Almost sharp global well-posedness for the defocusing Hartree equation
with radial data in $\mathbb R^5$
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 $E(Iu)(t)$.
