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Almost optimal local well-posedness for the Maxwell-Klein-Gordon system with data in Fourier-Lebesgue spaces. (English) Zbl 1437.35651

Summary: 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.

MSC:

35Q61 Maxwell equations
35L70 Second-order nonlinear hyperbolic equations
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35B65 Smoothness and regularity of solutions to PDEs
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References:

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