Abstract
We consider the hyperbolic Yang-Mills equation on the Minkowski space $\mathbb{R}^{4+1}$. Our main result asserts that this problem is globally well-posed for all initial data whose energy is sufficiently small. This solves a longstanding open problem.
-
[BH1] I. Bejenaru and S. Herr, On global well-posedness and scattering for the massive Dirac-Klein-Gordon system, 2014.
@MISC{BH1,
author = {Bejenaru, Ioan and Herr, Sebastian},
arxiv = {1409.1778},
title = {On global well-posedness and scattering for the massive {D}irac-{K}lein-{G}ordon system},
year = {2014},
} -
[ChoChr]
Y. Choquet-Bruhat and D. Christodoulou, "Existence of global solutions of the Yang-Mills, Higgs and spinor field equations in $3+1$ dimensions," Ann. Sci. École Norm. Sup., vol. 14, iss. 4, pp. 481-506 (1982), 1981.
@ARTICLE{ChoChr, mrkey = {0654209},
number = {4},
issn = {0012-9593},
author = {Choquet-Bruhat, Yvonne and Christodoulou, Demetrios},
mrclass = {81E10 (58D25 58G16 83C05)},
url = {http://www.numdam.org/item?id=ASENS_1981_4_14_4_481_0},
journal = {Ann. Sci. École Norm. Sup.},
zblnumber = {0499.35076},
volume = {14},
mrnumber = {0654209},
fjournal = {Annales Scientifiques de l'École Normale Supérieure. Quatrième Série},
mrreviewer = {Philippe-A. Dionne},
coden = {ASENAH},
title = {Existence of global solutions of the {Y}ang-{M}ills, {H}iggs and spinor field equations in {$3+1$} dimensions},
year = {1981},
pages = {481--506 (1982)},
} -
[MR649158]
D. M. Eardley and V. Moncrief, "The global existence of Yang-Mills-Higgs fields in $4$-dimensional Minkowski space. I. Local existence and smoothness properties," Comm. Math. Phys., vol. 83, iss. 2, pp. 171-191, 1982.
@ARTICLE{MR649158, mrkey = {0649158},
number = {2},
issn = {0010-3616},
author = {Eardley, Douglas M. and Moncrief, Vincent},
mrclass = {35Q20 (58D25 58E20 81E10)},
doi = {10.1007/BF01976040},
journal = {Comm. Math. Phys.},
zblnumber = {0496.35061},
volume = {83},
mrnumber = {0649158},
fjournal = {Communications in Mathematical Physics},
mrreviewer = {R. Glassey},
coden = {CMPHAY},
title = {The global existence of {Y}ang-{M}ills-{H}iggs fields in {$4$}-dimensional {M}inkowski space. {I}. {L}ocal existence and smoothness properties},
year = {1982},
pages = {171--191},
} -
[MR649159]
D. M. Eardley and V. Moncrief, "The global existence of Yang-Mills-Higgs fields in $4$-dimensional Minkowski space. II. Completion of proof," Comm. Math. Phys., vol. 83, iss. 2, pp. 193-212, 1982.
@ARTICLE{MR649159, mrkey = {0649159},
number = {2},
issn = {0010-3616},
author = {Eardley, Douglas M. and Moncrief, Vincent},
mrclass = {35Q20 (58D25 58E20 81E10)},
doi = {10.1007/BF01976041},
journal = {Comm. Math. Phys.},
zblnumber = {0496.35062},
volume = {83},
mrnumber = {0649159},
fjournal = {Communications in Mathematical Physics},
mrreviewer = {R. Glassey},
coden = {CMPHAY},
title = {The global existence of {Y}ang-{M}ills-{H}iggs fields in {$4$}-dimensional {M}inkowski space. {II}. {C}ompletion of proof},
year = {1982},
pages = {193--212},
} -
[Klainerman:1995]
S. Klainerman and M. Machedon, "Finite energy solutions of the Yang-Mills equations in $\Bbb R^{3+1}$," Ann. of Math., vol. 142, iss. 1, pp. 39-119, 1995.
