Unconditional uniqueness for the energy-critical nonlinear Schrödinger equation on

Abstract

We consider the $\mathbb {T}^{4}$ cubic nonlinear Schrödinger equation (NLS), which is energy-critical. We study the unconditional uniqueness of solutions to the NLS via the cubic Gross–Pitaevskii hierarchy, an uncommon method for NLS analysis which is being explored [24, 35] and does not require the existence of a solution in Strichartz-type spaces. We prove U-V multilinear estimates to replace the previously used Sobolev multilinear estimates. To incorporate the weaker estimates, we work out new combinatorics from scratch and compute, for the first time, the time integration limits, in the recombined Duhamel–Born expansion. The new combinatorics and the U-V estimates then seamlessly conclude the $H^{1}$ unconditional uniqueness for the NLS under the infinite-hierarchy framework. This work establishes a unified scheme to prove $H^{1}$ uniqueness for the $ \mathbb {R}^{3}/\mathbb {R}^{4}/\mathbb {T}^{3}/\mathbb {T}^{4}$ energy-critical Gross–Pitaevskii hierarchies and thus the corresponding NLS.

Dynamics of interacting bosons: a compact review

“…For β = 1 a new proof of uniqueness of the hierarchy has been given by Chen, Hainzl, Pavlović and Seiringer in [38]. Other recent, related works include [4,88,51].…”

Quantitative Derivation and Scattering of the 3D Cubic NLS in the Energy Space

“…With the help of the extended KM board game in [27] which allows the application of the U-V space-time estimates, and a X 1 2 + frequency localized version of the KM estimates, we expect improving (1.15) up to β = 1 once we put in the correlation structures we had in [24]. (Of course, (1.11) has to be changed as well.)…”

“…established, for the T 3 quintic energy-critical GP hierarchy, a H 1 -type uniqueness theorem which was neither conditional nor unconditional but implies the H 1 unconditional uniqueness for the T 3 quintic energy-critical NLS. More recently, in [27], X.C. and J.H.…”

“…worked out an extended KM board game from scratch to enable, finally, the application of dispersive norms like U-V and X s,b in the field and proved the H 1 unconditional uniqueness for the T 4 cubic energy-critical NLS, an unanticipated "odd ball" case in the R 3 /R 4 /T 3 /T 4 energy-critical sequence, with the hierarchy approach. The proof in [27] went so smoothly that, X.Chen, Shen, and Zhang completely and unifiedly solved the unconditional uniqueness for R d and T d cubic and quintic energy-supercritical NLS in [28].…”

Infinite-energy solutions to energy-critical nonlinear Schrödinger equations in modulation spaces

“…For further reading, we also refer to the very recent contribution [11], in which unconditional uniqueness of solutions in C([0, T ], H 1 (X d )) for energy critical Schrödinger equations is proved for d ∈ {3, 4} and X ∈ {T, R}. Chen and Holmer [11] use the Gross-Pitaevskii hierarchy, which was previously considered by Herr and Sohinger [19] to cover the whole subcritical range for d = 4; see also references therein.…”