### Global existence and uniqueness results for weak solutions of the focusing mass-critical non-linear Schr\"odinger equationGlobal existence and uniqueness results for weak solutions of the focusing mass-critical non-linear Schr\"odinger equation

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 Author Tao, Terence Source arXiv.org Content type Text File Format PDF Date of Submission 2008-07-16 Language English
 Subject Domain (in DDC) Natural sciences & mathematics ♦ Mathematics Subject Keyword Mathematics - Analysis of PDEs ♦ 35Q30 ♦ math Abstract We consider the focusing mass-critical NLS $iu_t + \Delta u = - u ^{4/d} u$ in high dimensions $d \geq 4$, with initial data $u(0) = u_0$ having finite mass $M(u_0) = \int_{\R^d} u_0(x) ^2 dx < \infty$. It is well known that this problem admits unique (but not global) strong solutions in the Strichartz class $C^0_{t,\loc} L^2_x \cap L^2_{t,\loc} L^{2d/(d-2)}_x$, and also admits global (but not unique) weak solutions in $L^\infty_t L^2_x$. In this paper we introduce an intermediate class of solution, which we call a \emph{semi-Strichartz class solution}, for which one does have global existence and uniqueness in dimensions $d \geq 4$. In dimensions $d \geq 5$ 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 $t \mapsto M(u(t))$. Educational Use Research Learning Resource Type Article Page Count 25