Mathematics – Analysis of PDEs
Scientific paper
2008-07-16
Mathematics
Analysis of PDEs
25 pages, 1 table, to appear, Analysis & PDE. Typo in definition of semi-Strichartz solution corrected
Scientific paper
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 $t \mapsto M(u(t))$.
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