Small data scattering for semi-relativistic equations with Hartree type nonlinearity

TL;DR

This paper proves global well-posedness and scattering for a semi-relativistic Hartree equation with small initial data in certain Sobolev spaces, using advanced harmonic analysis techniques.

Contribution

It establishes near-optimal global existence and scattering results for semi-relativistic equations with Hartree nonlinearity, including radially symmetric cases, employing endpoint Strichartz and bilinear estimates.

Findings

Global well-posedness for small data in H^s with s>1/2 for m>0.

Extended results to s>0 for radially symmetric data including the massless case.

Solutions scatter to free waves as t approaches infinity.

Abstract

We prove that the initial value problem for the equation \[ - i\partial_t u + \sqrt{m^2-\Delta} \, u= (\frac{e^{-\mu_0 |x|}}{|x|} \ast |u|^2)u \ \text{in} \ \mathbb R^{1+3}, \quad m\ge 0, \ \mu_0 >0\] is globally well-posed and the solution scatters to free waves asymptotically as $t \to \pm \infty$ if we start with initial data which is small in $H^s(\mathbb R^{3})$ for $s>\frac12$, and if $m>0$. Moreover, if the initial data is radially symmetric we can improve the above result to $m\ge 0$ and $s>0$, which is almost optimal, in the sense that $L^2(\mathbb R^{3})$ is the critical space for the equation. The main ingredients in the proof are certain endpoint Strichartz estimates, $L^2(\mathbb R^{1+3})$ bilinear estimates for free waves and the application of the $U^p$ and $V^p$ function spaces.