arXiv subjects

Younghun Hong

Publications and source records attributed to Younghun Hong.

At least 37 records · Page 2Linked to original sources

arXiv2021

Orbital stability for the mass-critical and supercritical pseudo-relativistic nonlinear Schrodinger equation

For the one-dimensional mass-critical/supercritical pseudo-relativistic nonlinear Schrodinger equation, a stationary solution can be constructed as an energy minimizer under an additional kinetic energy constraint and the set of energy minimizers is orbitally stable in \cite{BGV}. In this study, we proved the local uniqueness and established the orbital stability of the solitary wave by improving that of the energy minimizer set. A key aspect thereof is the reformulation of the variational problem in the non-relativistic regime, which we consider to be more natural because the proof extensively relies on the subcritical nature of the limiting model. Thus, the role of the additional constraint is clarified, a more suitable Gagliardo-Nirenberg inequality is introduced, and the non-relativistic limit is proved. Subsequently, this limit is employed to derive the local uniqueness and orbital stability.

math.AP↗

arXiv2021

Semi-classical limit of quantum free energy minimizers for the gravitational Hartree equation

For the gravitational Vlasov-Poisson equation, Guo and Rein constructed a class of classical isotropic states as minimizers of free energies (or energy-Casimir functionals) under mass constraints. For the quantum counterpart, that is, the gravitational Hartree equation, isotropic states are constructed as free energy minimizers by Aki, Dolbeault and Sparber. In this paper, we are concerned with the correspondence between quantum and classical isotropic states. Precisely, we prove that as the Planck constant $\hbar$ goes to zero, free energy minimizers for the Hartree equation converge to those for the Vlasov-Poisson equation in terms of potential functions as well as via the Husimi transform and the Töplitz quantization.

math.AP↗

arXiv2019

Korteweg--de Vries limit for the Fermi--Pasta--Ulam system

In this paper, we develop dispersive PDE techniques for the Fermi--Pasta--Ulam (FPU) system with infinitely many oscillators, and we show that general solutions to the infinite FPU system can be approximated by counter-propagating waves governed by the Korteweg--de Vries (KdV) equation as the lattice spacing approaches zero. Our result not only simplifies the hypotheses but also reduces the regularity requirement in the previous study [45].

math.AP↗

arXiv2019

Finite difference scheme for two-dimensional periodic nonlinear Schrödinger equations

A nonlinear Schrödinger equation (NLS) on a periodic box can be discretized as a discrete nonlinear Schrödinger equation (DNLS) on a periodic cubic lattice, which is a system of finitely many ordinary differential equations. We show that in two spatial dimensions, solutions to the DNLS converge strongly in $L^2$ to those of the NLS as the grid size $h>0$ approaches zero. As a result, the effectiveness of the finite difference method (FDM) is justified for the two-dimensional periodic NLS.

math.AP↗

arXiv2018

Convergence rate of the finite element approximation for extremizers of Sobolev inequalities

In this paper, we are concerned with the convergence rate of a FEM based numerical scheme approximating extremal functions of the Sobolev inequality. We prove that when the domain is polygonal and convex in $\R^2$, the convergence of a finite element solution to an exact extremal function in $L^2$ and $H^1$ norms has the rates $O(h^2)$ and $O(h)$ respectively, where $h$ denotes the mesh size of a triangulation of the domain.

math.NA↗

arXiv2018

Strong Convergence for Discrete Nonlinear Schrödinger equations in the Continuum Limit

We consider discrete nonlinear Schrödinger equations (DNLS) on the lattice $h\mathbb{Z}^d$ whose linear part is determined by the discrete Laplacian which accounts only for nearest neighbor interactions, or by its fractional power. We show that in the continuum limit $h\to 0$, solutions to DNLS converge strongly in $L^2$ to those to the corresponding continuum equations, but a precise rate of convergence is also calculated. In particular cases, this result improves weak convergence in Kirkpatrick, Lenzmann and Staffilani \cite{KLS}. Our proof is based on a suitable adjustment of dispersive PDE techniques to a discrete setting. Notably, we employ uniform-in-$h$ Strichartz estimates for discrete linear Schrödinger equations in \cite{HY}, which quantitatively measure dispersive phenomena on the lattice. Our approach could be adapted to a more general setting like \cite{KLS} as long as the desired Strichartz estimates are obtained.

