On the growth of Sobolev norms for Hartree equation

This paper is devoted to the growth of high-order Sobolev norms of solutions in time in the context of the Hartree equation in 2- or 3-dimensional spaces. By combining modified energies with appropriate Strichartz estimates, we derive polynomial-time bounds for the \(H^s(s\ge 2)\) $H^s(s\ge 2)$ norms. Compared to the bound in 2 dimensions due to harmonic analysis techniques in the work (Sohinger in Discr Cont Dyn A 32(10):3733–3771, 2012), we obtain a more refined growth result of \(H^s\) $H^s$ norm for \(s \le 21/5\) $s\le 21/5$ .