arXiv subjects

Gleison N. Santos

Publications and source records attributed to Gleison N. Santos.

2 recordsLinked to original sources

arXiv2018

On a class of solutions to the generalized derivative Schrödinger equations II

In this note we shall continue our study on the initial value problem associated for the generalized derivative Schrödinger (gDNLS) equation $$ \partial_tu=i\partial_x^2u + μ\,|u|^α\partial_x u, \hskip10pt x,t\in\mathbb{R}, \hskip5pt 0<α\le 1\;\; {\rm and}\;\; |μ|=1. $$ Inspiring by Cazenave-Naumkin's works we shall establish the local well-posedness for a class of data of arbitrary size in an appropriate weighted Sobolev space, thus removing the size restriction on the data required in our previous work. The main new tool in the proof is the homogeneous and inhomogeneous versions of the Kato smoothing effect for the linear Schrödinger equation with lower order variable coefficients established by Kenig-Ponce-Vega.

math.AP↗

arXiv2017

On a class of solutions to the generalized derivative Schrödinger equations

In this work we shall consider the initial value problem associated to the generalized derivative Schrödinger equations \begin{equation*} \p_tu=i\p_x^2u + μ\,|u|^{\a}\p_xu, \hskip10pt x,t\in\R, \hskip5pt 0<\a \le 1\;\, {\rm and}\;\, |μ|=1, \end{equation*} and \begin{equation*} \p_tu=i\p_x^2u + μ\,\p_x\big(|u|^{\a}u\big), \hskip10pt x,t\in\R, \hskip5pt 0<\a \le 1\;\, {\rm and}\;\, |μ|=1. \end{equation*} Following the argument introduced by Cazenave and Naumkin \cite{Cazenave} we shall establish the local well-posedness for a class of small data in an appropriate weighted Sobolev space. The other main tools in the proof include the homogeneous and inhomogeneous versions of the Kato smoothing effect for the linear Schrödinger equation established by Kenig-Ponce-Vega in \cite{KPV1}.

math.AP↗