Almost critical local well-posedness for the space-time Monopole equation in Lorenz gauge

2013/08/20 by Achenef Tesfahun, Tesfahun, Achenef
Mathematics · #35L70 #35Q40 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35L70 #msc:35Q40

paper · pdf · doi:10.48550/arxiv.1308.4285

15 pages

arxiv created 2013/08/20 · arxiv updated 2013/08/21

Abstract

Recently, Candy and Bournaveas proved local well-posedness of the space-time monopole equation in Lorenz gauge for initial data in H^s with s>\frac14. The equation is L²-critical, and hence a \frac14 derivative gap is left between their result and the scaling prediction. In this paper, we consider initial data in the Fourier-Lebesgue space H_p^s for 1<p≤ 2 which coincides with H^s when p=2 but scales like lower regularity Sobolev spaces for 1<p< 2. In particular, we will see that as p→ 1⁺, the critical exponent s^c_p→ 1^-, in which case H₁₊^1- is the critical space. We shall prove almost optimal local well-posedness to the space-time monopole equation in Lorenz gauge with initial data in the aforementioned spaces that correspond to p close to 1.

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