Global well-posedness for the massive Maxwell-Klein-Gordon equation with\n small critical Sobolev data

2016/10/11 by Cristian Gavrus, Gavrus, Cristian
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.1610.03581

openalex publication_date 2016/10/11 · openalex created_date 2023/03/15 · openalex updated_date 2026/07/28

Abstract

In this paper we prove global well-posedness and modified scattering for the\nmassive Maxwell-Klein-Gordon equation in the Coulomb gauge on\n\ℝ^1+d (d \≥ 4) for data with small critical Sobolev norm. This\nextends to the general case m² > 0 the results of Krieger-Sterbenz-Tataru\n(d=4,5 ) and Rodnianski-Tao ( d \≥ 6 ), who considered the case m=0.\n We proceed by generalizing the global parametrix construction for the\ncovariant wave operator and the functional framework from the massless case to\nthe Klein-Gordon setting. The equation exhibits a trilinear cancelation\nstructure identified by Machedon-Sterbenz. To treat it one needs sharp L² \nnull form bounds, which we prove by estimating renormalized solutions in null\nframes spaces similar to the ones considered by Bejenaru-Herr. To overcome\nlogarithmic divergences we rely on an embedding property of Box^-1 in\nconjunction with endpoint Strichartz estimates in Lorentz spaces.\n

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