We show that the cubic Dirac equation with zero mass is globally well-posed for small data in the scale invariant space H^{\frac{n-1}{2}}(R^n) for n=2, 3. The proof proceeds by using the Fierz identities to rewrite the equation in a form where the null structure of the system is readily apparent. This null structure is then exploited via bilinear estimates in spaces based on the null frame spaces of Tataru. We hope that the spaces and estimates used here can be applied to other nonlinear Dirac equations in the scale invariant setting. Our work complements recent results of Bejenaru-Herr who proved a similar result for n=3 in the massive case.
On the cubic Dirac equation with potential and the Lochak--Majorana condition
研究了$\mathbb{R}\times\mathbb{R}^{3}$ \begin{equation*} i \partial _t u + \mathcal{D} u + V(x) u = \langle βu,u \rangle βu \end{equation*}上的三次Dirac方程受几乎临界正则性的大扰动.我们证明了小初始数据在$H^{1}$中的整体存在性和散射性，并增加了角正则性。主要工具是扰动狄拉克流的一个端点Eschhartz估计。特别地，该结果覆盖了具有小$H^{1}$范数的球对称数据的情况。当潜在的$V$有一个合适的结构，我们证明了全球存在性和散射的\emph{large}初始数据具有一个小的手征成分，有关的Lochak-Majorana条件。
