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Almost sharp global well-posedness for the defocusing Hartree equation with radial data in $\mathbb R^5$
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  • Xingjian Mu,
  • Xingdong Tang,
  • Guixiang Xu,
  • Jianwei Yang
Xingjian Mu
Beijing Normal University
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Xingdong Tang
Nanjing University of Information Science and Technology
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Guixiang Xu
Beijing Normal University
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Jianwei Yang
Beijing Institute of Technology
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Abstract

We show global well-posedness and scattering for the defocusing, energy-subcritical Hartree equation \begin{equation*} iu_t + \Delta u =F(u), (t,x)\in\mathbb{R}\times\mathbb{R}^5 \end{equation*} where $F(u)= \big( V* |u|^2 \big) u$, $V(x)=|x|^{-\gamma}$, $3< \gamma< 4$, and initial data $u_0(x)$ is radial in almost sharp Sobolev space $ H^{s}\left(\R^5\right)$ for $s>s_c=\gamma/2-1$. Main difficulty is the lack of the conservation law. The main stategy is to use I-method together with the radial Sobolev inequality, the interaction Morawetz estimate, long-time Strichartz estimate and local smoothing effect to control the energy transfer of the solution and obtain the increment estimate of the modified energy $E(Iu)(t)$.