For d≥ 3, consider the mass-critical Hartree equation i partial_t u+Delta u=mu bigl(|x|^-2|u|^2 bigr)u on R^d, with u(0)=u_0∈ L^2(R^d) and mu∈+1,-1. If mu=+1 (the defocusing case), then every initial datum u_0∈ L^2(R^d) generates a global solution that scatters in L^2. If mu=-1 (the focusing case) and |u_0|_L^2<|Q|_L^2, where Q is the ground state, then the solution is likewise global and scatters: there exist u_+,u_-∈ L^2(R^d) such that |u(t)-e^it Deltau_pm|_L^2→0 as t→pm∞.

Mass-critical Hartree scattering conjecture

For $d\ge 3$ , consider the mass-critical Hartree equation $i\partial_t u+\Delta u=\mu\bigl(|x|^{-2}*|u|^2\bigr)u$ on $\mathbb{R}^d$ , with $u(0)=u_0\in L^2(\mathbb{R}^d)$ and $\mu\in\{+1,-1\}$ . If $\mu=+1$ (the defocusing case), then every initial datum $u_0\in L^2(\mathbb{R}^d)$ generates a global solution that scatters in $L^2$ . If $\mu=-1$ (the focusing case) and $\|u_0\|_{L^2}<\|Q\|_{L^2}$ , where $Q$ is the ground state, then the solution is likewise global and scatters: there exist $u_+,u_-\in L^2(\mathbb{R}^d)$ such that $\|u(t)-e^{it\Delta}u_\pm\|_{L^2}\to0$ as $t\to\pm\infty$ .

References

Primary source

arXiv

Additional references

Global well-posedness and scattering for mass-critical Hartree equation — arXiv — Zuyu Ma, Changxing Miao, Matthew Rosenzweig, Jiqiang Zheng

harmonic-analysis nonlinear-pde

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A September 2026 preprint claims the conjecture for arbitrary data, removing the radial restriction, but the result is not yet verified.

The conjecture concerns global existence and scattering for the mass-critical Hartree equation at scaling-critical $L^2$ regularity. A new preprint by Zuyu Ma, Changxing Miao, Matthew Rosenzweig, and Jiqiang Zheng claims the full nonradial result.

Known results

Radial data: global existence and scattering in the defocusing case, and below the ground-state mass in the focusing case (2008).

September 2026 nonradial preprint

On September 23, 2026, the preprint “Global well-posedness and scattering for the mass-critical Hartree equation” claimed global well-posedness and scattering without radial symmetry at scaling-critical $L^2$ regularity. If correct, this settles the stated conjecture in the broader nonradial class; the preprint is unrefereed.

Current status (as of September 2026): The radial theorem is established, while the full nonradial conjecture is claimed solved by a new preprint but remains unverified.

Sources

No solutions have been posted yet.