On the optimal local regularity for the YangMills equations in
Authors:
Sergiu Klainerman and Daniel Tataru
Journal:
J. Amer. Math. Soc. 12 (1999), 93116
MSC (1991):
Primary 58E15, 35B65, 35Q40
MathSciNet review:
1626261
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Additional Information
Abstract: The aim of the paper is to develop the Fourier Analysis techniques needed in the study of optimal wellposedness and global regularity properties of the YangMills equations in Minkowski spacetime , for the case of the critical dimension . We introduce new functional spaces and prove new bilinear estimates for solutions of the homogeneous wave equation, which can be viewed as generalizations of the wellknown StrichartzPecher inequalities.
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Machedon, Spacetime estimates for null forms and the local
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Machedon, Finite energy solutions of the YangMills equations in
𝐑³⁺¹, Ann. of Math. (2) 142
(1995), no. 1, 39–119. MR 1338675
(96i:58167), http://dx.doi.org/10.2307/2118611
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Klainerman and M.
Machedon, Smoothing estimates for null forms and applications,
Duke Math. J. 81 (1995), no. 1, 99–133 (1996).
A celebration of John F. Nash, Jr. MR 1381973
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S. Klainerman and M. Machedon, Estimates for null forms and the spaces , International Math. Research Notices 17 (1996), 853866. CMP 97:04
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Sergiu
Klainerman and Matei
Machedon, On the regularity properties of a model problem related
to wave maps, Duke Math. J. 87 (1997), no. 3,
553–589. MR 1446618
(98e:35118), http://dx.doi.org/10.1215/S0012709497087184
 [KM8]
S. Klainerman and M. Machedon, On the optimal local regularity for gauge field theories, Differential and Integral Equations 10 (1997), no. 6, 10191030. CMP 98:09
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S. Klainerman and S. Selberg, Remark on the optimal regularity for equations of Wave Maps type, Comm. P.D.E. 22 (1997), no. 56, 901918. CMP 97:13
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Robert
S. Strichartz, Restrictions of Fourier transforms to quadratic
surfaces and decay of solutions of wave equations, Duke Math. J.
44 (1977), no. 3, 705–714. MR 0512086
(58 #23577)
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Tataru, The 𝑋^{𝑠}_{𝜃} spaces and unique
continuation for solutions to the semilinear wave equation, Comm.
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 [B]
 J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, I, II, Geom. Funct. Analysis 3 (1993), 107156, 202262. MR 95d:35160a, MR 95d:35160b
 [Br]
 P. Brenner, On estimates for the wave equations, Math. Z. 145 (1975), 251254. MR 52:8658
 [GV]
 J. Ginibre, G. Velo, Generalized Strichartz inequality for the wave equation, J. Funct. Anal. 133 (1995), no. 1, 5068. MR 97a:46047
 [KT]
 M. Keel, T. Tao, Endpoints Strichartz Estimates, to appear in Amer. Jour. of Math.
 [KPV]
 C. Kenig, G. Ponce, L. Vega, The Cauchy problem for the KortewegDe Vries equation in Sobolev spaces of negative indices, Duke Math Journal 71, No. 1, pp. 121 (1994). MR 94g:35196
 [KM1]
 S. Klainerman and M. Machedon, Spacetime estimates for null forms and the local existence theorem, Comm. Pure Appl. Math 46 (1993), 12211268. MR 94h:35137
 [KM2]
 S. Klainerman and M. Machedon, On the MaxwellKleinGordon equation with finite energy, Duke Math. J. 74 (1994), no. 1, 1944. MR 95f:35210
 [KM3]
 S. Klainerman and M. Machedon, Finite energy solutions for the YangMills solutions in , Annals of Math. 142, 1995, 39119. MR 96i:58167
 [KM4]
 S. Klainerman and M. Machedon, Smoothing estimates for null forms and applications, Duke Math J. 81 (1995), 99103. MR 97h:35022
 [KM5]
 S. Klainerman and M. Machedon, with appendices by J. Bourgain and D. Tataru, Remark on Strichartz type inequalities, International Math. Research Notices, 1996, no. 5, 201220. MR 97g:46037
 [KM6]
 S. Klainerman and M. Machedon, Estimates for null forms and the spaces , International Math. Research Notices 17 (1996), 853866. CMP 97:04
 [KM7]
 S. Klainerman and M. Machedon, On the regularity properties of a model problem related to wave maps, Duke Math. Jour. 87 (1997), no. 3, 553589. MR 98e:35118
 [KM8]
 S. Klainerman and M. Machedon, On the optimal local regularity for gauge field theories, Differential and Integral Equations 10 (1997), no. 6, 10191030. CMP 98:09
 [KS]
 S. Klainerman and S. Selberg, Remark on the optimal regularity for equations of Wave Maps type, Comm. P.D.E. 22 (1997), no. 56, 901918. CMP 97:13
 [S1]
 R. S. Strichartz, Restrictions of Fourier transform to quadratic surfaces and decay of solutions of Wave Equations, Duke Math. J. 44 (1977), 705714. MR 58:23577
 [Ta]
 D. Tataru, On the spaces and unique continuation for semilinear hyperbolic equations, Comm. PDE 21 (1996), no. 56. MR 97i:35012
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Additional Information
Sergiu Klainerman
Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Daniel Tataru
Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544
DOI:
http://dx.doi.org/10.1090/S0894034799002829
PII:
S 08940347(99)002829
Keywords:
YangMills,
wellposedness,
regularity,
Strichartz
Received by editor(s):
April 1, 1997
Received by editor(s) in revised form:
March 3, 1998
Additional Notes:
The first author’s research was partially supported by NSF grant DMS9400258.
The second author’s research was partially supported by NSF grant DMS9622942 and by an Alfred P. Sloan fellowship.
Article copyright:
© Copyright 1999
American Mathematical Society
