论文标题
具有三阶导数非线性的四阶schrödinger方程的适合度
Well-posedness for the fourth-order Schrödinger equation with third order derivative nonlinearities
论文作者
论文摘要
我们研究了半线性四阶schrödinger方程的库奇问题：\ begin {equation} \ label {0-1} \ tag {4nls} \ begin u+\ \ partial_x^4u = g \ left（\ left \ {\ partial_x^{k} u \ right \} _ {k \leγ}，\ left \ left \ {\ partial_x^{ x\in \mathbb{R}, \\ \ \ \ u|_{t=0}=u_0\in H^s(\mathbb{R}), \end{cases} \end{equation} where $γ\in \{1,2,3\}$ and the unknown function $u=u(t,x)$ is complex valued.在本文中，我们考虑了多项式的非线性$ g $ [ g（z）= g（z_1，\ cdots，z_ {2（γ+1）}） ：= \ sum_ {m \ le |α| \ le l}c_αz^α，\ \]对于$ z \ in \ mathbb {c}^{2（γ+1）} $，其中$ m，l \ in \ in \ in \ mathbb {n} $带有$ 3 \ le m \ le l l $和$c_α} $α\ in（\ mathbb {n} \ cup \ {0 \}）^{2（γ+1）} $是一个常数。本文的目的是在低阶Sobolev Space $ H^S（\ Mathbb {r}）$或比以前的结果更一般的非线性证明问题（\ ref {0-1}）的适应性（\ ref {0-1}）。我们的主要结果证明是基于D. Pornnopparath（2018）采用的合适功能空间的收缩映射原理。为了获得关键的线性和双线性估计，我们构建了I. Bejenaru，A。D. Ionescu，C。E. Kenig和D. Tataru（2011）引入的Duhamel术语的合适分解。此外，我们讨论了全球解决方案的散射以及针对我们适当性结果的规律性的最佳性，即，我们证明在某些情况下流量图不是平滑的。
We study the Cauchy problem to the semilinear fourth-order Schrödinger equations: \begin{equation}\label{0-1}\tag{4NLS} \begin{cases} i\partial_t u+\partial_x^4u=G\left(\left\{\partial_x^{k}u\right\}_{k\le γ},\left\{\partial_x^{k}\bar{u}\right\}_{k\le γ}\right), & t>0,\ x\in \mathbb{R}, \\ \ \ \ u|_{t=0}=u_0\in H^s(\mathbb{R}), \end{cases} \end{equation} where $γ\in \{1,2,3\}$ and the unknown function $u=u(t,x)$ is complex valued. In this paper, we consider the nonlinearity $G$ of the polynomial \[ G(z)=G(z_1,\cdots,z_{2(γ+1)}) :=\sum_{m\le |α|\le l}C_αz^α, \] for $z\in \mathbb{C}^{2(γ+1)}$, where $m,l\in\mathbb{N}$ with $3\le m\le l$ and $C_α\in \mathbb{C}$ with $α\in (\mathbb{N}\cup \{0\})^{2(γ+1)}$ is a constant. The purpose of the present paper is to prove well-posedness of the problem (\ref{0-1}) in the lower order Sobolev space $H^s(\mathbb{R})$ 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 D. Pornnopparath (2018). To obtain the key linear and bilinear estimates, we construct a suitable decomposition of the Duhamel term introduced by I. Bejenaru, A. D. Ionescu, C. E. Kenig, and D. Tataru (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.
