Null structure and local well-posedness in the energy class for the Yang-Mills equations in Lorenz gauge

Abstract

We demonstrate null structure in the Yang-Mills equations in Lorenz gauge. Such structure was found in Coulomb gauge by Klainerman and Machedon, who used it to prove global well-posedness for finite-energy data. Compared with Coulomb gauge, Lorenz gauge has the advantage---shared with the temporal gauge---that it can be imposed globally in space even for large solutions. Using the null structure and bilinear space-time estimates, we also prove local-in-time well-posedness of the equations in Lorenz gauge, for data with finite energy. The time of existence depends on the initial energy and on the $H^s\times H^{s-1}$-norm of the initial potential, for some $s<1$.

Minor typos corrected, references updated

References in corpus (4)

• Finite energy global well-posedness of the Yang-Mills equations on ${\mathbb{R}}^{1+3}$: An approach using the Yang-Mills heat flow
• Low regularity local well-posedness for the Maxwell-Klein-Gordon equations in Lorenz gauge
• Gauge choice for the Yang-Mills equations using the Yang-Mills heat flow and local well-posedness in $H^1$
• Low regularity solutions to the Chern-Simons-Dirac and the Chern-Simons-Higgs equations in the Lorenz gauge

Cited by in corpus (3)

• Large data decay of Yang-Mills-Higgs fields on Minkowski and de Sitter spacetimes
• Local well-posedness of Yang-Mills equations in Lorenz gauge below the energy norm
• Finite energy local well-posedness for the Yang-Mills-Higgs equations in Lorenz gauge