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Kiyeon Lee

Publications and source records attributed to Kiyeon Lee.

18 recordsLinked to original sources
arXiv2026

Long-range scattering for 2D Dirac-Hartree equations

We investigate the long-time behavior of small solutions to the Dirac-Hartree equation in two spatial dimensions. This model describes the mean-field dynamics of relativistic fermions interacting through the three-dimensional Coulomb potential $|x|^{-1}$, which gives rise to long-range effects in the scattering dynamics. We prove global well-posedness and long-range scattering (modified scattering) for small initial data in weighted Sobolev spaces. In this setting, long-range scattering means that, unlike linear scattering, an additional logarithmic phase correction is required to describe the precise asymptotics of nonlinear solutions. Our approach relies on the space-time resonance method, combined with special null structures inherent in the equation. Compared to the three-dimensional case \cite{CKLY2022,cloos}, the novelty lies in overcoming the weaker time decay inherent to the two dimensional problem.

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arXiv2025

Global solution and asymptotic behavior for the kinetic derivative NLS on $\mathbb R$

In this paper we investigate the global well-posedness and long-term behavior of solutions to the kinetic derivative nonlinear Schr\"odinger equation (KDNLS) on the real line. The equation incorporates both local cubic nonlinearities with derivative terms and a non-local term arising from the Hilbert transform, modeling interactions in plasma physics. We establish global existence for small initial data in the weighted Sobolev space $H^2 \cap H^{1,1}$ and optimal time decay effect. Using energy methods and a frequency-localized gauge transformation, we overcome the difficulties posed by the non-local nonlinearities and provide a rigorous analysis of the asymptotic behavior. Our results also describe modified scattering phenomena with a suitable phase modification, showing that the solutions exhibit a precise asymptotic profile as $t \to \infty$.

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arXiv2025

Local and global well-posedness for the kinetic derivative NLS on $\mathbb{R}$

We investigate the local and global well-posedness of the kinetic derivative nonlinear Schr\"odinger equation (KDNLS) on $\mathbb{R}$, described by \[ i\partial_t u + \partial_x^2 u = i\alpha \partial_x (|u|^2 u) + i\beta \partial_x (H(|u|^2) u), \] where $\alpha, \beta \in \mathbb{R}$, and $H$ represents the Hilbert transformation. For KDNLS, the $L^2$ norm of a solution is decreasing (resp. increasing, conserved) when $\beta$ is negative (resp. positive, zero). Focusing on the Sobolev spaces $H^2$ and $H^2 \cap H^{1,1}$, we establish local well-posedness via the energy method combined with gauge transformations to address resonant interactions in both cases of negative and positive $\beta$. For the dissipative case $\beta < 0$, we further demonstrate global well-posedness by deriving an a priori bound in $H^2$.

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arXiv2024

The global dynamics for the Maxwell-Dirac system

In this paper, we study the (1+3) dimensional massive Maxwell-Dirac system in the context of global existence and asymptotic behavior of solutions under the Lorenz gauge condition, as well as the modified and linear scattering phenomena for the Dirac spinor and the electromagnetic potential, respectively. We employ a vector fields energy method combined with a detailed analysis of the space-time resonance argument. This approach allows us to establish decay estimates and energy bounds crucial for proving the main theorems. Especially, we provide the explicit phase correction arising from the strong nonlinear resonances.

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arXiv2023

Scattering results for the (1+4) dimensional massive Maxwell-Dirac system under Lorenz gauge condition

This paper investigates the \emph{massive} Maxwell-Dirac system under the Lorenz gauge condition in (4+1) dimensional Minkowski space. The focus is on establishing global existence and scattering results for small solutions on the weighted Sobolev class. The imposition of the Lorenz gauge condition transforms the Maxwell-Dirac system into a set of Dirac equations coupled with an electromagnetic potential derived from five quadratic wave equations. To achieve a comprehensive understanding of the global solution and its behavior, we employ various energy estimates based on the space-time resonance argument. This involves addressing diverse resonance functions arising from the free Dirac and wave propagators. Additionally, we identify the space-time resonant sets associated with the \emph{massive} Maxwell-Dirac system.