@ARTICLE{Klainerman:1995, mrkey = {1338675},
number = {1},
issn = {0003-486X},
author = {Klainerman, S. and Machedon, M.},
mrclass = {58G16 (35L70 58E15)},
doi = {10.2307/2118611},
journal = {Ann. of Math.},
zblnumber = {0827.53056},
volume = {142},
mrnumber = {1338675},
fjournal = {Annals of Mathematics. Second Series},
mrreviewer = {David M. A. Stuart},
coden = {ANMAAH},
title = {Finite energy solutions of the {Y}ang-{M}ills equations in {$\bold R\sp {3+1}$}},
year = {1995},
pages = {39--119},
} -
[Klainerman:1999do]
S. Klainerman and D. Tataru, "On the optimal local regularity for Yang-Mills equations in ${\bf R}^{4+1}$," J. Amer. Math. Soc., vol. 12, iss. 1, pp. 93-116, 1999.
@ARTICLE{Klainerman:1999do, mrkey = {1626261},
number = {1},
issn = {0894-0347},
author = {Klainerman, Sergiu and Tataru, Daniel},
mrclass = {58J45 (35L70 35Q75)},
doi = {10.1090/S0894-0347-99-00282-9},
journal = {J. Amer. Math. Soc.},
zblnumber = {0924.58010},
volume = {12},
mrnumber = {1626261},
fjournal = {Journal of the American Mathematical Society},
mrreviewer = {Thierry Cazenave},
title = {On the optimal local regularity for {Y}ang-{M}ills equations in {${\bf R}\sp {4+1}$}},
year = {1999},
pages = {93--116},
} -
[KL]
J. Krieger and J. Lührmann, "Concentration compactness for the critical Maxwell-Klein-Gordon equation," Ann. PDE, vol. 1, iss. 1, 2015.
@ARTICLE{KL, mrkey = {3479062},
number = {1},
issn = {2199-2576},
author = {Krieger, Joachim and L{ü}hrmann, Jonas},
mrclass = {35Q60 (35B45 35B65 35P25)},
journal = {Ann. PDE},
volume = {1},
mrnumber = {3479062},
fjournal = {Annals of PDE. Journal Dedicated to the Analysis of Problems from Physical Sciences},
title = {Concentration compactness for the critical {M}axwell-{K}lein-{G}ordon equation},
year = {2015},
note = {Art. 5, 208 pp.},
doi = {10.1007/s40818-015-0004-y},
} -
[Krieger:2009uy]
J. Krieger and W. Schlag, Concentration Compactness for Critical Wave Maps, European Mathematical Society (EMS), Zürich, 2012.
@BOOK{Krieger:2009uy, mrkey = {2895939},
author = {Krieger, Joachim and Schlag, Wilhelm},
mrclass = {58J45 (35L52 58E20)},
series = {EMS Monogr. Math.},
isbn = {978-3-03719-106-4},
publisher = {European Mathematical Society (EMS), Zürich},
doi = {10.4171/106},
zblnumber = {06004782},
mrnumber = {2895939},
mrreviewer = {Terence Tao},
title = {Concentration Compactness for Critical Wave Maps},
year = {2012},
pages = {vi+484},
} -
[Krieger:2005wh]
J. Krieger and J. Sterbenz, "Global regularity for the Yang-Mills equations on high dimensional Minkowski space," Mem. Amer. Math. Soc., vol. 223, iss. 1047, p. vi, 2013.
@ARTICLE{Krieger:2005wh, mrkey = {3087010},
number = {1047},
issn = {0065-9266},
author = {Krieger, Joachim and Sterbenz, Jacob},
mrclass = {35Q41 (35B65 35L70 70S15)},
isbn = {978-0-8218-4489-2},
doi = {10.1090/S0065-9266-2012-00566-1},
journal = {Mem. Amer. Math. Soc.},
zblnumber = {1304.35005},
volume = {223},
mrnumber = {3087010},
fjournal = {Memoirs of the American Mathematical Society},
mrreviewer = {Thierry Cazenave},
title = {Global regularity for the {Y}ang-{M}ills equations on high dimensional {M}inkowski space},
year = {2013},
pages = {vi+99},
} -
[MKG]
J. Krieger, J. Sterbenz, and D. Tataru, "Global well-posedness for the Maxwell-Klein-Gordon equation in $4+1$ dimensions: small energy," Duke Math. J., vol. 164, iss. 6, pp. 973-1040, 2015.
