m at h . A P ] 1 6 O ct 2 00 6 MINIMAL - MASS BLOWUP SOLUTIONS OF THE MASS - CRITICAL NLS
نویسنده
چکیده
We consider the minimal mass m0 required for solutions to the mass-critical nonlinear Schrödinger (NLS) equation iut + ∆u = μ|u|4/du to blow up. If m0 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 Lx(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, [17], in dimensions 1, 2 and Begout and Vargas, [2], in dimensions d ≥ 3 for the mass-critical NLS and by Kenig and Merle, [18], in the energy-critical case. In a subsequent paper we shall use this compactness result to establish global existence and scattering in Lx(R d) for the defocusing NLS in three and higher dimensions with spherically symmetric data.
منابع مشابه
Nonlinear Schrödinger Equations at Critical Regularity
1 2 ROWAN KILLIP AND MONICA VIS¸AN Contents 1. Introduction 3 1.1. Where are we? And how did we get there? 6 1.2. Notation 9 2. Symmetries 9 2.1. Hamiltonian formulation 9 2.2. The symmetries 10 2.3. Group therapy 13 2.4. Complete integrability 14 3. The local theory 15 3.1. Dispersive and Strichartz inequalities 15 3.2. The ˙ H s x critical case 16 3.3. Stability: the mass-critical case 20 3.4...
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