The Unconditional Uniqueness for the Energy-critical Nonlinear Schrödinger Equation on ${\mathbb{T}}^4$

Abstract

We consider the ${\mathbb{T}}^4$ cubic NLS which is energy-critical. We study the unconditional uniqueness of solution to the NLS via the cubic Gross-Pitaevskii hierarchy, an uncommon method, and does not require the existence of solution in Strichartz type spaces. We prove $U$-$V$ multilinear estimates to replace the previously used Sobolev multilinear estimates, which fail on ${\mathbb{T}}^4$. 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 schemes 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.

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Cited by in corpus (3)

• The unconditional uniqueness for the energy-supercritical NLS
• Well/Ill-posedness of the Boltzmann Equation with Soft Potential
• On the mean-field and semiclassical limit from quantum $N$-body dynamics