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

Global existence versus finite time blowup dichotomy for the system of nonlinear Schrödinger equations

We construct an extremizer for the kinetic energy inequality (except the endpoint cases) developing the concentration-compactness technique for operator valued inequality in the formulation of the profile decomposition. Moreover, we investigate the properties of the extremizer, such as the system of Euler-Lagrange equations, regularity and summability. As an application, we study a dynamical consequence of a system of nonlinear Schrödinger equations with focusing cubic nonlinearities in three dimension when each wave function is restricted to be orthogonal. Using the critical element of the kinetic energy inequality, we establish a global existence versus finite time blowup dichotomy. This result extends the single particle result of Holmer and Roudenko to infinitely many particles system.

math.AP↗

arXiv2017

Semilinear elliptic equations with the pseudo-relativistic operator on a bounded domain

We study the Dirichlet problem for the semilinear equations involving the pseudo-relativistic operator on a bounded domain, (\sqrt{-Δ+ m^2} - m)u =|u|^{p-1}u \quad \textrm{in}~Ω, with the Dirichlet boundary condition $u=0$ on $\partial Ω$. Here, $p \in (1,\infty)$ and the operator $(\sqrt{-Δ+ m^2} - m)$ is defined in terms of spectral decomposition. In this paper, we investigate existence and nonexistence of a nontrivial solution, depending on the choice of $p$, $m$ and $Ω$. Precisely, we show that $(i)$ if $p$ is not $H^1$ subcritical ($p \geq \frac{n+2}{n-2}$) and $Ω$ is star-shaped, the equation has no nontrivial solution for all $m > 0$; $(ii)$ if $p$ is not $H^{1/2}$ supercritical ($1 0$ and any bounded domain $Ω$; $(iii)$ finally, in the intermediate range ($\frac{n+1}{n-1}<p<\frac{n+2}{n-2}$), the problem has a nontrivial solution, provided that $m$ is sufficiently large and the problem -Δu = |u|^{p-1}u \quad \textrm{in}~Ω, \qquad u =0\quad \textrm{on}~\partial Ωadmits a non-degenerate nontrivial solution, for example, when $Ω$ is a ball or an annulus.

math.AP↗

arXiv2017

Uniqueness and symmetry of ground states for higher-order equations

We establish uniqueness and radial symmetry of ground states for higher-order nonlinear Schrödinger and Hartree equations whose higher-order differentials have small coefficients. As an application, we obtain error estimates for higher-order approximations to the pseudo-relativistic ground state. Our proof adapts the strategy of Lenzmann using local uniqueness near the limit of ground states in a variational problem. However, in order to bypass difficulties from lack of symmetrization tools for higher-order differential operators, we employ the contraction mapping argument in our earlier work to construct radially symmetric real-valued solutions, as well as improving local uniqueness near the limit.

math.AP↗

arXiv2017

On the Strichartz estimates for orthonormal systems of initial data with regularity

The classical Strichartz estimates for the free Schrödinger propagator have recently been substantially generalised to estimates of the form \[ \bigg\|\sum_jλ_j|e^{itΔ}f_j|^2\bigg\|_{L^p_tL^q_x}\lesssim\|λ\|_{\ell^α} \] for orthonormal systems $(f_j)_j$ of initial data in $L^2$, firstly in work of Frank--Lewin--Lieb--Seiringer and later by Frank--Sabin. The primary objective is identifying the largest possible $α$ as a function of $p$ and $q$, and in contrast to the classical case, for such estimates the critical case turns out to be $(p,q) = (\frac{d+1}{d},\frac{d+1}{d-1})$. We consider the case of orthonormal systems $(f_j)_j$ in the homogeneous Sobolev spaces $\dot{H}^s$ for $s \in (0,\frac{d}{2})$ and we establish the sharp value of $α$ as a function of $p$, $q$ and $s$, except possibly an endpoint in certain cases, at which we establish some weak-type estimates. Furthermore, at the critical case $(p,q) = (\frac{d+1}{d-2s},\frac{d(d+1)}{(d-1)(d-2s)})$ for general $s$, we show the veracity of the desired estimates when $α= p$ if we consider frequency localised estimates, and the failure of the (non-localised) estimates when $α= p$; this exhibits the difficulty of upgrading from frequency localised estimates in this context, again in contrast to the classical setting.

math.FA↗

arXiv2017

On critical and supercritical pseudo-relativistic nonlinear Schrödinger equations

