The threshold conjecture for the energy critical hyperbolic Yang–Mills equation

Abstract

This article represents the fourth and final part of a four-paper sequence whose aim is to prove the Threshold Conjecture as well as the more general Dichotomy Theorem for the energy critical $4+1$-dimensional hyperbolic Yang–Mills equation. The Threshold Theorem asserts that topologically trivial solutions with energy below twice the ground state energy are global and scatter. The Dichotomy Theorem applies to solutions in arbitrary topological class with large energy, and provides two exclusive alternatives: Either the solution is global and scatters, or it bubbles off a soliton in either finite time or infinite time.

Using the caloric gauge developed in the first paper, the continuation/scattering criteria established in the second paper, and the large data analysis in an arbitrary topological class at optimal regularity in the third paper, here we perform a blow-up analysis that shows that the failure of global well-posedness and scattering implies either the existence of a soliton with at most the same energy bubbling off, or the existence of a non-trivial self-similar solution. The proof is completed by showing that the latter solutions do not exist.

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      mrclass = {35J15 (35B60 35R05)},
      mrnumber = {1809741},
      mrreviewer = {Florin Iacob},
      url = {https://doi.org/c6wrkd},
      zblnumber = {1033.35025},
      }
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    @ARTICLE{KST2,
      author = {Krieger, Joachim and Schlag, Wilhelm and Tataru, D.},
      title = {Renormalization and blow up for the critical {Y}ang-{M}ills problem},
      journal = {Adv. Math.},
      fjournal = {Advances in Mathematics},
      volume = {221},
      year = {2009},
      number = {5},
      pages = {1445--1521},
      issn = {0001-8708},
      mrclass = {58E15},
      mrnumber = {2522426},
      mrreviewer = {Michael Ruzhansky},
      doi = {10.1016/j.aim.2009.02.017},
      url = {https://doi.org/10.1016/j.aim.2009.02.017},
      zblnumber = {1183.35203},
      }
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    @ARTICLE{KL,
      author = {Krieger, Joachim and Lührmann, Jonas},
      title = {Concentration compactness for the critical {M}axwell-{K}lein-{G}ordon equation},
      journal = {Ann. PDE},
      fjournal = {Annals of PDE. Journal Dedicated to the Analysis of Problems from Physical Sciences},
      volume = {1},
      year = {2015},
      pages={208 pp.},
      number = {1},
      issn = {2524-5317},
      mrclass = {35Q60 (35B45 35B65 35P25)},
      mrnumber = {3479062},
      doi = {10.1007/s40818-015-0004-y},
      url = {https://doi.org/10.1007/s40818-015-0004-y},
      zblnumber = {1406.35181},
      }
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    @BOOK{KriSch,
      author = {Krieger, Joachim and Schlag, Wilhelm},
      title = {Concentration Compactness for Critical Wave Maps},
      series = {EMS Monogr. Math.},
      publisher = {European Math. Soc. (EMS), Zürich},
      year = {2012},
      pages = {vi+484},
      isbn = {978-3-03719-106-4},
      mrclass = {58J45 (35L52 58E20)},
      mrnumber = {2895939},
      mrreviewer = {Terence Tao},
      doi = {10.4171/106},
      url = {https://doi.org/10.4171/106},
      zblnumber = {1387.35006},
      }
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    @BOOK{KS,
      author = {Krieger, Joachim and Sterbenz, Jacob},
      title = {Global Regularity for the {Y}ang-{M}ills Equations on High Dimensional {M}inkowski Space},
      series = {Mem. Amer. Math. Soc.},
      publisher={Amer. Math. Soc., Providence, RI},
      volume = {223},
      year = {2013},
      number = {1047},
      pages = {vi+99},
      issn = {0065-9266},
      isbn = {978-0-8218-4489-2},
      mrclass = {35Q41 (35B65 35L70 70S15)},
      mrnumber = {3087010},
      mrreviewer = {Thierry Cazenave},
      doi = {10.1090/S0065-9266-2012-00566-1},
      url = {https://doi.org/10.1090/S0065-9266-2012-00566-1},
      zblnumber = {1304.35005},
      }
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    @ARTICLE{KST,
      author = {Krieger, Joachim and Sterbenz, Jacob and Tataru, Daniel},
      title = {Global well-posedness for the {M}axwell-{K}lein-{G}ordon equation in {$4+1$} dimensions: small energy},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {164},
