Abstract
We give a sharp counterexample to local existence of low regularity solutions to Einstein equations in wave coordinates. We show that there are initial data in $H^2$ satisfying the wave coordinate condition such that there is no solution in $H^2$ to Einstein equations in wave coordinates for any positive time. This result is sharp since Klainerman-Rodnianski and Smith-Tataru proved existence for the same equations with slightly more regular initial data.
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@MISC{C,
author = {Czimek, S.},
ARXIV={1609.08814},
title = {Extensions of vacuum initial data in weak regularity},
year = {2016},
} -
[KM]
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AUTHOR = {Klainerman, S. and Machedon, M.},
TITLE = {Smoothing estimates for null forms and applications. {A} celebration of {J}ohn {F}. {N}ash, {J}r.},
JOURNAL = {Duke Math. J.},
FJOURNAL = {Duke Mathematical Journal},
VOLUME = {81},
YEAR = {1995},
NUMBER = {1},
PAGES = {99--133},
ISSN = {0012-7094},
CODEN = {DUMJAO},
MRCLASS = {35B65 (35L05 35L70 35Q53)},
MRNUMBER = {1381973},
MRREVIEWER = {Vladimir Georgiev},
DOI = {10.1215/S0012-7094-95-08109-5},
zblnumber = {0909.35094},
} -
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@ARTICLE{KR, mrkey = {2180400},
number = {3},
issn = {0003-486X},
author = {Klainerman, Sergiu and Rodnianski, Igor},
mrclass = {58J45 (35Q75)},
doi = {10.4007/annals.2005.161.1143},
journal = {Ann. of Math.},
zblnumber = {1089.83006},
volume = {161},
mrnumber = {2180400},
fjournal = {Annals of Mathematics. Second Series},
mrreviewer = {Lars {\AA}ke Andersson},
coden = {ANMAAH},
title = {Rough solutions of the {E}instein-vacuum equations},
year = {2005},
pages = {1143--1193},
} -
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@ARTICLE{KRS, mrkey = {3402797},
number = {1},
issn = {0020-9910},
author = {Klainerman, Sergiu and Rodnianski, Igor and Szeftel, Jeremie},
mrclass = {83C05 (35A01 35A09 35Q76 58J45)},
doi = {10.1007/s00222-014-0567-3},
journal = {Invent. Math.},
zblnumber = {1330.53089},
volume = {202},
mrnumber = {3402797},
fjournal = {Inventiones Mathematicae},
mrreviewer = {Piotr T. Chru{\'s}ciel},
title = {The bounded {$L\sp 2$} curvature conjecture},
year = {2015},
pages = {91--216},
} -
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zblnumber = {0855.35080},
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coden = {AJMAAN},
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@ARTICLE{L2, mrkey = {1666844},
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@ARTICLE{LR, mrkey = {2134337},
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@ARTICLE{ST, mrkey = {2178963},
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year = {2005},
pages = {291--366},
} -
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mrclass = {46.38 (26.00)},
series = {Princeton Math. Ser.},
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arxiv = {1408.3780},
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