Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case
Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case
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DOI：
10.1007/s00222-006-0011-4
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发表时间：
2006-10
影响因子：
3.1
通讯作者：
C. Kenig;F. Merle
中科院分区：
文献类型：
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作者：
C. Kenig;F. Merle
我们证明，对于能量临界的聚焦NLS，对于能量小于驻波能量的数据，其齐次Sobolev范数H^1小于驻波的齐次Sobolev范数并且是径向的，我们在3维、4维和5维上具有整体适定性和散射。这是尖锐的，因为如果数据在非齐次索伯列夫空间H^1中，能量小于驻波，但具有较大的齐次H^1范数，我们就在有限时间内爆破。该结果来自我们引入此类关键问题的通用方法。通过集中紧性，我们产生了一个临界元素，它模方程的对称性是紧的，在那些没有我们定理的结论的元素中具有最小的能量。此外，我们证明了对于这个解，对称性中的伸缩参数可以取严格正的，然后我们建立了一个刚性定理，证明了这种紧的模对称对象不可能存在。只有在这一步，我们才使用径向假设，同样的分析，以简化的形式，也适用于散焦的情况，给Bourgain和Tao的结果一个新的证明。
We prove, for the energy critical, focusing NLS, that for data whose energy is smaller than that of the standing wave, and whose homogeneous Sobolev norm H^1 is smaller than that of the standing wave and which is radial, we have global well-posedness and scattering in dimensions 3, 4 and 5. This is sharp since if the data is in the inhomogeneous Sobolev space H^1, of energy smaller than the standing wave but of larger homogeneous H^1 norm, we have blow-up in finite time. The result follows from a general method that we introduce into this type of critical problem. By concentration-compactness we produce a critical element, which modulo the symmetries of the equation is compact, has minimal energy among those which fail to have the conclusion of our theorem. In addition, we show that the dilation parameter in the symmetry, for this solution, can be taken strictly positive.We then establish a rigidity theorem that shows that no such compact, modulo symmetries, object can exist. It is only at this step that we use the radial hypothesis.The same analysis, in a simplified form, applies also to the defocusing case, giving a new proof of results of Bourgain and Tao.