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Long-time Behavior of Solutions to Cubic Dirac Equation with Hartree Type Nonlinearity in ℝ1+2
International Mathematics Research Notices ( IF 0.9 ) Pub Date :  , DOI: 10.1093/imrn/rny217
Achenef Tesfahun 1
Affiliation  

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:

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。
更新日期:2020-10-27
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