SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2969-3003, January 2020.
We prove that the initial value problem for the Dirac equation $(-i\gamma^\mu \partial_\mu + m) \psi= ( \frac{e^{- |x|}}{|x|} \ast ( \overline \psi \psi)) \psi \quad \text{in } \ \mathbb{R}^{1+3}$ is globally well-posed and the solution scatters to free waves asymptotically as $t \rightarrow \pm \infty$ if we start with initial data that are small in $H^s$ for $s>0$. This is an almost critical well-posedness result in the sense that $L^2$ is the critical space for the equation. The main ingredients in the proof are Strichartz estimates, space-time bilinear null-form estimates for free waves in $L^2$, and an application of the $U^p$ and $V^p$ function spaces.

中文翻译：

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$ \ mathbb {R} ^ {1 + 3} $中具有Hartree型非线性的三次Dirac方程的小数据散射

SIAM数学分析杂志，第52卷，第3期，第2969-3003页，2020年1月。
我们证明Dirac方程的初值问题$（-i \ gamma ^ \ mu \ partial_ \ mu + m）\ psi = （\ frac {e ^ {-| x |}} {| x |} \ ast（\ overline \ psi \ psi））\ psi \ quad \ text {in} \ \ mathbb {R} ^ {1 + 3}如果我们从$ H ^ s $中小的$ s> 0 $的初始数据开始，则$在全球范围内位置良好，并且解以$ t \ rightarrow \ pm \ infty $的形式渐渐分散到自由波上。在$ L ^ 2 $是方程式的临界空间的意义上，这是几乎临界的适定性结果。证明中的主要成分是Strichartz估计，$ L ^ 2 $中自由波的时空双线性零形式估计以及$ U ^ p $和$ V ^ p $函数空间的应用。