arXiv subjects

Changhun Yang

Publications and source records attributed to Changhun Yang.

21 records · Page 2Linked to original sources

arXiv2018

Small data scattering of semirelativistic Hartree equation

In this paper we study the small data scattering of Hartree type semirelativistic equation in space dimension $3$. The Hartree type nonlinearity is $[V * |u|^2]u$ and the potential $V$ which generalizes the Yukawa has some growth condition. We show that the solution scatters to linear solution if an initial data given in $ H^{s,1}$ is sufficiently small and $s>\frac14$. Here, $H^{s, 1}$ is Sobolev type space taking in angular regularity with norm defined by $\|φ\|_{ H^{s, 1}} = \|φ\|_{ H^{s}} + \|\nabla_{\mathbb S} φ\|_{H^{s}}$. To establish the results we employ the recently developed Strichartz estimate which is $L_θ^2$-averaged on the unit sphere $\mathbb S^{2}$ and construct the resolution space based on $U^p$-$V^p$ space.

math.AP↗

arXiv2018

Uniform Strichartz estimates on the lattice

In this paper, we investigate Strichartz estimates for discrete linear Schrödinger and discrete linear Klein-Gordon equations on a lattice $h\mathbb{Z}^d$ with $h>0$, where $h$ is the distance between two adjacent lattice points. As for fixed $h>0$, Strichartz estimates for discrete Schrödinger and one-dimensional discrete Klein-Gordon equations are established by Stefanov-Kevrekidis \cite{SK2005}. Our main result shows that such inequalities hold uniformly in $h\in(0,1]$ with additional fractional derivatives on the right hand side. As an application, we obtain local well-posedness of a discrete nonlinear Schrödinger equation with a priori bounds independent of $h$. The theorems and the harmonic analysis tools developed in this paper would be useful in the study of the continuum limit $h\to 0$ for discrete models, including our forthcoming work \cite{HY} where strong convergence for a discrete nonlinear Schrödinger equation is addressed.

math.AP↗

arXiv2017

On the modified scattering of $3$-d Hartree type fractional Schrödinger equations with Coulomb potential

In this paper we study 3-d Hartree type fractional Schrödin-ger equations: \begin{equation} i\partial_{t}u-|\nabla|^αu = λ\left(|x|^{-γ} *| u|^{2} \right)u,\;\;1 < α< 2,\;\;0 < γ< 3,\;\; λ\in \mathbb R \setminus \{0\}. \end{equation} In \cite{cho} it is known that no scattering occurs in $L^2$ for the long range ($0 < γ\le 1$). In \cite{c0, chooz2, cho1} the short-range scattering ($1 < γ< 3$) was treated for the scattering in $H^s$. In this paper we consider the critical case ($γ= 1$) and prove a modified scattering in $L^\infty$ on the frequency to the Cauchy problem with small initial data. For this purpose we investigate the global behavior of $x e^{it\nabla} u$, $x^2 e^{it\nabla} u$ and $\langleξ\rangle^5 \widehat{e^{it\nabla} u}$. Due to the non-smoothness of $\nabla$ near zero frequency the range of $α$ is restricted to $(\frac{17}{10}, 2)$.

math.AP↗