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Global Well-Posedness for the Massive Maxwell–Klein–Gordon Equation with Small Critical Sobolev Data

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Abstract

In this paper we prove global well-posedness and modified scattering for the massive Maxwell–Klein–Gordon equation in the Coulomb gauge on \(\mathbb {R}^{1+d}\) \((d \ge 4)\) for data with small critical Sobolev norm. This extends to the general case \( m^2 > 0 \) the results of Krieger–Sterbenz–Tataru (\(d=4,5 \)) and Rodnianski–Tao (\( d \ge 6 \)), who considered the case \( m=0\). We proceed by generalizing the global parametrix construction for the covariant wave operator and the functional framework from the massless case to the Klein–Gordon setting. The equation exhibits a trilinear cancelation structure identified by Machedon–Sterbenz. To treat it one needs sharp \(L^{2}\) null form bounds, which we prove by estimating renormalized solutions in null frames spaces similar to the ones considered by Bejenaru–Herr. To overcome logarithmic divergences we rely on an embedding property of \( \Box ^{-1} \) in conjunction with endpoint Strichartz estimates in Lorentz spaces.

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Notes

  1. Here, \( A^{free} \) is the free solution \( \Box A^{free}=0 \) with \(A^{free}_{x}[0] = A_{x}[0]\) and \( A^{free}_{0} = 0\).

  2. \(\delta _{1} \) is the admissible frequency envelope constant.

  3. \( \mathcal {L}\) denotes any translation invariant bilinear form with bounded mass kernel.

  4. Notice that this case does not occur when \( k_{\min }=0 \).

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Authors and Affiliations

  1. Department of Mathematics, The Johns Hopkins University, Baltimore, MD, 21218, USA

    Cristian Gavrus

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  1. Cristian Gavrus
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Correspondence to Cristian Gavrus.

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The author thanks Daniel Tataru for many fruitful conversations and thanks Sung-Jin Oh for suggesting this problem. The author was supported in part by the NSF Grant DMS-1266182 as a graduate student at UC Berkeley.

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Gavrus, C. Global Well-Posedness for the Massive Maxwell–Klein–Gordon Equation with Small Critical Sobolev Data. Ann. PDE 5, 10 (2019). https://doi.org/10.1007/s40818-019-0065-4

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