The unconditional uniqueness for the energy-supercritical NLS

Abstract

We consider the cubic and quintic nonlinear Schrödinger 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.

Revised per the referee reports

References in corpus (4)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• Rigorous Derivation of the Gross-Pitaevskii Equation
• On the uniqueness of solutions to the Gross-Pitaevskii hierarchy
• On the unconditional uniqueness of solutions to the infinite radial Chern-Simons-Schrödinger hierarchy

Cited by in corpus (3)

• Well/Ill-posedness of the Boltzmann Equation with Soft Potential
• Well/ill-posedness bifurcation for the Boltzmann equation with constant collision kernel
• On the mean-field and semiclassical limit from quantum $N$-body dynamics