Small data scattering for a cubic Dirac equation with Hartree type nonlinearity in ${\mathbb{R}}^{1+3}$

Abstract

We prove that the initial value problem for the Dirac equation $\left(-i{\gamma }^{\mu }{\partial }_{\mu }+m\right)\psi =\left(\frac{e^{-|x|}}{|x|}\ast (\overline{\psi }\psi )\right)\psi \text{in }{\mathbb{R}}^{1+3}$ is globally well-posed and the solution scatters to free waves asymptotically as $t\to \pm \infty$, if we start with initial data that is 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.

34 pages: To appear in SIAM J. Math. Analysis

References in corpus (1)

• The cubic Dirac equation: Small initial data in $H^{\frac{1}{2}}({\mathbb{R}}^2)$