Finite Energy Local Well-Posedness for the Yang–Mills–Higgs Equations in Lorenz Gauge

We introduce a novel approach to the problem of gauge choice for the Yang–Mills equations on the Minkowski space ℝ1+3, which uses the Yang–Mills heat flow in a crucial way. As this approach does not possess the drawbacks of the previous approaches, it is expected to be more robust and easily adaptable to other settings. As a first application, we give an alternative proof of the local well-posedness of the Yang–Mills equations for initial data [Formula: see text], which is a classical result of Klainerman and Machedon (1995) that had been proved using a different method (local Coulomb gauges). The new proof does not involve localization in space–time, which had been the key drawback of the previous method. Based on the results proved in this paper, a new proof of finite energy global well-posedness of the Yang–Mills equations, also using the Yang–Mills heat flow, is established in a companion article.

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Finite energy global well-posedness of the Yang–Mills equations on $\mathbb{R}^{1+3}$ : An approach using the Yang–Mills heat flow

Duke Mathematical Journal ◽

10.1215/00127094-3119953 ◽

2015 ◽

Vol 164 (9) ◽

pp. 1669-1732 ◽

Cited By ~ 15

Author(s):

Sung-Jin Oh

Keyword(s):

Heat Flow ◽

Finite Energy ◽

Well Posedness ◽

Yang Mills

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Finite-Energy Global Well-Posedness of the Maxwell–Klein–Gordon System in Lorenz Gauge

Communications in Partial Differential Equations ◽

10.1080/03605301003717100 ◽

2010 ◽

Vol 35 (6) ◽

pp. 1029-1057 ◽

Cited By ~ 24

Author(s):

Sigmund Selberg ◽

Achenef Tesfahun

Keyword(s):

Finite Energy ◽

Well Posedness ◽

Klein Gordon ◽

Lorenz Gauge

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FINITE ENERGY ONE-HALF MONOPOLE SOLUTIONS

Modern Physics Letters A ◽

10.1142/s0217732312502331 ◽

2012 ◽

Vol 27 (40) ◽

pp. 1250233 ◽

Cited By ~ 5

Author(s):

ROSY TEH ◽

BAN-LOONG NG ◽

KHAI-MING WONG

Keyword(s):

Magnetic Field ◽

Magnetic Fields ◽

Magnetic Flux ◽

Topological Charge ◽

Magnetic Monopole ◽

Magnetic Charge ◽

Finite Energy ◽

Spatial Infinity ◽

Yang Mills ◽

The Magnetic Field

We present finite energy SU(2) Yang–Mills–Higgs particles of one-half topological charge. The magnetic fields of these solutions at spatial infinity correspond to the magnetic field of a positive one-half magnetic monopole at the origin and a semi-infinite Dirac string on one-half of the z-axis carrying a magnetic flux of [Formula: see text] going into the origin. Hence the net magnetic charge is zero. The gauge potentials are singular along one-half of the z-axis, elsewhere they are regular.

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