For every initial datum u_0∈dot H^1/2( R^3) and every time interval I containing 0, if u,v∈ C(I;dot H^1/2( R^3)) are solutions of the cubic nonlinear Schrödinger equation i partial_t w+Delta w=|w|^2w on I with u(0)=v(0)=u_0, then u=v on I.

Unconditional uniqueness for the cubic nonlinear Schrödinger equation in dot H^1/2( R^3)

The problem asks whether solutions of the cubic nonlinear Schrödinger equation in the critical space $\dot H^{1/2}(\mathbb{R}^3)$ are unconditionally unique. No proposer or original date is identified in the retrieved material.

Known results

Herr and Sohinger (2018) proved unconditional uniqueness on a three-dimensional rectangular torus for $s>\frac{4}{5}$ , not at the stated endpoint.
Their discussion records earlier unconditional-uniqueness results on $\mathbb{R}^d$ beginning with Kato and subsequent authors, but gives no theorem at $\dot H^{1/2}(\mathbb{R}^3)$ .

October 2026 claimed resolution

Yongming Luo and Jiayu Zheng claim unconditional uniqueness at the scaling-critical regularity using a critical Besov-space bootstrap and a second Duhamel iteration. The retrieved preprint is unrefereed, and no independent mathematical assessment or verification was found.

Current status (as of October 2026): unconditional uniqueness at $\dot H^{1/2}(\mathbb{R}^3)$ is claimed by a preprint but remains unverified.

Unconditional uniqueness for the cubic nonlinear Schrödinger equation in \dot H^{1/2} $\mathbb R^3$

Primary source

Additional references

Progress summary

Solutions 0