1, 2},
      VENUE={{W}arsaw, 1983},
      PAGES = {1209--1215},
      PUBLISHER = {PWN},
      ADDRESS = {Warsaw},
      YEAR = {1984},
      MRCLASS = {35B40 (35G20)},
      MRNUMBER = {804771},
      ZBLNUMBER = {0581.35052},
      }
  • [K2] S. Klainerman, "The null condition and global existence to nonlinear wave equations," in Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1, Providence, RI: Amer. Math. Soc., 1986, vol. 23, pp. 293-326.
    @incollection {K2, MRKEY = {837683},
      AUTHOR = {Klainerman, Sergiu},
      TITLE = {The null condition and global existence to nonlinear wave equations},
      BOOKTITLE = {Nonlinear Systems of Partial Differential Equations in Applied Mathematics, {P}art 1},
      VENUE={{S}anta {F}e, {NM},
      1984},
      SERIES = {Lectures in Appl. Math.},
      VOLUME = {23},
      PAGES = {293--326},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1986},
      MRCLASS = {35L70 (81E99)},
      MRNUMBER = {87h:35217},
      MRREVIEWER = {Philippe-A. Dionne},
      ZBLNUMBER = {0599.35105},
      }
  • [K-N1] S. Klainerman and F. Nicolò, The Evolution Problem in General Relativity, Boston, MA: Birkhäuser, 2003.
    @book {K-N1, MRKEY = {1946854},
      AUTHOR = {Klainerman, Sergiu and Nicol{ò},
      Francesco},
      TITLE = {The Evolution Problem in General Relativity},
      SERIES = {Prog. Math. Phys.},
      NUMBER = {25},
      PUBLISHER = {Birkhäuser},
      ADDRESS = {Boston, MA},
      YEAR = {2003},
      PAGES = {xiv+385},
      ISBN = {0-8176-4254-4},
      MRCLASS = {58J45 (35Q75 83C05)},
      MRNUMBER = {2004f:58036},
      MRREVIEWER = {Lars {\AA}ke Andersson},
      ZBLNUMBER = {1010.83004},
      }
  • [K-N2] Go to document S. Klainerman and F. Nicolò, "Peeling properties of asymptotically flat solutions to the Einstein vacuum equations," Classical Quantum Gravity, vol. 20, iss. 14, pp. 3215-3257, 2003.
    @article {K-N2, MRKEY = {1992002},
      AUTHOR = {Klainerman, Sergiu and Nicol{ò},
      Francesco},
      TITLE = {Peeling properties of asymptotically flat solutions to the {E}instein vacuum equations},
      JOURNAL = {Classical Quantum Gravity},
      FJOURNAL = {Classical and Quantum Gravity},
      VOLUME = {20},
      YEAR = {2003},
      NUMBER = {14},
      PAGES = {3215--3257},
      ISSN = {0264-9381},
      CODEN = {CQGRDG},
      MRCLASS = {83C30 (83C05)},
      MRNUMBER = {2005b:83038a},
      MRREVIEWER = {Lars {\AA}ke Andersson},
      DOI = {10.1088/0264-9381/20/14/319},
      ZBLNUMBER = {1045.83016},
      }
  • [L1] Go to document H. Lindblad, "On the lifespan of solutions of nonlinear wave equations with small initial data," Comm. Pure Appl. Math., vol. 43, iss. 4, pp. 445-472, 1990.
    @article {L1, MRKEY = {1047332},
      AUTHOR = {Lindblad, Hans},
      TITLE = {On the lifespan of solutions of nonlinear wave equations with small initial data},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {43},
      YEAR = {1990},
      NUMBER = {4},
      PAGES = {445--472},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {35L70 (35B05)},
      MRNUMBER = {91i:35129},
      MRREVIEWER = {Vladimir Georgiev},
      DOI = {10.1002/cpa.3160430403},
      ZBLNUMBER = {0719.35005},
      }