@ARTICLE{MKG, mrkey = {3336839},
number = {6},
issn = {0012-7094},
author = {Krieger, Joachim and Sterbenz, Jacob and Tataru, Daniel},
mrclass = {35L70 (35B30 70S15)},
doi = {10.1215/00127094-2885982},
journal = {Duke Math. J.},
zblnumber = {1329.35209},
volume = {164},
mrnumber = {3336839},
fjournal = {Duke Mathematical Journal},
mrreviewer = {N. Duruk Mutluba{\c{s}}},
title = {Global well-posedness for the {M}axwell-{K}lein-{G}ordon equation in {$4+1$} dimensions: small energy},
year = {2015},
pages = {973--1040},
} -
[Machedon:2004cu]
M. Machedon and J. Sterbenz, "Almost optimal local well-posedness for the $(3+1)$-dimensional Maxwell-Klein-Gordon equations," J. Amer. Math. Soc., vol. 17, iss. 2, pp. 297-359, 2004.
@ARTICLE{Machedon:2004cu, mrkey = {2051613},
number = {2},
issn = {0894-0347},
author = {Machedon, Matei and Sterbenz, Jacob},
mrclass = {35Q60 (35B30 35L70)},
doi = {10.1090/S0894-0347-03-00445-4},
journal = {J. Amer. Math. Soc.},
zblnumber = {1048.35115},
volume = {17},
mrnumber = {2051613},
fjournal = {Journal of the American Mathematical Society},
mrreviewer = {Nikos M. Stavrakakis},
title = {Almost optimal local well-posedness for the {$(3+1)$}-dimensional {M}axwell-{K}lein-{G}ordon equations},
year = {2004},
pages = {297--359},
} -
[Oh12]
S. Oh, "Gauge choice for the Yang-Mills equations using the Yang-Mills heat flow and local well-posedness in $H^1$," J. Hyperbolic Differ. Equ., vol. 11, iss. 1, pp. 1-108, 2014.
@ARTICLE{Oh12, mrkey = {3190112},
number = {1},
issn = {0219-8916},
author = {Oh, Sung-Jin},
mrclass = {35L71 (35B30 58E15 81T13)},
doi = {10.1142/S0219891614500015},
journal = {J. Hyperbolic Differ. Equ.},
zblnumber = {1295.35328},
volume = {11},
mrnumber = {3190112},
fjournal = {Journal of Hyperbolic Differential Equations},
mrreviewer = {Yisong Yang},
title = {Gauge choice for the {Y}ang-{M}ills equations using the {Y}ang-{M}ills heat flow and local well-posedness in {$H\sp 1$}},
year = {2014},
pages = {1--108},
} -
[OT1]
S. Oh and D. Tataru, "Local well-posedness of the $(4 + 1)$-dimensional Maxwell-Klein-Gordon equation at energy regularity," Ann. PDE, vol. 2, iss. 1, 2016.
@ARTICLE{OT1, mrkey = {3462105},
number = {1},
issn = {2199-2576},
author = {Oh, Sung-Jin and Tataru, Daniel},
mrclass = {35Q60 (35B30 35B65 35Q61)},
doi = {10.1007/s40818-016-0006-4},
journal = {Ann. PDE},
volume = {2},
mrnumber = {3462105},
fjournal = {Annals of PDE. Journal Dedicated to the Analysis of Problems from Physical Sciences},
title = {Local well-posedness of the {$(4 + 1)$}-dimensional {M}axwell-{K}lein-{G}ordon equation at energy regularity},
year = {2016},
note = {Art. 2, 70 pp.},
} -
[OT2] S. Oh and D. Tataru, Energy dispersed solutions for the $(4+1)$-dimensional Maxwell-Klein-Gordon equation at energy regularity, 2015.
