PDE-Applied Math Archives for Academic Year 2020
The Unconditional Uniqueness for the Energy-critical Nonlinear Schrödinger Equation on T^4
When: Thu, September 24, 2020 - 3:30pm
Speaker: Xuwen Chen (University of Rochester) -
Abstract: We consider the 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 solutions in Strichartz type spaces. We prove U-V multilinear estimates to replace the previously used Sobolev multilinear estimates, which fail on 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 scheme to prove H^1 uniqueness for the R^3/R^4/T^3/T^4 energy-critical Gross-Pitaevskii hierarchies and thus the corresponding NLS.