Local well-posedness of Yang-Mills equations in Lorenz gauge below the energy norm

2014/08/22 by Achenef Tesfahun, Tesfahun, Achenef
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP

paper · pdf · doi:10.48550/arxiv.1408.5363

26 pages

arxiv created 2014/08/22 · arxiv updated 2014/08/25

Abstract

We prove that the Yang-Mills equations in the Lorenz gauge (YM-LG) is locally well-posed for data below the energy norm, in particular, we can take data for the gauge potential A and the associated curvature F in H^s× H^s-1 and H^r× H^r-1 for s=(\frac67+,-\frac114 +), respectively. This extends a recent by Selberg and the present author on the local well-posedness of YM-LG for finite energy data (specifically, for (s, r)=(1-, 0)). We also prove unconditional uniqueness of the energy class solution, that is, uniqueness in the classical space C([-T, T]; X₀), where X₀ is the energy data space. The key ingredient in the proof is the fact that most bilinear terms in YM-LG contain null structure some of which uncovered in the present paper.

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