论文标题
$ \ mathbb {t}^{4} $在$ \ mathbb上的无条件唯一性的无条件唯一性
The Unconditional Uniqueness for the Energy-critical Nonlinear Schrödinger Equation on $\mathbb{T}^{4}$
论文作者
论文摘要
我们考虑$ \ mathbb {t}^{4} $ cubic NLS，它是批判能量的。我们通过Cubic Gross-Pitaevskii层次结构（一种罕见的方法）研究了对NLS解决方案的无条件唯一性，并且不需要在Strichartz型空间中存在溶液。我们证明了$ u $ - $ V $多线性估算值以替换先前使用的SOBOLEV多线估计，该估计失败了，$ \ m athbb {t}^{4} $。为了结合较弱的估计值，我们首次从头开始和计算重新组合Duhamel出生的扩展中，从头开始和计算进行了新的组合制剂。新组合物和$ u $ - $ V $估算值然后无缝地结论了$ h^{1} $在无限层次结构框架下的NLS无条件唯一性。这项工作建立了一个统一的计划，以证明$ h^{1} $唯一性的唯一性的唯一性相应的NLS。
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.
