Are there any time-decay results for the solution of the Hartree equation \begin{equation}\frac{1}{i}\partial_t\phi-\Delta\phi=-(|x|^{-1}\ast|\phi|^2)\phi,\quad x\in\mathbb{R}^3\end{equation} in $L^p$-spaces which would ensure e.g. $\phi\in L^2((0,\infty);L^3(\mathbb{R}^3))$ or $\phi\in L^2((0,\infty);L^4(\mathbb{R}^3))$?

According to the information at DispersiveWiki, Coulomb potential is the borderline case and there is no scattering results (in the sense of asymptotic completeness) (??) for the solutions of the above equation. There is a paper by Hayashi and Ozawa on time-decay for Hartree equation with Coulomb or more singular potentials which makes use of pseudo-conformal invariance. This work implies that one can get rates like $\|\phi\|_4\lesssim t^{-3/8}$ or $\|\phi\|_3\lesssim t^{-1/4}$, where $\phi$ is the solution of the above equation. Those are slower rates compared to the rates at which the free solution decays. Are there more recent publications which might imply better rates? Do you know of any $L^\infty$-decay results (for the above equation) which might be interpolated by mass conservation to get faster $L^p$-decay?


1 Answer 1


You may find what you're looking for in the 1998 AJM paper of Hayashi and Naumkin:


They get the sharp $L^\infty$-decay rate of $t^{-3/2}$ for small data in weighted Sobolev spaces. By interpolation, this would give a decay rate of $t^{-1/2}$ in $L_x^3$, thus putting the solution in $L_t^{2,\infty} L_x^3$.


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