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On the optimal local regularity for the Yang-Mills equations in $\mathbb{R}^{4+1}$

Author(s): Sergiu Klainerman; Daniel Tataru
Journal: J. Amer. Math. Soc. 12 (1999), 93-116.
MSC (1991): Primary 58E15, 35B65, 35Q40
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Abstract: The aim of the paper is to develop the Fourier Analysis techniques needed in the study of optimal well-posedness and global regularity properties of the Yang-Mills equations in Minkowski space-time $\mathbb{R}^{n+1}$, for the case of the critical dimension $n=4$. 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 well-known Strichartz-Pecher inequalities.

References:

[B]
J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to non-linear evolution equations, I, II, Geom. Funct. Analysis 3 (1993), 107-156, 202-262. MR 95d:35160a, MR 95d:35160b

[Br]
P. Brenner, On $L^p-L^{p'}$ estimates for the wave equations, Math. Z. 145 (1975), 251-254. MR 52:8658

[G-V]
J. Ginibre, G. Velo, Generalized Strichartz inequality for the wave equation, J. Funct. Anal. 133 (1995), no. 1, 50-68. MR 97a:46047
[K-T]
M. Keel, T. Tao, Endpoints Strichartz Estimates, to appear in Amer. Jour. of Math.

[K-P-V]
C. Kenig, G. Ponce, L. Vega, The Cauchy problem for the Korteweg-De Vries equation in Sobolev spaces of negative indices, Duke Math Journal 71, No. 1, pp. 1-21 (1994). MR 94g:35196

[K-M1]
S. Klainerman and M. Machedon, Space-time estimates for null forms and the local existence theorem, Comm. Pure Appl. Math 46 (1993), 1221-1268. MR 94h:35137

[K-M2]
S. Klainerman and M. Machedon, On the Maxwell-Klein-Gordon equation with finite energy, Duke Math. J. 74 (1994), no. 1, 19-44. MR 95f:35210
[K-M3]
S. Klainerman and M. Machedon, Finite energy solutions for the Yang-Mills solutions in $\mathbb R^{3+1}$, Annals of Math. 142, 1995, 39-119. MR 96i:58167

[K-M4]
S. Klainerman and M. Machedon, Smoothing estimates for null forms and applications, Duke Math J. 81 (1995), 99-103. MR 97h:35022

[K-M5]
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, 201-220. MR 97g:46037

[K-M6]
S. Klainerman and M. Machedon, Estimates for null forms and the spaces $H_{s, \delta}$, International Math. Research Notices 17 (1996), 853-866. CMP 97:04

[K-M7]
S. Klainerman and M. Machedon, On the regularity properties of a model problem related to wave maps, Duke Math. Jour. 87 (1997), no. 3, 553-589. MR 98e:35118
[K-M8]
S. Klainerman and M. Machedon, On the optimal local regularity for gauge field theories, Differential and Integral Equations 10 (1997), no. 6, 1019-1030. CMP 98:09

[K-S]
S. Klainerman and S. Selberg, Remark on the optimal regularity for equations of Wave Maps type, Comm. P.D.E. 22 (1997), no. 5-6, 901-918. 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), 705-714. MR 58:23577
[Ta]
D. Tataru, On the $X^s_\theta$ spaces and unique continuation for semilinear hyperbolic equations, Comm. PDE 21 (1996), no. 5-6. 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: 10.1090/S0894-0347-99-00282-9
PII: S 0894-0347(99)00282-9
Keywords: Yang-Mills, well-posedness, 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 DMS-9400258.
The second author's research was partially supported by NSF grant DMS-9622942 and by an Alfred P. Sloan fellowship.
Copyright of article: Copyright 1999, American Mathematical Society

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