Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\mathbb{R}^2$
Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\mathbb{R}^2$
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DOI：
10.3934/dcds.2008.21.665
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发表时间：
2008-04
影响因子：
1.1
通讯作者：
J. Colliander;M. Keel;G. Staffilani;H. Takaoka;T. Tao
中科院分区：
文献类型：
--
作者：
J. Colliander;M. Keel;G. Staffilani;H. Takaoka;T. Tao
研究了三次离焦非线性薛定谔方程$i\partial_t u +\Delta u =| u|对于初始数据在H ^s（\mathbb {R}^2）中的情形，证明了平面上的^2 u $是全局适定的，只要s> 1/2 $.同样的结果也适用于类似的聚焦问题，只要初始数据的质量小于基态的质量。证明依赖于一个几乎守恒量构造使用多线性校正项。主要的新困难是控制共振相互作用对这些修正项的贡献。共振相互作用是显着的，由于多维设置的问题和一些正交性问题出现。
The initial value problem for the cubic defocusing nonlinear Schrodinger equation $i \partial_t u + \Delta u = |u|^2 u$ on theplane is shown to be globally well-posed for initial data in $H^s (\mathbb{R}^2)$ provided $s>1/2$. The same result holds true for theanalogous focusing problem provided the mass of the initial data issmaller than the mass of the ground state. The proof relies upon analmost conserved quantity constructed using multilinear correctionterms. The main new difficulty is to control the contribution ofresonant interactions to these correction terms. The resonantinteractions are significant due to the multidimensional setting ofthe problem and some orthogonality issues which arise.