      year = {2015},
      number = {6},
      pages = {973--1040},
      issn = {0012-7094},
      mrclass = {35L70 (35B30 70S15)},
      mrnumber = {3336839},
      mrreviewer = {N. Duruk Mutluba\c{s}},
      doi = {10.1215/00127094-2885982},
      url = {https://doi.org/10.1215/00127094-2885982},
      zblnumber = {1329.35209},
      }
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    @ARTICLE{KT,
      author = {Krieger, Joachim and Tataru, Daniel},
      title = {Global well-posedness for the {Y}ang-{M}ills equation in {$4+1$} dimensions. {S}mall energy},
      journal = {Ann. of Math. (2)},
      fjournal = {Annals of Mathematics. Second Series},
      volume = {185},
      year = {2017},
      number = {3},
      pages = {831--893},
      issn = {0003-486X},
      mrclass = {35Q40 (35B30 58E15 58J45 81T13)},
      mrnumber = {3664812},
      mrreviewer = {César R. de Oliveira},
      doi = {10.4007/annals.2017.185.3.3},
      url = {https://doi.org/10.4007/annals.2017.185.3.3},
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      }
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      author = {Lawrie, Andrew and Oh, Sung-Jin},
      title = {A refined threshold theorem for {$(1+2)$}-dimensional wave maps into surfaces},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {342},
      year = {2016},
      number = {3},
      pages = {989--999},
      issn = {0010-3616},
      mrclass = {58E20},
      mrnumber = {3465437},
      mrreviewer = {Yuan-Jen Chiang},
      doi = {10.1007/s00220-015-2513-7},
      url = {https://doi.org/10.1007/s00220-015-2513-7},
      zblnumber = {1336.58017},
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    @ARTICLE{Oh1,
      author = {Oh, Sung-Jin},
      title = {Gauge choice for the {Y}ang-{M}ills equations using the {Y}ang-{M}ills heat flow and local well-posedness in {$H^1$}},
      journal = {J. Hyperbolic Differ. Equ.},
      fjournal = {Journal of Hyperbolic Differential Equations},
      volume = {11},
      year = {2014},
      number = {1},
      pages = {1--108},
      issn = {0219-8916},
      mrclass = {35L71 (35B30 58E15 81T13)},
      mrnumber = {3190112},
      mrreviewer = {Yisong Yang},
      doi = {10.1142/S0219891614500015},
      url = {https://doi.org/10.1142/S0219891614500015},
      zblnumber = {1295.35328},
      }
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    @ARTICLE{Oh2,
      author = {Oh, Sung-Jin},
      title = {Finite energy global well-posedness of the {Y}ang-{M}ills equations on {$\Bbb{R}^{1+3}$}: an approach using the {Y}ang-{M}ills heat flow},
      journal = {Duke Math. J.},
      fjournal = {Duke Mathematical Journal},
      volume = {164},
      year = {2015},
      number = {9},
      pages = {1669--1732},
      issn = {0012-7094},
      mrclass = {35R01 (35B30 35Q40 70S15)},
      mrnumber = {3357182},
      mrreviewer = {Ralf Hofmann},
      doi = {10.1215/00127094-3119953},
      url = {https://doi.org/10.1215/00127094-3119953},
      zblnumber = {1325.35180},
      }
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    @ARTICLE{OT3,
      author = {Oh, Sung-Jin and Tataru, Daniel},
      title = {Global well-posedness and scattering of the {$(4+1)$}-dimensional {M}axwell-{K}lein-{G}ordon equation},
      journal = {Invent. Math.},
      fjournal = {Inventiones Mathematicae},
      volume = {205},
      year = {2016},
      number = {3},
      pages = {781--877},
      issn = {0020-9910},
      mrclass = {35Q60 (35B30 35B44 35C06 35P25 81U10)},
      mrnumber = {3539926},
      mrreviewer = {Gaetano Siciliano},
      doi = {10.1007/s00222-016-0646-8},
      url = {https://doi.org/10.1007/s00222-016-0646-8},
      zblnumber = {1364.35198},
      }
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    @ARTICLE{OT1,
      author = {Oh, Sung-Jin and Tataru, Daniel},
      title = {Local well-posedness of the {$(4 + 1)$}-dimensional {M}axwell--{K}lein--{G}ordon equation at energy regularity},
      journal = {Ann. PDE},
      fjournal = {Annals of PDE Journal Dedicated to the Analysis of Problems from Physical Sciences},
      volume = {2},
      year = {2016},
      number = {1},
      pages = {Art. 2, 70},
      issn = {2524-5317},
      mrclass = {35Q60 (35B30 35B65 35Q61)},
      mrnumber = {3462105},
      mrreviewer = {Luc Paquet},
      doi = {10.1007/s40818-016-0006-4},
