Terence Tao Global Existence and Uniqueness Results for Weak Solutions of the Focusing Mass-critical Nonlinear Schrödinger Equation

We consider the focusing mass-critical NLS iu t + u = −|u| 4/d u in high dimensions d ≥ 4, with initial data u(0) = u 0 having finite mass M(u 0) = ‫ޒ‬ d |u 0 (x)| 2 d x < ∞. It is well known that this problem admits unique (but not global) strong solutions in the Strichartz class C 0 t,loc L 2 x ∩ L 2 t,loc L 2d/(d−2) x , and also admits global (but not unique) weak solutions in L ∞ t L 2 x. In this paper we introduce an intermediate class of solution, which we call a semi-Strichartz class solution, for which one does have global existence and uniqueness in dimensions d ≥ 4. In dimensions d ≥ 5 and assuming spherical symmetry, we also show the equivalence of the Strichartz class and the strong solution class (and also of the semi-Strichartz class and the semi-strong solution class), thus establishing unconditional uniqueness results in the strong and semi-strong classes. With these assumptions we also characterise these solutions in terms of the continuity properties of the mass function t → M(u(t)).