On the analytic smoothing effect for the Hartree equation with a short range interaction potential
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. \(1 < \sigma < 2\)).
scientific article; zbMATH DE number 6759262
scientific article