      url = {https://doi.org/10.1007/s40818-016-0006-4},
      zblnumber = {1402.35273},
      }
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    @ARTICLE{OT2,
      author = {Oh, Sung-Jin and Tataru, Daniel},
      title = {Energy dispersed solutions for the {$(4+1)$}-dimensional {M}axwell-{K}lein-{G}ordon equation},
      journal = {Amer. J. Math.},
      fjournal = {American Journal of Mathematics},
      volume = {140},
      year = {2018},
      number = {1},
      pages = {1--82},
      issn = {0002-9327},
      mrclass = {35Q61 (35B30)},
      mrnumber = {3749190},
      mrreviewer = {Nikolaos L. Tsitsas},
      doi = {10.1353/ajm.2018.0000},
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      }
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    @MISC{OTYM1,
      author = {Oh, Sung-Jin and Tataru, Daniel},
      title = {The {Y}ang--{M}ills heat flow and the caloric gauge},
      year = {2017},
      arxiv = {1709.08599},
      zblnumber = {},
      }
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    @ARTICLE{OTYM2,
      author = {Oh, Sung-Jin and Tataru, Daniel},
      title = {The hyperbolic {Y}ang--{M}ills equation in the caloric gauge: local well-posedness and control of energy-dispersed solutions},
      journal = {Pure Appl. Anal.},
      fjournal = {Pure and Applied Analysis},
      volume = {2},
      year = {2020},
      number = {2},
      pages = {233--384},
      issn = {2578-5885},
      mrclass = {35L70 (70S15)},
      mrnumber = {4113787},
      mrreviewer = {Dongbing Zha},
      doi = {10.2140/paa.2020.2.233},
      url = {https://doi.org/10.2140/paa.2020.2.233},
      zblnumber = {1446.35150},
      }
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    @MISC{OTYM2.5,
      author = {Oh, Sung-Jin and Tataru, Daniel},
      title = {The hyperbolic {Y}ang-{M}ills equation for connections in an arbitrary topological class},
      year = {2017},
      arxiv = {1709.08604},
      zblnumber = {},
      }
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    @ARTICLE{OTYM0,
      author = {Oh, Sung-Jin and Tataru, Daniel},
      title = {The threshold theorem for the {$(4+1)$}-dimensional {Y}ang-{M}ills equation: an overview of the proof},
      journal = {Bull. Amer. Math. Soc. (N.S.)},
      fjournal = {Amer. Math. Soc.. Bulletin. New Series},
      volume = {56},
      year = {2019},
      number = {2},
      pages = {171--210},
      issn = {0273-0979},
      mrclass = {81T13 (35Q61 70S15)},
      mrnumber = {3923343},
      doi = {10.1090/bull/1640},
      url = {https://doi.org/10.1090/bull/1640},
      zblnumber = {1420.35273},
      }
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    @ARTICLE{PetRiv,
      author = {Petrache, Mircea and Rivière, Tristan},
      title = {Global gauges and global extensions in optimal spaces},
      journal = {Anal. PDE},
      fjournal = {Analysis \& PDE},
      volume = {7},
      year = {2014},
      number = {8},
      pages = {1851--1899},
      issn = {2157-5045},
      mrclass = {46E35 (35B45 35J57 58J05 70S15)},
      mrnumber = {3318742},
      mrreviewer = {Yoichi Miyazaki},
      doi = {10.2140/apde.2014.7.1851},
      url = {https://doi.org/10.2140/apde.2014.7.1851},
      zblnumber = {1328.46034},
      }
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    @ARTICLE{MR2929728,
      author = {Raphaël, Pierre and Rodnianski, Igor},
      title = {Stable blow up dynamics for the critical co-rotational wave maps and equivariant {Y}ang-{M}ills problems},
      journal = {Publ. Math. Inst. Hautes \'{E}tudes Sci.},
      fjournal = {Publ. Mathématiques. Institut de Hautes \'{E}tudes Scientifiques},
      volume = {115},
      year = {2012},
      pages = {1--122},
      issn = {0073-8301},
      mrclass = {58E20 (35A20 35B44 35L70 58E15 81T13)},
      mrnumber = {2929728},
      mrreviewer = {Andreas Gastel},
      doi = {10.1007/s10240-011-0037-z},
      url = {https://doi.org/10.1007/s10240-011-0037-z},
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      }
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    @ARTICLE{RT,
      author = {Rodnianski, Igor and Tao, Terence},
      title = {Global regularity for the {M}axwell-{K}lein-{G}ordon equation with small critical {S}obolev norm in high dimensions},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {251},
      year = {2004},
      number = {2},
      pages = {377--426},
