Poincaré-Dulac Normal Form Reduction for Unconditional Well-Posedness of the Periodic Cubic NLS
Poincaré-Dulac Normal Form Reduction for Unconditional Well-Posedness of the Periodic Cubic NLS
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周期性三次 NLS 无条件适定性的 Poincaré-Dulac 范式约简
DOI：
10.1007/s00220-013-1755-5
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发表时间：
2011
影响因子：
2.4
通讯作者：
Tadahiro Oh
中科院分区：
文献类型：
--
作者：
Zihua Guo;Soonsik Kwon;Tadahiro Oh
We implement an infinite iteration scheme of Poincaré-Dulac normal form reductions to establish an energy estimate on the one-dimensional cubic nonlinear Schrödinger equation (NLS) in CtL2(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C_tL^2(\mathbb{T})}$$\end{document}, without using any auxiliary function space. This allows us to construct weak solutions of NLS in CtL2(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C_tL^2(\mathbb{T})}$$\end{document} with initial data in L2(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^2(\mathbb{T})}$$\end{document} as limits of classical solutions. As a consequence of our construction, we also prove unconditional well-posedness of NLS in Hs(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H^s(\mathbb{T})}$$\end{document} for s≥16\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${s \geq \frac{1}{6}}$$\end{document}.
We implement an infinite iteration scheme of Poincaré-Dulac normal form reductions to establish an energy estimate on the one-dimensional cubic nonlinear Schrödinger equation (NLS) in CtL2(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C_tL^2(\mathbb{T})}$$\end{document}, without using any auxiliary function space. This allows us to construct weak solutions of NLS in CtL2(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${C_tL^2(\mathbb{T})}$$\end{document} with initial data in L2(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^2(\mathbb{T})}$$\end{document} as limits of classical solutions. As a consequence of our construction, we also prove unconditional well-posedness of NLS in Hs(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${H^s(\mathbb{T})}$$\end{document} for s≥16\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${s \geq \frac{1}{6}}$$\end{document}.