In this paper, we investigate existence and non-existence of a nontrivial solution to the pseudo-relativistic nonlinear Schrödinger equation $$\left( \sqrt{-c^2Δ+ m^2 c^4}-mc^2\right) u + μu = |u|^{p-1}u\quad \textrm{in}~\mathbb{R}^n~(n \geq 2)$$ involving an $H^{1/2}$-critical/supercritical power-type nonlinearity, i.e., $p \geq \frac{n+1}{n-1}$. We prove that in the non-relativistic regime, there exists a nontrivial solution provided that the nonlinearity is $H^{1/2}$-critical/supercritical but it is $H^1$-subcritical. On the other hand, we also show that there is no nontrivial bounded solution either $(i)$ if the nonlinearity is $H^{1/2}$-critical/supercritical in the ultra-relativistic regime or $(ii)$ if the nonlinearity is $H^1$-critical/supercritical in all cases.

math.AP↗

arXiv2017

On the scattering problem for infinitely many fermions in dimensions $d\geq3$ at positive temperature

In this paper, we study the dynamics of a system of infinitely many fermions in dimensions $d\geq3$ near thermal equilibrium and prove scattering in the case of small perturbation around equilibrium in a certain generalized Sobolev space of density operators. This work is a continuation of our previous paper, and extends the important recent result of M. Lewin and J. Sabin of a similar type for dimension d=2. In the work at hand, we establish new, improved Strichartz estimates that allow us to control the case $d\geq3$.

math.AP↗

arXiv2016

Optimal convergence rate of nonrelativistic limit for the nonlinear pseudo-relativistic equations

In this paper, we are concerned with the nonrelativistic limit of the following pseudo-relativistic equation with Hartree nonlinearity or power type nonlinearity \[ \left(\sqrt{-\hbar^2c^2 Δ+m^2c^4} - mc^2 \right) u + μu = \mathcal{N}(u), \] where $c$ denotes the speed of light. We prove that the ground states of this equation converges to the ground state of its nonrelativistic counterpart \[ -\frac{\hbar^2}{2m}Δu + μu = \mathcal{N}(u) \] with an explicit convergence rate $1/c^2$ in arbitrary order as $c \to \infty$. Moreover, we show that this rate is optimal.

math.AP↗

arXiv2016

On dispersive blow-ups for the nonlinear Schrödinger equation

In this article, we provide a simple method for constructing dispersive blow-up solutions to the nonlinear Schrödinger equation. Our construction mainly follows the approach in Bona, Ponce, Saut and Sparber [2]. However, we make use of the dispersive estimate to enjoy the smoothing effect of the Schrödinger propagator in the integral term appearing in Duhamel's formula. In this way, not only do we simplify the argument, but we also reduce the regularity requirement to construct dispersive blow-ups. In addition, we provide more examples of dispersive blow-ups by constructing solutions that blow up on a straight line and on a sphere.

math.AP↗

arXiv2016

Global Well-posedness of the NLS System for infinitely many fermions

In this paper, we study the mean field quantum fluctuation dynamics for a system of infinitely many fermions with delta pair interactions in the vicinity of an equilibrium solution (the Fermi sea) at zero temperature, in dimensions $d=2,3$, and prove global well-posedness of the corresponding Cauchy problem. Our work extends some of the recent important results obtained by M. Lewin and J. Sabin, who addressed this problem for more regular pair interactions.

math.AP↗

arXiv2015

A Remark on Littlewood-Paley projections

We establish the kernel estimates for the Littlewood-Paley projections associated with a Schrödinger operator H=-Δ+V in \mathbb{R}^3 for a large class of short-range potentials V(x). As a corollary, we prove the homogeneous Sobolev inequality.

math.AP↗

arXiv2015

A new class of Traveling Solitons for cubic Fractional Nonlinear Schrodinger equations

We consider the one-dimensional cubic fractional nonlinear Schrödinger equation $$i\partial_tu-(-Δ)^σu+|u|^{2}u=0,$$ where $σ\in (\frac12,1)$ and the operator $(-Δ)^σ$ is the fractional Laplacian of symbol $|ξ|^{2σ}$. Despite of lack of any Galilean-type invariance, we construct a new class of traveling soliton solutions of the form $$u(t,x)=e^{-it(|k|^{2σ}-ω^{2σ})}Q_{ω,k}(x-2tσ|k|^{2σ-2}k),\quad k\in\mathbb{R},\ ω>0$$ by a rather involved variational argument.

math.AP↗