Almost optimal local well-posedness for the Maxwell-Klein-Gordon system with data in Fourier-Lebesgue spaces

Hartmut Pecher

doi:10.3934/cpaa.2020146

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2020, Volume 19, Issue 6: 3303-3321. Doi: 10.3934/cpaa.2020146

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Almost optimal local well-posedness for the Maxwell-Klein-Gordon system with data in Fourier-Lebesgue spaces

Hartmut Pecher

Fakultät für Mathematik und Naturwissenschaften, Bergische Universität Wuppertal, Gaußstr. 20, 42119 Wuppertal, Germany

Received: July 31, 2019

Revised: October 31, 2019

Published: March 2020

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Abstract

We prove a low regularity local well-posedness result for the Maxwell-Klein-Gordon system in three space dimensions for data in Fourier - Lebesgue spaces $ \widehat{H}^{s,r} $, where $ \|f\|_{\widehat{H}^{s,r}} = \|\langle \xi \rangle^s \widehat{f}(\xi)\|_{\widehat{L}^{r'}} $, $ \frac{1}{r}+\frac{1}{r'} = 1 $. The assumed regularity for the data is almost optimal with respect to scaling as $ r \to 1 $. This closes the gap between what is known in the case $ r = 2 $, namely $ s > \frac{3}{4} $, and the critical value $ s_c = \frac{1}{2} $ with respect to scaling.

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Mathematics Subject Classification: 35Q61, 35L70.

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