2017/10/31 by Pecher, Hartmut
#35L70 #35Q40 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1710.11399
The Maxwell-Klein-Gordon equation ∂α Fαβ = -Im(ΦDβ Φ) , DμDμ Φ= m2 Φ , where Fαβ = ∂α Aβ - ∂β Aα, Dμ = ∂μ - iAμ , in the (3+1)-dimensional case is known to be unconditionally well-posed in energy space, i.e. well-posed in the natural solution space. This was proven by Klainerman-Machedon and Masmoudi-Nakanishi in Coulomb gauge and by Selberg-Tesfahun in Lorenz gauge. The main purpose of the present paper is to establish that for both gauges this also holds true for data Φ(0) in Sobolev spaces Hs with less regularity, i.e. s < 1, but s sufficently close to 1. This improves the (conditional) well-posedness results in both cases, i.e. uniqueness in smaller solution spaces of Bourgain-Klainerman-Machedon type, which were essentially known by Cuccagna, Selberg and the author for s > (3)/(4) , and which in Coulomb gauge is also contained in the present paper. In fact, the proof consists in demonstrating that any solution in the natural solution space for some s > s0 belongs to a Bourgain-Klainerman-Machedon space where uniqueness is known. Here s0 ≈ 0.914 in Coulomb gauge and s0 ≈ 0.907 in Lorenz gauge.