@MISC{OT2,
author = {Oh, Sung-Jin and Tataru, Daniel},
note = {preprint},
title = {Energy dispersed solutions for the $(4+1)$-dimensional {M}axwell-{K}lein-{G}ordon equation at energy regularity},
year = {2015},
arxiv = {1503.01561},
} -
[OT3]
S. Oh and D. Tataru, "Global well-posedness and scattering of the $(4+1)$-dimensional Maxwell-Klein-Gordon equation," Invent. Math., vol. 205, iss. 3, pp. 781-877, 2016.
@ARTICLE{OT3, mrkey = {3539926},
number = {3},
issn = {0020-9910},
author = {Oh, Sung-Jin and Tataru, Daniel},
mrclass = {35Q60 (35B30 35P25 81U10)},
doi = {10.1007/s00222-016-0646-8},
journal = {Invent. Math.},
zblnumber = {06641858},
volume = {205},
mrnumber = {3539926},
fjournal = {Inventiones Mathematicae},
coden = {INVMBH},
title = {Global well-posedness and scattering of the {$(4+1)$}-dimensional {M}axwell-{K}lein-{G}ordon equation},
year = {2016},
pages = {781--877},
} -
[MR2100060]
I. Rodnianski and T. Tao, "Global regularity for the Maxwell-Klein-Gordon equation with small critical Sobolev norm in high dimensions," Comm. Math. Phys., vol. 251, iss. 2, pp. 377-426, 2004.
@ARTICLE{MR2100060, mrkey = {2100060},
number = {2},
issn = {0010-3616},
author = {Rodnianski, Igor and Tao, Terence},
mrclass = {35Q60 (35L15 81Q05 81T13)},
doi = {10.1007/s00220-004-1152-1},
journal = {Comm. Math. Phys.},
zblnumber = {1106.35073},
volume = {251},
mrnumber = {2100060},
fjournal = {Communications in Mathematical Physics},
mrreviewer = {Tohru Ozawa},
coden = {CMPHAY},
title = {Global regularity for the {M}axwell-{K}lein-{G}ordon equation with small critical {S}obolev norm in high dimensions},
year = {2004},
pages = {377--426},
} -
[2013arXiv1309.1977S]
S. Selberg and A. Tesfahun, "Null structure and local well-posedness in the energy class for the Yang-Mills equations in Lorenz gauge," J. Eur. Math. Soc. $($JEMS$)$, vol. 18, iss. 8, pp. 1729-1752, 2016.
@ARTICLE{2013arXiv1309.1977S, mrkey = {3519539},
number = {8},
issn = {1435-9855},
author = {Selberg, Sigmund and Tesfahun, Achenef},
mrclass = {35Q40 (35B30 35L70 70S15)},
doi = {10.4171/JEMS/627},
journal = {J. Eur. Math. Soc. $($JEMS$)$},
zblnumber = {06617139},
volume = {18},
mrnumber = {3519539},
fjournal = {Journal of the European Mathematical Society (JEMS)},
title = {Null structure and local well-posedness in the energy class for the {Y}ang-{M}ills equations in {L}orenz gauge},
year = {2016},
pages = {1729--1752},
} -
[MR2657817]
J. Sterbenz and D. Tataru, "Energy dispersed large data wave maps in $2+1$ dimensions," Comm. Math. Phys., vol. 298, iss. 1, pp. 139-230, 2010.
@ARTICLE{MR2657817, mrkey = {2657817},
number = {1},
issn = {0010-3616},
author = {Sterbenz, Jacob and Tataru, Daniel},
mrclass = {58J45 (35L60)},
doi = {10.1007/s00220-010-1061-4},
journal = {Comm. Math. Phys.},
zblnumber = {1218.35129},
volume = {298},
mrnumber = {2657817},
fjournal = {Communications in Mathematical Physics},
mrreviewer = {Michael Ruzhansky},
coden = {CMPHAY},
title = {Energy dispersed large data wave maps in {$2+1$} dimensions},
year = {2010},
pages = {139--230},
} -
[MR2657818]
J. Sterbenz and D. Tataru, "Regularity of wave-maps in dimension $2+1$," Comm. Math. Phys., vol. 298, iss. 1, pp. 231-264, 2010.
