Holmer–Roudenko weak conjecture for three-dimensional focusing cubic NLS

Prove that every three-dimensional focusing cubic nonlinear Schrödinger equation solution in H^1, without radial-symmetry or finite-variance assumptions, satisfies \(\|\nabla u(t)\|_{L^2}\to\infty\) as \(t\to+\infty\) whenever it is a global solution on the relevant grow-up branch, thereby upgrading divergence along a sequence of times to divergence along every diverging sequence.

Background

The paper compares its finite-time blow-up result for the four-dimensional quadratic Schrödinger system with the corresponding unresolved issue for the three-dimensional focusing cubic nonlinear Schrödinger equation. Holmer and Roudenko established that an H1 solution, without radial-symmetry or finite-variance assumptions, either blows up in finite time or has unbounded gradient norm along a sequence of times tending to positive infinity.

The stronger assertion that the gradient norm diverges for all times tending to positive infinity is identified as the Holmer–Roudenko weak conjecture. The paper explains that resolving this conjecture in the nonradial setting is more difficult because spatial translation of the solution must also be controlled, and suggests that a translation-adapted virial-defect argument might be relevant.

References

They further formulated the stronger assertion

|\nabla u(t)|_{L2}\longrightarrow\infty \qquad\text{as }t\to+\infty

as their weak conjecture. Thus, in the nonradial setting, the problem is to upgrade grow-up along some diverging sequence to grow-up along every diverging sequence.

— Finite-time blow-up for the four-dimensional mass-critical quadratic nonlinear Schrödinger system without mass resonance  (2608.18453 - Nguyen et al., 19 Aug 2026) in Remark “Relation to the three-dimensional cubic NLS,” Section 1 (Introduction)