Abstract
AbstractWe 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 proveU-Vmultilinear 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 theU-Vestimates 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.
Highlights
The cubic nonlinear Schrödinger equation (NLS) in four dimensions i∂t u = −Δu ± |u|2 u in R × Λ, (1.1)u(0, x) = u0, where Λ = R4 or T4, is called energy-critical, because it is invariant under the H1 scaling u(t, x) ↦→ uλ(t, x) = 1 λ u t λ2, x λ if Λ = R4
Before moving into the proof of Proposition 3.7, we remark that the extra 2T does not imply that the estimate is critical or subcritical; this T appears only once
We notice that all μ j, sgn j are tamed and that wild moves, unlike KM moves, do change the tree skeleton, but this change is restricted to shuffling nodes along a left branch, subject to the restrictions that the ordering of the plus nodes and minus nodes remain intact
Summary
With everything ready by the extended KM board game, the quantum de Finetti theorem from [6], the U-V space techniques from [50], the trilinear estimates proved using the scale-invariant Stichartz estimates and l2-decoupling theorem in [4, 43] and the HUFL properties from [24], all work together seamlessly in Section 5 to establish Theorem 3.1 and provide a unified proof of large-solution uniqueness for the R3/T3 quintic and the R4/T4 cubic energy-critical GP hierarchies, and the corresponding NLS The discovery of such an unexpected close and effective collaboration of these previously independent deep theorems is the main novelty of this paper. Dispersive estimate technology to bear on various type of hierarchies of equations and related problems, and this is our first example of it. (An immediate step has been taken [26].)
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