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arXiv2023

The modified scattering of 2 dimensional semi-relativistic Hartree equations

In this paper, we consider the asymptotic behaviors of small solutions to the semi-relativistic Hartree equations in two dimension. The nonlinear term is convolved with the Coulomb potential 1/|x|, and it produces the long-range interaction in the sense of scattering phenomenon. From this observation, one anticipates that small solutions converge to a modified scattering states, although they decay as linear solutions. We show the global well-posedness and the modified scattering for small solutions in weighted Sobolev spaces. Our proof follows a road map of exploiting the space-time resonance developed by Germain, Masmoudi, and Shatah. Compared to the result in three dimensional case by Pusateri, weaker time decay in two dimension is one of the main obstacles.

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arXiv2023

Energy solutions for the fifth-order modified Korteweg de-Vries equations

We consider the Cauchy problem for the fifth-order modified Korteweg-de Vries equation (mKdV) under the periodic boundary condition. The fifth-order mKdV is an asymptotic model for shallow surface waves, and (in the perspective of integrable systems) the second equation in the mKdV hierarchy as well. In strong contrast with the non-periodic case, periodic solutions for dispersive equations do not have a (local) smoothing effect, and this observation becomes a major obstacle to considering the Cauchy problem for dispersive equations under the periodic condition, consequently, the periodic fifth-order mKdV shows a quasilinear phenomenon, while the non-periodic case can be considered as a semilinear equation. In this paper, we mainly establish the global well-posedness of the fifth-order mKdV in the energy space ($H^2(\mathbb T)$), which is an improvement of the former result by the first author (2018). The main idea to overcome the lack of (local) smoothing effect is to introduce a suitable (frequency dependent) short-time space originally motivated by the work by Ionescu, Kenig, and Tataru (2008). The new idea is to combine the (frequency) localized modified energy with additional weight in the spaces, which eventually handles the logarithmic divergence appearing in the energy estimates. Moreover, by using examples localized in low and very high frequencies, we show that the flow map of the fifth-order mKdV equation is not $C^3$, which implies that the Picard iterative method is not available for the local theory. This weakly concludes the quasilinear phenomenon of the periodic fifth-order mKdV.

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arXiv2023

Cauchy problem for Dirac equations with Chern-Simons-Proca gauge field

In this paper, we consider the Cauchy problem of Dirac equations with Chern-Simons-Proca (CSP) gauge field. We investigate global well-posedness and scattering theory for the solutions with small initial data. The main difficulties come from the fact that Strichartz estimate does not work and an absence of spinorial null structure which disturbs to show the global existence. To overcome these obstacles, we employ the space-time resonance argument introduced by Germain-Masmoudi-Shatah (2009, 2012, 2012), as well as various resonance functions derived from Dirac operators and the normal form approach. Our argument enables us to establish the global in time existence of solutions to Dirac equations with the CSP gauge field. As a byproduct of our argument, we obtain the scattering results.

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arXiv2022

Almost Optimal Local Well-Posedness of the Chern-Simons-Dirac System in the Coulomb Gauge

In this paper, we study the local well-posedness of Chern-Simons-Dirac system in the Coulomb gauge for initial data in $H^s(\mathbb R^2)$ for $s>0$. The novelty of this paper is to prove almost critical regularity by using the bilinear estimates of wave type localized in a thickened null cone, given by \cite{selb} via null structure. We also prove the Dirac spinor flow of Chern-Simons-Dirac system cannot be $C^3$ at the origin in $H^s$ if $s<0$.

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arXiv2022

The modified scattering for Dirac equations of scattering-critical nonlinearity

In this paper, we consider the Maxwell-Dirac system in 3 dimension under zero magnetic field. We prove the global well-posedness and modified scattering for small solutions in the weighted Sobolev class. Imposing the Lorenz gauge condition, (and taking the Dirac projection operator), it becomes a system of Dirac equations with Hartree type nonlinearity with a long range potential as $|x|^{-1} $. We perform the weighted energy estimates. In this procedure, we have to deal with various resonance functions that stem from the Dirac projections. We use the spacetime resonance argument of Germain-Masmoudi-Shatah, as well as the spinorial null-structure. On the way, we recognize a long range interaction which is responsible for a logarithmic phase correction in the modified scattering statement.

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arXiv2022

Conditional large-data global well-posedness of Dirac equation with Hartree-type nonlinearity

We study the Cauchy problems for the Hartree-type nonlinear Dirac equations with Yukawa-type potential in two and three spatial dimensions. This paper improves our previous results \cite{chohlee,cholee}; we establish global well-posedness and scattering for large data with a certain condition. Firstly we investigate the long-time behavior of solutions to the Dirac equation satisfies good control provided that a particular dispersive norm of solutions is bounded. The key of our proof relies on modifying multilinear estimates obtained in our previous papers. Secondly, we obtain large data scattering by exploiting the Majorana condition.

