The unconditional uniqueness for the energy-supercritical NLS

We consider the cubic and quintic nonlinear Schr"{o}dinger equations (NLS) under the ${\mathbb{R}}^d$ and ${\mathbb{T}}^d$ energy-supercritical setting. Via a newly developed unified scheme, we prove the unconditional uniqueness for solutions to NLS at critical regularity for all dimensions. Thus, together with [18,19], the unconditional uniqueness problems for $H^1$-critical and $H^1$-supercritical cubic and quintic NLS are completely and uniformly resolved at critical regularity for these domains. One application of our theorem is to prove that defocusing blowup solutions of the type in [54] is the only possible $C([0,T);{\dot{H}}^{s_c})$ solution if exist in these domains.