Abstract

In this paper we study the long-time behavior of solutions to the Dirac equation $$\begin{equation*} \big ( -i\gamma^\mu \partial_\mu + m \big) \psi= \left(V \ast ( \overline \psi \psi)\right) \psi \ \ \textrm{in } \ {\mathbb{R}}^{1+2},\end{equation*}$$where $V$ is the Yukawa potential in ${\mathbb{R}}^{2}$. It is proved that if $m>0$ and the initial data is small in $H^s({\mathbb{R}}^2)$ for $s>0$, the corresponding initial value problem is globally well posed and the solution scatters to free waves asymptotically as $t \rightarrow \pm \infty $. The main ingredients in the proof are Strichartz estimates and space-time $L^2$-bilinear null-form estimates for free waves.

中文翻译：

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Har1 + 2中具Hartree型非线性的三次Dirac方程解的长时间行为

摘要

在本文中，我们研究Dirac方程$$ \ begin {equation *} \ big（-i \ gamma ^ \ mu \ partial_ \ mu + m \ big）\ psi = \ left（V \ ast（\ overline \ psi \ psi）\ right）\ psi \ \ \ textrm {in} \ {\ mathbb {R}} ^ {1 + 2}，\ end {equation *} $$其中$ V $是$ {\ mathbb {R}} ^ {2} $中的Yukawa势。证明如果$ m> 0 $且$ s> 0 $的初始数据在$ H ^ s（{\ mathbb {R}} ^ 2）$中较小，则相应的初始值问题在全局上是合适的，并且解决方案渐近地以$ t \ rightarrow \ pm \ infty $的形式散射到自由波。证明中的主要成分是Strichartz估计和自由波的时空$ L ^ 2 $-双线性零形式估计。