      issn = {0010-3616},
      mrclass = {35Q60 (35L15 81Q05 81T13)},
      mrnumber = {2100060},
      mrreviewer = {Tohru Ozawa},
      doi = {10.1007/s00220-004-1152-1},
      url = {https://doi.org/10.1007/s00220-004-1152-1},
      zblnumber = {1106.35073},
      }
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    @ARTICLE{ST1,
      author = {Sterbenz, Jacob and Tataru, Daniel},
      title = {Energy dispersed large data wave maps in {$2+1$} dimensions},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {298},
      year = {2010},
      number = {1},
      pages = {139--230},
      issn = {0010-3616},
      mrclass = {58J45 (35L60)},
      mrnumber = {2657817},
      mrreviewer = {Michael Ruzhansky},
      doi = {10.1007/s00220-010-1061-4},
      url = {https://doi.org/10.1007/s00220-010-1061-4},
      zblnumber = {1218.35129},
      }
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      author = {Sterbenz, Jacob and Tataru, Daniel},
      title = {Regularity of wave-maps in dimension {$2+1$}},
      journal = {Comm. Math. Phys.},
      fjournal = {Communications in Mathematical Physics},
      volume = {298},
      year = {2010},
      number = {1},
      pages = {231--264},
      issn = {0010-3616},
      mrclass = {58E20},
      mrnumber = {2657818},
      mrreviewer = {Michael Ruzhansky},
      doi = {10.1007/s00220-010-1062-3},
      url = {https://doi.org/10.1007/s00220-010-1062-3},
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      }
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    @ARTICLE{Tao2,
      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},
      mrclass = {58J45 (35B60 35B65 35L15 58J47)},
      mrnumber = {1869874},
      mrreviewer = {Joachim Krieger},
      doi = {10.1007/PL00005588},
      url = {https://doi.org/10.1007/PL00005588},
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    @INCOLLECTION{Tao-caloric,
      author = {Tao, Terence},
      title = {Geometric renormalization of large energy wave maps},
      booktitle = {Journées {``\'{E}}quations aux {D}{é}riv{é}es {P}artielles''},
      pages = {Exp. No. XI, 32},
      publisher = {\'{E}cole Polytech., Palaiseau},
      year = {2004},
      mrclass = {58J45 (35L05 53C44)},
      mrnumber = {2135366},
      doi = {10.5802/jedp.11},
      url = {https://doi.org/10.5802/jedp.11},
      zblnumber = {1087.58019},
      }
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    @MISC{Tao:2008wn,
      author = {Tao, Terence},
      title = {Global regularity of wave maps {III}. {L}arge energy from {${\mathbf R}^{1+2}$} to hyperbolic spaces},
      year = {2008},
      arxiv = {0805.4666},
      zblnumber = {},
      }
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    @MISC{Tao:2008tz,
      author = {Tao, Terence},
      title = {Global regularity of wave maps {IV}. {A}bsence of stationary or self-similar solutions in the energy class},
      year = {2008},
      arxiv = {0806.3592},
      zblnumber = {},
      }
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    @MISC{Tao:2008wo,
      author = {Tao, Terence},
      title = {Global regularity of wave maps {V}. {L}arge data local wellposedness and perturbation theory in the energy class},
      year = {2008},
      arxiv = {0808.0368},
      zblnumber = {},
      }
  • [Tao:2009ta] T. Tao, Global regularity of wave maps VI. Abstract theory of minimal-energy blowup solutions, 2009.
    @MISC{Tao:2009ta,
      author = {Tao, Terence},
      title = {Global regularity of wave maps {VI}. {A}bstract theory of minimal-energy blowup solutions},
      year = {2009},
      arxiv = {0906.2833},
      zblnumber = {},
      }
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    @MISC{Tao:2009ua,
      author = {Tao, Terence},
      title = {Global regularity of wave maps {VII}. {C}ontrol of delocalised or dispersed solutions},
      year = {2009},
      arxiv = {0908.0776},
      zblnumber = {},
      }
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    @ARTICLE{Tat,
      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},
      mrclass = {58J45 (35L70 35P25)},
      mrnumber = {1827277},
      mrreviewer = {Kenji Nakanishi},
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Authors

Sung-Jin Oh

Department of Mathematics, University of California, Berkeley, Berkeley, CA

Daniel Tataru

Department of Mathematics, University of California, Berkeley, Berkeley, CA