CPAA
On Fractional Schrödinger Equations in sobolev spaces
Younghun Hong Yannick Sire
Let $\sigma \in (0,1)$ with $\sigma \neq \frac{1}{2}$. We investigate the fractional nonlinear Schrödinger equation in $\mathbb R^d$: \begin{eqnarray} i\partial_tu+(-\Delta)^\sigma u+\mu|u|^{p-1}u=0, u(0)=u_0\in H^s, \end{eqnarray} where $(-\Delta)^\sigma$ is the Fourier multiplier of symbol $|\xi|^{2\sigma}$, and $\mu=\pm 1$. This model has been introduced by Laskin in quantum physics [23]. We establish local well-posedness and ill-posedness in Sobolev spaces for power-type nonlinearities.
keywords: Sobolev spaces ill-posedness. Fractional Schrödinger Local and global well-posedness
CPAA
Scattering for a nonlinear Schrödinger equation with a potential
Younghun Hong
We consider a 3d cubic focusing nonlinear Schrödinger equation with a potential $$i\partial_t u+\Delta u-Vu+|u|^2u=0,$$ where $V$ is a real-valued short-range potential having a small negative part. We find criteria for global well-posedness analogous to the homogeneous case $V=0$ [10, 5]. Moreover, by the concentration-compactness approach, we prove that if $V$ is repulsive, such global solutions scatter.
keywords: scattering Nonlinear Schrödinger equation potential perturbation.
DCDS
Strichartz estimates for $N$-body Schrödinger operators with small potential interactions
Younghun Hong

In this paper, we prove Strichartz estimates for $N$-body Schrödinger operators, provided that interaction potentials are small enough. Our tools are new Strichartz estimates with frozen spatial variables, and its improvement in the $V_S^p$-norm of Koch and Tataru [19]. As an application, we prove scattering for $N$-body Schrödinger operators.

keywords: Many-body Schrödinger equation Strichartz estimates scattering

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