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

Abstract

We consider the focusing mass-critical NLS $i{u}_t+\Delta u=-|u{|}^{4∕d}u$ in high dimensions $d\ge 4$, with initial data $u\left(0\right)=u_0$ having finite mass $M\left(u_0\right)={\int }_{{ℝ}^d}|u_0\left(x\right){|}^{2}dx<\infty$. It is well known that this problem admits unique (but not global) strong solutions in the Strichartz class $C_{t,loc}^0{L}_x^2\cap L_{t,loc}^2{L}_x^{2d∕\left(d-2\right)}$, and also admits global (but not unique) weak solutions in $L_t^{\infty }{L}_x^2$. 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 $d\ge 4$. In dimensions $d\ge 5$ 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 $t\mapsto M\left(u\left(t\right)\right)$.