Well-posedness in a critical space of Chern-Simons-Dirac system in the Lorenz gauge

2019/12/14 by Cho, Yonggeun, Hong, Seokchang
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1912.06790

Abstract

In this paper, we consider the Cauchy problem of local well-posedness of the Chern-Simons-Dirac system in the Lorenz gauge for B^\frac14_2,1 initial data. We improve the low regularity well-posedness, compared to Huh-Oh \citehuhoh and Okamoto \citeoka, by using the localization of space-time Fourier side and bilinear estimates given by Selberg \citeselb, whereas the authors of \citehuhoh, oka used global estimates of \citedanfoselb. Then we show the Dirac spinor flow of Chern-Simons-Dirac system is not C² at the origin in H^s if s < \frac14. From this point of view, the space B_2,1^\frac14 can be regarded as a critical space for the local well-posedness. We apply the argument for failure of smoothness to the Dirac equation decoupled from Chern-Simons-Dirac system and show the flow is not C³ in H^s, s < 0.

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