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Younghun Hong

Publications and source records attributed to Younghun Hong.

43 records · Page 3Linked to original sources
arXiv2015

A Spectral Multiplier Theorem associated with a Schrödinger Operator

We establish a spectral multiplier theorem associated with a Schrödinger operator H=-Δ+V(x) in \mathbb{R}^3. We present a new approach employing the Born series expansion for the resolvent. This approach provides an explicit integral representation for the difference between a spectral multiplier and a Fourier multiplier, and it allows us to treat a large class of Schrödinger operators without Gaussian heat kernel estimates. As an application to nonlinear PDEs, we show the local-in-time well-posedness of a 3d quintic nonlinear Schrödinger equation with a potential.

math.AP↗
arXiv2015

On Fractional Schrodinger Equations in sobolev spaces

Let $σ\in(0,1)$ with $σ\neq\frac{1}{2}$. We investigate the fractional nonlinear Schrödinger equation in $\mathbb R^d$: $$i\partial_tu+(-Δ)^σu+μ|u|^{p-1}u=0,\, u(0)=u_0\in H^s,$$ where $(-Δ)^σ$ is the Fourier multiplier of symbol $|ξ|^{2σ}$, and $μ=\pm 1$. This model has been introduced by Laskin in quantum physics \cite{laskin}. We establish local well-posedness and ill-posedness in Sobolev spaces for power-type nonlinearities.

math.AP↗
arXiv2014

Uniqueness of solutions to the 3D quintic Gross-Pitaevskii Hierarchy

In this paper, we study solutions to the three-dimensional quintic Gross-Pitaevskii hierarchy. We prove unconditional uniqueness among all small solutions in the critical space $\mathfrak{H}^1$ (which corresponds to $H^1$ on the NLS level). With slight modifications to the proof, we also prove unconditional uniqueness of solutions to the Hartree hierarchy without smallness condition. Our proof uses the quantum de Finetti theorem, and is an extension of the work by Chen-Hainzl-Pavlović-Seiringer \cite{CHPS}, and our previous work \cite{UniqueLowReg}.

math.AP↗
arXiv2014

Scattering for a Nonlinear Schrödinger Equation with a Potential

We consider a 3d cubic focusing nonlinear Schrödinger equation with a potential $$i\partial_t u+Δu-Vu+|u|^2u=0,$$ 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 $V=0$ (Duyckaerts-Holmer-Roudenko). Moreover, by the concentration-compactness approach, we prove that if $V$ is repulsive, such global solutions scatter.

math.AP↗
arXiv2014

Unconditional Uniqueness of the cubic Gross-Pitaevskii Hierarchy with Low Regularity

In this paper, we establish the unconditional uniqueness of solutions to the cubic Gross-Pitaevskii hierarchy on $\mathbb{R}^d$ in a low regularity Sobolev type space. More precisely, we reduce the regularity $s$ down to the currently known regularity requirement for unconditional uniqueness of solutions to the cubic nonlinear Schrödinger equation ($s\ge\frac{d}{6}$ if $d=1,2$ and $s>s_c=\frac{d-2}{2}$ if $d\ge 3$). In such a way, we extend the recent work of Chen-Hainzl-Pavlović-Seiringer.

math.AP↗
arXiv2013

Global well-posedness of NLS with a rough potential below the energy norm

We show that in the presence of a rough external potential, a 3d cubic defocusing NLS is globally well-posed in H^s for s>5/6. The proof is based on the approach of Colliander-Keel-Staffilani-Takaoka-Tao, called the I-method, but in order to deal with a rough potential, we modify harmonic analysis tools by spectral theory.

math.AP↗