Well-posedness for the fourth-order Schr\"odinger equation with third order derivative nonlinearities
Well-posedness for the fourth-order Schr\"odinger equation with third order derivative nonlinearities
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具有三阶导数非线性的四阶薛定格方程的适定性
DOI：
10.1007/s00030-021-00707-6
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发表时间：
2021
期刊：
影响因子：
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通讯作者：
Tanaka Tomoyuki
中科院分区：
文献类型：
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作者：
Hirayama Hiroyuki;Ikeda Masahiro;Tanaka Tomoyuki
研究了一类半线性四阶Schr dinger方程i ∈ tu+ n x4 u= G n xkuk≤ γ，n xkuk ≤ γ，t> 0，x∈ R，u的Cauchy问题|t= 0= u 0∈ Hs（R），（4 NLS）其中γ∈{1，2，3}，未知函数u= u（t，x）是复值函数.本文考虑多项式G（z）= G（z1，...，z2（γ+ 1））：=∑ m≤| α| ≤ lC α z α，其中m，l∈ N且3≤ m≤ l，C α∈ C且α∈（N <${0}）2（γ+ 1）为常数.本文的目的是证明问题（4 NLS）在低阶Sobolev空间HS（R）中的适定性，或者比以前的结果具有更一般的非线性.我们对主要结果的证明是基于Pornopparath在一个合适的函数空间上使用的压缩映射原理（J Differ Equ，265：3792-3840，2018）。为了获得关键的线性和双线性估计，我们构建了一个合适的分解的Duhamel项介绍Bejenaru等人。(Ann Math 173：1443-1506，2011）。此外，我们还讨论了整体解的分散性和我们的适定性结果的正则性的最优性，即我们证明了流映射在几种情况下是不光滑的。
We study the Cauchy problem to the semilinear fourth-order Schrödinger equations: i∂ tu+∂ x 4 u= G∂ xkuk≤ γ,∂ xku¯ k≤ γ, t> 0, x∈ R, u| t= 0= u 0∈ H s (R),(4 NLS) where γ∈{1, 2, 3} and the unknown function u= u (t, x) is complex valued. In this paper, we consider the nonlinearity G of the polynomial G (z)= G (z 1,…, z 2 (γ+ 1)):=∑ m≤| α|≤ l C α z α, for z∈ C 2 (γ+ 1), where m, l∈ N with 3≤ m≤ l and C α∈ C with α∈(N∪{0}) 2 (γ+ 1) is a constant. The purpose of the present paper is to prove well-posedness of the problem (4NLS) in the lower order Sobolev space H s (R) or with more general nonlinearities than previous results. Our proof of the main results is based on the contraction mapping principle on a suitable function space employed by Pornnopparath (J Differ Equ, 265: 3792–3840, 2018). To obtain the key linear and bilinear estimates, we construct a suitable decomposition of the Duhamel term introduced by Bejenaru et al.(Ann Math 173: 1443–1506, 2011). Moreover we discuss scattering of global solutions and the optimality for the regularity of our well-posedness results, namely we prove that the flow map is not smooth in several cases.