Global existence and uniqueness results for weak solutions of the focusing mass-critical non-linear 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 $u(0)=u_0$ having finite mass $M(u_0)={\int }_{{\mathbb{R}}^d}|u_0(x){|}^{2}dx<\infty$. 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 $L_t^{\infty }{L}_x^2$. 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 $d\ge 4$. 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 $t\mapsto M(u(t))$.

25 pages, 1 table, to appear, Analysis & PDE. Typo in definition of semi-Strichartz solution corrected

References in corpus (3)

• The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher
• The cubic nonlinear Schrödinger equation in two dimensions with radial data
• A pseudoconformal compactification of the nonlinear Schrödinger equation and applications