Scattering for the YangMills equations
Author:
John C. Baez
Journal:
Trans. Amer. Math. Soc. 315 (1989), 823832
MSC:
Primary 58G30; Secondary 35P25, 47F05, 58G25, 81E13
MathSciNet review:
949897
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Abstract: We construct wave and scattering operators for the YangMills equations on Minkowski space, . Sufficiently regular solutions of the YangMills equations on are known to extend uniquely to solutions of the corresponding equations on the universal cover of its conformal compactification, . Moreover, the boundary of as embedded in is the union of "lightcones at future and past infinity", . We construct wave operators as continuous maps from a space of timezero Cauchy data for the YangMills equations to Hilbert spaces of Goursat data on . The scattering operator is then a homeomorphism such that , where is the linear isomorphism arising from a conformal transformation of mapping onto . The maps and are shown to be smooth in a certain sense.
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 S. M. Paneitz and I. E. Segal, Analysis in spacetime bundles. I, J. Funct. Anal. 47 (1982), 78142. MR 663834 (83k:22042)
 [2]
 Y. ChoquetBruhat and D. Christodoulou, Existence of global solutions the YangMills, Higgs, and spinor field equations in dimensions, Ann. Ecole. Norm. Sup. (4) 14 (1981), 481506. MR 654209 (84c:81041)
 [3]
 D. Eardley and V. Moncrief, The global existence of YangMillsHiggs fields on dimensional Minkowski space, Comm. Math. Phys. 83 (1982), 121191, 193212. MR 649159 (83e:35106b)
 [4]
 Y. ChoquetBruhat, S. M. Paneitz and I. E. Segal, The YangMills equations on the universal cosmos, J. Funct. Anal. 53 (1983), 112150. MR 722506 (85i:58111)
 [5]
 I. E. Segal, Reduction of scattering to an invariant finite displacement in an ambient spacetime, Proc. Nat. Acad. Sci. U.S.A. 81 (1984), 72667268. MR 768161 (86b:81090)
 [6]
 J. Baez, I. E. Segal and Z. Zhou, The global Goursat problem and scattering for nonlinear wave equations (to appear).
 [7]
 J. Baez, Scattering and the geometry of the solution manifold of , J. Funct. Anal. (to appear).
 [8]
 D. Bleecker, Gauge theory and variational principles, AddisonWesley, Reading, Mass. 1981. MR 643361 (83h:53049)
 [9]
 V. Moncrief, Reduction of the YangMills equations, Differential Geometrical Methods in Mathematical Physics, Lecture Notes in Math., vol. 836, Springer, New York 1980. MR 607702 (83a:58041)
 [10]
 I. E. Segal, Nonlinear semigroups, Ann. of Math. 78 (1964), 339364. MR 0152908 (27:2879)
 [11]
 , Differential operators in the manifold of solutions of a nonlinear differential equation, J. Math. Pures Appl. 44 (1965), 71132. MR 0192369 (33:594)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198909498977
PII:
S 00029947(1989)09498977
Article copyright:
© Copyright 1989
American Mathematical Society
