Small data scattering of semirelativistic Hartree equation

TL;DR

This paper proves that solutions to the semirelativistic Hartree equation in three dimensions scatter to linear solutions when initial data is sufficiently small in a specific Sobolev space with angular regularity.

Contribution

It establishes small data scattering for the semirelativistic Hartree equation using advanced Strichartz estimates and $U^p$-$V^p$ space techniques, generalizing previous results.

Findings

Solutions scatter to linear solutions for small initial data.

The results hold under specific growth conditions on the potential V.

The approach employs $L_ heta^2$-averaged Strichartz estimates and $U^p$-$V^p$ spaces.

Abstract

In this paper we study the small data scattering of Hartree type semirelativistic equation in space dimension $3$. The Hartree type nonlinearity is $[V * |u|^2]u$ and the potential $V$ which generalizes the Yukawa has some growth condition. We show that the solution scatters to linear solution if an initial data given in $ H^{s,1}$ is sufficiently small and $s>\frac14$. Here, $H^{s, 1}$ is Sobolev type space taking in angular regularity with norm defined by $\|\varphi\|_{ H^{s, 1}} = \|\varphi\|_{ H^{s}} + \|\nabla_{\mathbb S} \varphi\|_{H^{s}}$. To establish the results we employ the recently developed Strichartz estimate which is $L_\theta^2$-averaged on the unit sphere $\mathbb S^{2}$ and construct the resolution space based on $U^p$-$V^p$ space.