Global existence and uniqueness results for weak solutions of the focusing mass-critical nonlinear Schrödinger equation

Abstract

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, 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)).

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