On the analytic smoothing effect for the Hartree equation with a short range interaction potential
type: Item
instance of: scholarly article
title: On the analytic smoothing effect for the Hartree equation with a short range interaction potential
zbMATH Open document ID: 1379.35297
author: Hironobu Sasaki
published in: Journal of Mathematical Analysis and Applications
publication date: 2017-08-10T00:00:00Z
review text: The author discusses the Cauchy problem for the Hartree equation in space dimension \(d\geq 2\). Under consideration is the analytic smoothing effect of global in time solutions to \[ \begin{cases} iu_t+\Delta u=F(u),\\ u(0,x)=\phi(x). \end{cases} \] It is assumed that the interaction potential \(V\) belongs to the weak space \(L^{d/\sigma}\) where \(1 < \sigma < d\); i.e., \[ \sup_{\lambda>0} \lambda \mu \left( \{ x\in\mathbb{R}^d:|V(x)|>\lambda \} \right)^{\sigma/d} \] Since \(1 < \sigma < d\), the interaction potential \(V\) is short range (when \(0 < \sigma \leq 1\), \(V\) becomes long range). It is shown that if \(2\leq\sigma<d\) (resp. \(1<\sigma<2\)), the initial data \(\phi\) is small in the sense of the homogeneous Sobolev space \(\dot H^{\sigma/2-1}\) (resp. the homogeneous weighted Sobolev space \(\mathcal{F}\dot H^{1-\sigma/2}\)) and the Fourier transform \(\mathcal{F}\phi\) satisfies a real-analytic condition, then the corresponding solution \(u(t)\) is also real-analytic for any \(t\not =0\). Interestingly, no \(\dot H^{\sigma/2-1}\) (resp. \(\mathcal{F}\dot H^{1-\sigma/2}\)) smallness condition is imposed on first and higher order partial derivatives of \(\mathcal{F}\phi\) when \(2 \leq\sigma < d\) (resp. \(1 < \sigma < 2\)).
reviewed by: Joseph Shomberg
Mathematics Subject Classification ID: 35Q55
zbMATH DE Number: 6759262
zbMATH Keywords: Hartree equation
zbMATH Keywords: analytic smoothing effect
zbMATH Keywords: short range potential
MaRDI profile type: Publication
full work available at URL: j.jmaa.2017.05.067
OpenAlex ID: W2621014455
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DOI: 10.1016/J.JMAA.2017.05.067
description: scientific article; zbMATH DE number 6759262