Minimal-mass blowup solutions of the mass-critical NLS

Abstract

We consider the minimal mass $m_0$ required for solutions to the mass-critical nonlinear Schrödinger (NLS) equation $i{u}_t+\Delta u=\mu |u{|}^{4/d}u$ to blow up. If $m_0$ is finite, we show that there exists a minimal-mass solution blowing up (in the sense of an infinite spacetime norm) in both time directions, whose orbit in $L_x^2({\mathbb{R}}^d)$ is compact after quotienting out by the symmetries of the equation. A similar result is obtained for spherically symmetric solutions. Similar results were previously obtained by Keraani, \cite{keraani}, in dimensions 1, 2 and Begout and Vargas, \cite{begout}, in dimensions $d\ge 3$ for the mass-critical NLS and by Kenig and Merle, \cite{merlekenig}, in the energy-critical case. In a subsequent paper we shall use this compactness result to establish global existence and scattering in $L_x^2({\mathbb{R}}^d)$ for the defocusing NLS in three and higher dimensions with spherically symmetric data.

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