  • [L2] Go to document H. Lindblad, "Global solutions of nonlinear wave equations," Comm. Pure Appl. Math., vol. 45, iss. 9, pp. 1063-1096, 1992.
    @article {L2, MRKEY = {1177476},
      AUTHOR = {Lindblad, Hans},
      TITLE = {Global solutions of nonlinear wave equations},
      JOURNAL = {Comm. Pure Appl. Math.},
      FJOURNAL = {Communications on Pure and Applied Mathematics},
      VOLUME = {45},
      YEAR = {1992},
      NUMBER = {9},
      PAGES = {1063--1096},
      ISSN = {0010-3640},
      CODEN = {CPAMA},
      MRCLASS = {35L70 (35B05)},
      MRNUMBER = {94a:35080},
      MRREVIEWER = {Yoshihiro Shibata},
      DOI = {10.1002/cpa.3160450902},
      ZBLNUMBER = {0840.35065},
      }
  • [L3] Go to document H. Lindblad, "Blow-up for solutions of $\square u=\vert u\vert ^p$ with small initial data," Comm. Partial Differential Equations, vol. 15, iss. 6, pp. 757-821, 1990.
    @article {L3, MRKEY = {1070232},
      AUTHOR = {Lindblad, Hans},
      TITLE = {Blow-up for solutions of {$\square u=\vert u\vert \sp p$} with small initial data},
      JOURNAL = {Comm. Partial Differential Equations},
      FJOURNAL = {Communications in Partial Differential Equations},
      VOLUME = {15},
      YEAR = {1990},
      NUMBER = {6},
      PAGES = {757--821},
      ISSN = {0360-5302},
      CODEN = {CPDIDZ},
      MRCLASS = {35L70 (35B05)},
      MRNUMBER = {91k:35168},
      MRREVIEWER = {Michelangelo Franciosi},
      DOI = {10.1080/03605309908820708},
      ZBLNUMBER = {0712.35018},
      }
  • [L-R1] Go to document H. Lindblad and I. Rodnianski, "The weak null condition for Einstein’s equations," C. R. Math. Acad. Sci. Paris, vol. 336, iss. 11, pp. 901-906, 2003.
    @article {L-R1, MRKEY = {1994592},
      AUTHOR = {Lindblad, Hans and Rodnianski, Igor},
      TITLE = {The weak null condition for {E}instein's equations},
      JOURNAL = {C. R. Math. Acad. Sci. Paris},
      FJOURNAL = {Comptes Rendus Mathématique. Académie des Sciences. Paris},
      VOLUME = {336},
      YEAR = {2003},
      NUMBER = {11},
      PAGES = {901--906},
      ISSN = {1631-073X},
      MRCLASS = {83C05 (35Q75 58J45)},
      MRNUMBER = {2004h:83008},
      MRREVIEWER = {Norbert Noutchegueme},
      DOI = {10.1016/S1631-073X(03)00231-0},
      ZBLNUMBER = {1045.35101},
      }
  • [L-R2] Go to document H. Lindblad and I. Rodnianski, "Global existence for the Einstein vacuum equations in wave coordinates," Comm. Math. Phys., vol. 256, iss. 1, pp. 43-110, 2005.
    @article {L-R2, MRKEY = {2134337},
      AUTHOR = {Lindblad, Hans and Rodnianski, Igor},
      TITLE = {Global existence for the {E}instein vacuum equations in wave coordinates},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {256},
      YEAR = {2005},
      NUMBER = {1},
      PAGES = {43--110},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {83C05 (35Q75 58J45)},
      MRNUMBER = {2006b:83020},
      MRREVIEWER = {Marcel Dossa},
      DOI = {10.1007/s00220-004-1281-6},
      ZBLNUMBER = {1081.83003},
      }
  • [S] Go to document W. Shu, "Asymptotic properties of the solutions of linear and nonlinear spin field equations in Minkowski space," Comm. Math. Phys., vol. 140, iss. 3, pp. 449-480, 1991.
    @article {S, MRKEY = {1130694},