@ARTICLE{MR2657818, mrkey = {2657818},
number = {1},
issn = {0010-3616},
author = {Sterbenz, Jacob and Tataru, Daniel},
mrclass = {58E20},
doi = {10.1007/s00220-010-1062-3},
journal = {Comm. Math. Phys.},
zblnumber = {1218.35057},
volume = {298},
mrnumber = {2657818},
fjournal = {Communications in Mathematical Physics},
mrreviewer = {Michael Ruzhansky},
coden = {CMPHAY},
title = {Regularity of wave-maps in dimension {$2+1$}},
year = {2010},
pages = {231--264},
} -
[Tao:2001gb]
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:2001gb, mrkey = {1869874},
number = {2},
issn = {0010-3616},
author = {Tao, Terence},
mrclass = {58J45 (35B60 35B65 35L15 58J47)},
doi = {10.1007/PL00005588},
journal = {Comm. Math. Phys.},
zblnumber = {1020.35046},
volume = {224},
mrnumber = {1869874},
fjournal = {Communications in Mathematical Physics},
mrreviewer = {Joachim Krieger},
coden = {CMPHAY},
title = {Global regularity of wave maps. {II}. {S}mall energy in two dimensions},
year = {2001},
pages = {443--544},
arxiv = {0011173},
} -
[Tao:2008wn] T. Tao, Global regularity of wave maps III. Large energy from $\mathrm{R}^{1+2}$ to hyperbolic spaces, 2008.
@MISC{Tao:2008wn,
author = {Tao, Terence},
title = {Global regularity of wave maps {III. L}arge energy from {$\mathrm{R}^{1+2}$} to hyperbolic spaces},
year = {2008},
arxiv = {0805.4666},
} -
[Tao:2008tz] T. Tao, Global regularity of wave maps IV. Absence of stationary or self-similar solutions in the energy class, 2008.
@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},
} -
[Tao:2008wo] T. Tao, Global regularity of wave maps V. Large data local wellposedness and perturbation theory in the energy class, 2008.
@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},
} -
[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},
} -
[Tao:2009ua] T. Tao, Global regularity of wave maps VII. Control of delocalised or dispersed solutions, 2009.
@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},
} -
[MR1827277]
D. Tataru, "On global existence and scattering for the wave maps equation," Amer. J. Math., vol. 123, iss. 1, pp. 37-77, 2001.
@ARTICLE{MR1827277, mrkey = {1827277},
number = {1},
issn = {0002-9327},
author = {Tataru, Daniel},
mrclass = {58J45 (35L70 35P25)},
journal = {Amer. J. Math.},
zblnumber = {0979.35100},
volume = {123},
mrnumber = {1827277},
fjournal = {American Journal of Mathematics},
mrreviewer = {Kenji Nakanishi},
coden = {AJMAAN},
title = {On global existence and scattering for the wave maps equation},
year = {2001},
pages = {37--77},
doi = {10.1353/ajm.2001.0005},
} -
[wm]
D. Tataru, "Rough solutions for the wave maps equation," Amer. J. Math., vol. 127, iss. 2, pp. 293-377, 2005.
@ARTICLE{wm, mrkey = {2130618},
number = {2},
issn = {0002-9327},
author = {Tataru, Daniel},
mrclass = {58J45 (35B30 35L15)},
journal = {Amer. J. Math.},
zblnumber = {1330.58021},
volume = {127},
mrnumber = {2130618},
fjournal = {American Journal of Mathematics},
mrreviewer = {Terence C. Tao},
coden = {AJMAAN},
title = {Rough solutions for the wave maps equation},
year = {2005},
pages = {293--377},
doi = {10.1353/ajm.2005.0014},
}