Journal article
ON THE RIGIDITY OF MINIMAL MASS SOLUTIONS TO THE FOCUSING MASS-CRITICAL NLS FOR ROUGH INITIAL DATA
Electronic journal of differential equations, Vol.2009(78), pp.1-19
06/16/2009
Abstract
For the focusing mass-critical nonlinear Schrodinger equation iu(t)+ Delta u = -|u|(4/d)u, an important problem is to establish Liouville type results for solutions with ground state mass. Here the ground state is the positive solution to elliptic equation Delta Q-Q+Q(1+4d) = 0. Previous results in this direction were established in [12, 16, 17, 29] and they all require u(0) is an element of H(x)(1)(R(d)). In this paper, we consider the rigidity results for rough initial data u(0) is an element of H(x)(s)(R(d)) for any s > 0. We show that in dimensions d >= 4 and under the radial assumption, the only solution that does not scatter in both time directions (including the finite time blowup case) must be global and coincide with the solitary wave e(it)Q up to symmetries of the equation. The proof relies on a non-uniform local iteration scheme, the refined estimate involving the P(+/-) operator and a new smoothing estimate for spherically symmetric solutions.
Details
- Title: Subtitle
- ON THE RIGIDITY OF MINIMAL MASS SOLUTIONS TO THE FOCUSING MASS-CRITICAL NLS FOR ROUGH INITIAL DATA
- Creators
- Dong Li - Inst Adv Study, Princeton, NJ 08544 USAXiaoyi Zhang - Inst Adv Study, Princeton, NJ 08544 USA
- Resource Type
- Journal article
- Publication Details
- Electronic journal of differential equations, Vol.2009(78), pp.1-19
- Publisher
- TEXAS STATE UNIV
- ISSN
- 1072-6691
- eISSN
- 1072-6691
- Number of pages
- 19
- Grant note
- 973 in China Mathematics Department of University of Iowa DMS-0635607; 10601060 / National Science Foundation
- Language
- English
- Date published
- 06/16/2009
- Academic Unit
- Mathematics
- Record Identifier
- 9984240862402771
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