The unconditional uniqueness for the energy-supercritical NLS

TL;DR

This paper proves the unconditional uniqueness of solutions to energy-supercritical cubic and quintic nonlinear Schrödinger equations across all dimensions, resolving a key problem in the theory at critical regularity.

Contribution

It introduces a unified scheme to establish unconditional uniqueness for NLS at critical regularity in all dimensions, completing the understanding for these equations on both Euclidean and torus domains.

Findings

Unconditional uniqueness is established for all energy-supercritical NLS in all dimensions.

The results unify previous partial solutions, providing a comprehensive resolution.

Application to defocusing blowup solutions shows they are the only solutions of a certain type.

Abstract

We consider the cubic and quintic nonlinear Schr\"{o}dinger equations (NLS) under the $\mathbb{R}^{d}$ and $\mathbb{T}^{d}$ energy-supercritical setting. Via a newly developed unified scheme, we prove the unconditional uniqueness for solutions to NLS at critical regularity for all dimensions. Thus, together with [18,19], the unconditional uniqueness problems for $H^{1}$-critical and $H^{1}$-supercritical cubic and quintic NLS are completely and uniformly resolved at critical regularity for these domains. One application of our theorem is to prove that defocusing blowup solutions of the type in [54] is the only possible $C([0,T);\dot{H}^{s_{c}})$ solution if exist in these domains.