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arXiv2021

Small data scattering of 2d Hartree type Dirac equations

In this paper, we study the Cauchy problem of 2d Dirac equation with Hartree type nonlinearity $c(|\cdot|^{-γ} * \langle ψ, βψ\rangle)βψ$ with $c\in \mathbb R\setminus\{0\} $, $0 < γ< 2$. Our aim is to show the small data global well-posedness and scattering in $H^s$ for $s > γ-1$ and $1 < γ< 2$. The difficulty stems from the singularity of the low-frequency part $|ξ|^{-(2-γ)}χ_{\{|ξ|\le 1\}}$ of potential. To overcome it we adapt $U^p-V^p$ space argument and bilinear estimates of \cite{yang, tes2d} arising from the null structure. We also provide nonexistence result for scattering in the long-range case $0 < γ\le 1$.

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arXiv2021

Scattering and non-scattering of the Hartree-type nonlinear Dirac system at critical regularity

We consider Cauchy problem of the Hartree-type nonlinear Dirac equation with potentials given by $V_b(x) = \frac1{4π}\frac{e^{-b|x|}}{|x|}\, (b \ge 0)$. In previous works, a standard argument is to utilise null form estimates in order to prove global well-posedness for $H^s$-data, $s>0$. However, the null structure inside the equations is not enough to attain the critical regularity. We impose an extra regularity assumption with respect to the angular variable. Firstly, we prove global well-posedness and scattering of Dirac equations with Hartree-type nonlinearity for $b>0$ for small $L^2_x$-data with additional angular regularity. We also show that only small amount of angular regularity is required to obtain global existence of solutions. Secondly, we obtain non-scattering result for a certain class of solutions with the Coulomb potential $b=0$.

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arXiv2020

On the focusing energy-critical inhomogeneous NLS: weighted space approach

This paper is concerned with the global well-posedness and finite time blowup problem for the 3D focusing energy-critical inhomogeneous NLS. In the previous results \cite{chkl2, chkl3} the authors considered the same problems with the spatial inhomogeneity coefficient $g$ such that $g(x) \sim |x|^{-b}$ for $0 \le b < \frac43$. Here we extend the inhomogeneous index $b$ up to $\frac32$. For this purpose, we improve the local theory and develop a new profile decomposition based on weighted space.

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arXiv2019

On the GWP of focusing energy-ciritical inhomogeneous NLS

We consider the focussing energy-critical inhomogeneous nonlinear Schrödinger equation: $$ iu_t + Δu + g|u|^2u = 0, u(0)= φ\in \dot{H}^1,\;\; 0 \le g_i \le |x|g \le g_s.$$ On the road map of Kenig-Merle \cite{km} we show the global well-posedness and scattering of radial solutions under energy condition $$E_g(φ) < E_g(Q),\;\;\mbox{and}\;\; g_s\|φ\|_{\dot{H}^1}^2 < \|Q\|_{\dot{H}^1}^2,$$ where $Q$ is the solution of $ΔQ + |x|^{-1}Q^3 = 0$, together with scaling condition $|g(x)| + |x||\nabla g(x)| \lesssim |x|^{-1}$, variational condition $g_s(2-g_i) \le 1$, and rigidity condition $-g(x) \le x\cdot \nabla g(x)$. We also provide sharp finite time blowup results for nonradial and radial solutions. For this we utilize the localized virial identity.

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arXiv2019

Small data scattering of the inhomogeneous cubic-quintic NLS in 2 dimensions

The aim of this paper is to show the small data scattering for 2D ICQNLS: $$iu_t=-Δu + K_1(x)|u|^2u+K_2(x)|u|^4u.$$ Under the assumption that $\left| \partial^j K_l \right| \lesssim |x|^{b_l -j}$ for $j=0, 1, 2, l=1, 2$ and $0 \le b_l \le l - \frac23$, we prove the small data scattering in an angularly regular Sobolev space $H_θ^{1,1}$. We use the decaying property of angularly regular functions, which are defined as functions in Sobolev space $H_θ^{1, 1} \subset H^1$ with angular regularity such that $\|\partial_θf\|_{H^1} < \infty$, and also use the recently developed angularly averaged Strichartz estimates \cite{stri2, cholee, ghn}. In addition, we suggest a sufficient condition for non-existence of scattering.

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