#### Vol. 2, No. 1, 2009

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

### Terence Tao

Vol. 2 (2009), No. 1, 61–81
##### 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 $M\left({u}_{0}\right)={\int }_{{ℝ}^{d}}|{u}_{0}\left(x\right){|}^{2}\phantom{\rule{1em}{0ex}}dx<\infty$. 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 $t↦M\left(u\left(t\right)\right)$.

##### Keywords
Strichartz estimates, nonlinear Schrodinger equation, weak solutions, unconditional uniqueness
Primary: 35Q30