Abstract
The Yang–Mills and Yang–Mills–Higgs equations in temporal gauge are locally well-posed for small and rough initial data, which can be shown using the null structure of the critical bilinear terms. This carries over a similar result by Tao for the Yang–Mills equations in the (3+1)-dimensional case to the more general Yang–Mills–Higgs system and to general dimensions.
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Eardley D.M., Moncrief V.: The global existence of Yang–Mills–Higgs fields in 4-dimensional Minkowski space. I. Local existence and smoothness properties. Commun. Math. Phys. 83, 171–191 (1982)
Eardley D.M., Moncrief V.: The global existence of Yang–Mills–Higgs fields in 4-dimensional Minkowski space. II. Completion of the proof. Commun. Math. Phys. 83, 193–212 (1982)
Foschi, D., Klainerman, S.: Bilinear space-time estimates for homogeneous wave equations. Ann. Scient. ENS, 4e Ser. 33, 211–174 (2000)
Ginibre J., Velo G.: Generalized Strichartz inequalities for the wave equation. J. Funct. Anal. 133, 60–68 (1995)
Keel M.: Global existence for critical power Yang–Mills–Higgs equation in \({\mathbb{R}^{3+1}}\). Commun. Part. Diff. Equ. 22, 1161–1225 (1997)
Klainerman S., Machedon M.: On the Maxwell–Klein–Gordon equation with finite energy. Duke Math. J. 74, 19–44 (1994)
Klainerman S., Machedon M.: Finite energy solutions of the Yang–Mills equations in \({{\mathbb R}^{3+1}}\). Ann. Math. 142, 39–119 (1995)
Klainerman, S.; Machedon, M.: (Appendices by J. Bougain and D. Tataru): Remark on Strichartz-type inequalities. Int. Math. Res. Not. no.5, 201–220 (1996)
Klainerman S., Tataru D.: On the optimal local regularity for the Yang–Mills equations in \({\mathbb{R}^{4+1}}\). J. AMS. 12, 93–116 (1999)
Krieger, J., Sterbenz, J.: Global regularity for the Yang–Mills equations on high dimensional Minkowski space. Mem. AMS, vol. 223, No. 1047 (2013)
Krieger, J., Tataru, D.: Global well-posedness for the Yang–Mills equations in 4+1 dimensions. Small Energy. arXiv:1509.00751
Oh S.: Gauge choice for the Yang–Mills equations using the Yang–Mills heat flow and local well-posedness in H 1. J. Hyperbolic Differ. Equ. 11, 1–108 (2014)
Oh S.: Finite energy global well-posedness of the Yang–Mills equations on \({\mathbb{R}^{1+3}}\): an approach using the Yang–Mills heat flow. Duke Math. J. 164, 1669–1732 (2015)
Selberg, S.; Tesfahun, A.: Null structure and local well-posedness in the energy class for the Yang–Mills equations in Lorenz gauge. arXiv:1309.1977
Sterbenz J.: Global regularity and scattering for general non-linear wave equations. II. (4+1) dimensional Yang–Mills equations in the Lorentz gauge. Am. J. Math. 129, 611–664 (2007)
Tao T.: Multilinear weighted convolutions of L 2-functions and applications to non-linear dispersive equations. Am. J. Math. 123, 838–908 (2001)
Tao, T.: Local well-posedness of the Yang–Mills equation in the temporal gauge below the energy norm. arXiv:math/0005064v5
Tesfahun A.: Local well-posedness of Yang–Mills equations in Lorenz gauge below the energy norm. Nonlin. Differ. Equ. Appl. 22, 849–875 (2015)
Tesfahun, A.: Finite energy local well-posedness for the Yang–Mills–Higgs equations in Lorenz gauge. Int. Math. Res. Not. 5140–5161 (2015)
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Pecher, H. Local well-posedness for the (n + 1)-dimensional Yang–Mills and Yang–Mills–Higgs system in temporal gauge. Nonlinear Differ. Equ. Appl. 23, 40 (2016). https://doi.org/10.1007/s00030-016-0395-9
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DOI: https://doi.org/10.1007/s00030-016-0395-9