      AUTHOR = {Shu, Wei-Tong},
      TITLE = {Asymptotic properties of the solutions of linear and nonlinear spin field equations in {M}inkowski space},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {140},
      YEAR = {1991},
      NUMBER = {3},
      PAGES = {449--480},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {58G17 (35L45 35L70 35Q60 81R25 81T13)},
      MRNUMBER = {92m:58138},
      MRREVIEWER = {Piotr T. Chru{\'s}ciel},
      URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1104248093},
      ZBLNUMBER = {0735.53060},
      }
  • [S-Y] Go to document R. Schoen and S. T. Yau, "On the proof of the positive mass conjecture in general relativity," Comm. Math. Phys., vol. 65, iss. 1, pp. 45-76, 1979.
    @article {S-Y, MRKEY = {526976},
      AUTHOR = {Schoen, Richard and Yau, Shing Tung},
      TITLE = {On the proof of the positive mass conjecture in general relativity},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {65},
      YEAR = {1979},
      NUMBER = {1},
      PAGES = {45--76},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {83C99 (53C20 58E20)},
      MRNUMBER = {80j:83024},
      MRREVIEWER = {J. L. Kazdan},
      URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1103904790},
      ZBLNUMBER = {0405.53045},
      }
  • [Wa] R. M. Wald, General Relativity, Chicago, IL: University of Chicago Press, 1984.
    @book {Wa, MRKEY = {757180},
      AUTHOR = {Wald, Robert M.},
      TITLE = {General Relativity},
      PUBLISHER = {University of Chicago Press},
      ADDRESS = {Chicago, IL},
      YEAR = {1984},
      PAGES = {xiii+491},
      ISBN = {0-226-87032-4; 0-226-87033-2},
      MRCLASS = {83-02 (83-01)},
      MRNUMBER = {86a:83001},
      MRREVIEWER = {Arthur Greenspoon},
      ZBLNUMBER = {0549.53001},
      }
  • [Wi] Go to document E. Witten, "A new proof of the positive energy theorem," Comm. Math. Phys., vol. 80, iss. 3, pp. 381-402, 1981.
    @article {Wi, MRKEY = {626707},
      AUTHOR = {Witten, Edward},
      TITLE = {A new proof of the positive energy theorem},
      JOURNAL = {Comm. Math. Phys.},
      FJOURNAL = {Communications in Mathematical Physics},
      VOLUME = {80},
      YEAR = {1981},
      NUMBER = {3},
      PAGES = {381--402},
      ISSN = {0010-3616},
      CODEN = {CMPHAY},
      MRCLASS = {83C99 (58G25)},
      MRNUMBER = {83e:83035},
      MRREVIEWER = {Andrzej Trautman},
      URL = {http://projecteuclid.org/getRecord?id=euclid.cmp/1103919981},
      ZBLNUMBER = {1051.83532},
      }
  • [Z] N. Zipser, "The global nonlinear stability of the trivial solution of the Einstein-Maxwell equations," PhD Thesis , Harvard University, 2000.
    @phdthesis{Z,
      author={Zipser, N.},
      TITLE={The global nonlinear stability of the trivial solution of the Einstein-Maxwell equations},
      SCHOOL={Harvard University},
      YEAR={ 2000},
      }

Keywords
, , ,
DOI
10.4007/annals.2010.171.1401
MR
2680391
zbMATH
1192.53066
Milestones
Received: 20 December 2004
Accepted: 5 December 2005

Authors

Hans Lindblad

University of California, San Diego
Department of Mathematics
La Jolla CA 92093-0112
United States

Igor Rodnianski

Princeton University
Department of Mathematics
Fine Hall–Washington Road
Princeton NJ 08544
United States