Lorenz规范下Chern-Simons-Dirac和Chern-Simons-Higgs方程的低正则解
Low regularity solutions to the Chern-Simons-Dirac and the Chern-Simons-Higgs equations in the Lorenz gauge
摘要：
本文研究了Lorenz规范下Chern-Simons-Dirac（CSD）和Chern-Simons-Higgs（CSH）方程在低正则性初值下的局部适定性问题.我们的主要贡献之一是发现了（CSD）的一个空结构。结合$X^{s,b}$空间的标准机制，我们得到了（CSD）对于初始数据$a_μ, ψ\in H^{1/4+ε}_{x}$的局部适定性.此外，观察到应用于（CSH）的相同技术导致初始数据$a_μ \in H^{1/4+ε}_{x}$，$(ϕ, \partial_{t} ϕ) \in H^{3/4+ε}_{x} \times H^{-1/4+ε}_{x}$的局部适定性的快速证明，这改进了Selberg-Tesfahun（2012）的先前结果。
英文摘要：
In this paper, we address the problem of local well-posedness of the Chern-Simons-Dirac (CSD) and the Chern-Simons-Higgs (CSH) equations in the Lorenz gauge for low regularity initial data. One of our main contributions is the uncovering of a null structure of (CSD). Combined with the standard machinery of $X^{s,b}$ spaces, we obtain local well-posedness of (CSD) for initial data $a_μ, ψ\in H^{1/4+ε}_{x}$. Moreover, it is observed that the same techniques applied to (CSH) lead to a quick proof of local well-posedness for initial data $a_μ \in H^{1/4+ε}_{x}$, $(ϕ, \partial_{t} ϕ) \in H^{3/4+ε}_{x} \times H^{-1/4+ε}_{x}$, which improves the previous result of Selberg-Tesfahun (2012).
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