Global well-posedness and scattering for the mass critical nonlinear Schr{\"o}dinger equation with mass below the mass of the ground state
Global well-posedness and scattering for the mass critical nonlinear Schr{\"o}dinger equation with mass below the mass of the ground state
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DOI：
10.1016/j.aim.2015.04.030
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发表时间：
2011-04
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影响因子：
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通讯作者：
B. Dodson
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文献类型：
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作者：
B. Dodson
本文证明了d维质量临界非线性薛定谔初值问题对u 0∈ L2（Rd），<$u0 <$L2（Rd）<<$Q <$L2（Rd）是整体适定的和散射的，其中Q是基态，d≥ 1.我们首先建立了一个相互作用的Morawetz估计，当φ u 0 <$L2（Rd）<φ Q <$L2（Rd）时，该估计是正定的，并且具有适当的标度.接下来，我们将证明一个局限于低频的相互作用Morawetz估计，类似于[13]，[14]，[15]中的估计。另见[12]在能量临界情况下定位到高频。
In this paper we prove that the focusing, d-dimensional mass critical nonlinear Schrödinger initial value problem is globally well-posed and scattering for u 0∈ L 2 (R d),‖ u 0‖ L 2 (R d)<‖ Q‖ L 2 (R d), where Q is the ground state, and d≥ 1. We first establish an interaction Morawetz estimate that is positive definite when‖ u 0‖ L 2 (R d)<‖ Q‖ L 2 (R d), and has the appropriate scaling. Next, we will prove an interaction Morawetz estimate localized to low frequencies, similar to the estimates made in [13],[14],[15]. See also [12] for localization to high frequencies in the energy critical case.