arXiv subjects

Changhun Yang

Publications and source records attributed to Changhun Yang.

At least 19 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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arXiv2026

Modified Scattering for the Time-Dependent Kohn--Sham Equation

We study the long-time behavior of the (critical) Kohn--Sham equation in two and three dimensions, i.e.,\[ \mathrm{i} \partial_t {\gamma} = \Big[-\frac{1}{2}\Delta + \lambda \, |\cdot|^{-1} \ast \rho_{{\gamma}} + \mu \, \rho_{{\gamma}}^{1/d}, {\gamma} \Big] \quad \text{for} \quad d=2,3. \] By introducing a suitable ''square root'' of the density matrix and exploiting the pseudo-conformal transform, we establish global well-posedness for small initial data in an appropriate weighted Schatten norm. We also prove the optimal time decay of the particle density and establish modified scattering for small and localized solutions. In particular, our results provide a resolution to the open problems proposed by Pusateri and Sigal (2021) for the critical and subcritical regimes, rigorously proving their conjectures regarding modified scattering in the critical case and linear scattering in the subcritical cases. Our results place these scattering phenomena in the operator-valued setting of density matrices, thereby extending the classical scalar theory to a broader framework.

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arXiv2026

Mapping properties of the $S$-operator

In this paper, we study the $\ell^p\to \ell^r$ estimates for the $S$-operator arising in restriction problems for spheres over finite fields. We establish a necessary and sufficient condition for the boundedness of the $S$-operator. Furthermore, we investigate this problem under certain restrictions on test functions. In particular, we address the sharp results when test functions are restricted to radial functions.

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arXiv2025

On the dispersive estimates for the discrete Schr\"odinger equation on a honeycomb lattice

The discrete Schr\"odinger equation on a two-dimensional honeycomb lattice is a fundamental tight-binding approximation model that describes the propagation of waves on graphene. For free evolution, we first show that the degenerate frequencies of the dispersion relation are completely characterized by three symmetric periodic curves (Theorem 2.1), and that the three curves meet at Dirac points where conical singularities appear (see Figure 2.1). Based on this observation, we prove the $L^1\to L^\infty$ dispersion estimates for the linear flow depending on the frequency localization (Theorem 2.3). Collecting all, we obtain the dispersion estimate with $O(|t|^{-2/3})$ decay as well as Strichartz estimates. As an application, we prove small data scattering for a nonlinear model (Theorem 2.10). The proof of the key dispersion estimates is based on the associated oscillatory integral estimates with degenerate phases and conical singularities at Dirac points. Our proof is direct and uses only elementary methods.

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arXiv2025

Periodic FPU system: Continuum limit to KdV via regularization and Fourier analysis

The Fermi-Pasta-Ulam (FPU) system, initially introduced by Fermi for numerical simulations, models vibrating chains with fixed endpoints, where particles interact weakly, nonlinearly with their nearest neighbors. Contrary to the anticipated ergodic behavior, the simulation revealed nearly periodic (quasi-periodic) motion of the solutions, a phenomenon later referred to as the FPU paradox. A partial but remarkable explanation was provided by Zabusky and Kruskal [36], who formally derived the continuum limit of the FPU system, connecting it to the Korteweg-de Vries (KdV) equation. This formal derivation was later rigorously justified by Bambusi and Ponno [4]. In this paper, we revisit the problem studied in [4], specifically focusing on the continuum limit of the periodic FPU system for a broader class of initial data, as the number of particles N tends to infinity within a fixed domain. Unlike the non-periodic case discussed in [15], periodic FPU solutions lack a (local) smoothing effect, posing a significant challenge in controlling one derivative in the nonlinearity. This control is crucial not only for proving the (uniform in N) well-posedness for rough data but also for deriving the continuum limit. The main strategies to resolve this issue involve deriving L4-Strichartz estimates for FPU solutions, analogous to those previously derived for KdV solutions in [7], and regularizing the system via the normal form method introduced in [1].

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arXiv2025

On the Benjamin-Bona-Mahony regularization of the Korteweg-de Vries equation

The Benjamin-Bona-Mahony equation (BBM) is introduced as a regularization of the Korteweg-de Vries equation (KdV) for long water waves \cite{BBM1972}. In this paper, we establish the convergence from the BBM to the KdV for energy class solutions. As a consequence, employing the conservation laws, we extend the known temporal interval of validity for the BBM regularization.

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arXiv2024

On the Korteweg-de Vries limit for the Boussinesq equation

The Korteweg-de Vries (KdV) equation is known as a universal equation describing various long waves in dispersive systems. In this article, we prove that in a certain scaling regime, a large class of rough solutions to the Boussinesq equation are approximated by the sums of two counter-propagating waves solving the KdV equations. It extends the earlier result by \cite{Schneider1998} to slightly more regular than $L^2$-solutions. Our proof is based on robust Fourier analysis methods developed for the low regularity theory of nonlinear dispersive equations.

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arXiv2023

Scattering for 2d semi-relativistic Hartree equations with short range potential

We study the long time behavior of small solutions to the semi-relativistic Hartree equations in two dimension. The nonlinear term is convolved with the singular potential $|x|^{-\gamma}$ for $1<\gamma<2$, which is referred to as short-range interaction potential in the sense of scattering phenomenon. We establish the scattering results for small solutions in a weighted space, in other words, we prove that the nonlinear solutions exist globally and behave asymptotically like a linear solution whenever the initial data is sufficiently small. To achieve this, we should obtain time decay estimates for the nonlinear term which is integrable. A key observation is that the loss in time in the course of weighted energy estimates can be recovered by the space resonance method with the help of null structure.

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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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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

Strichartz estimates for higher-order Schrödinger equations and their applications

In this paper, we consider the higher-order linear Schrödinger equations, that is, a formal finite Taylor expansion of the linear pseudo-relativistic equation. We establish the global-in-time Strichartz estimates for these higher-order equations which hold uniformly in the speed of light. As nonlinear applications, we show that the higher-order Hartree(-Fock) equation approximates the corresponding pseudo-relativistic equation on an arbitrarily long time interval, with higher accuracy than the non-relativistic equation. We also prove small data scattering for the higher-order nonlinear Schrödinger equations.

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arXiv2022

On the continuum limit for the discrete Nonlinear Schrödinger equation on a large finite cubic lattice

In this study, we consider the nonlinear Schödinger equation (NLS) with the zero-boundary condition on a two- or three-dimensional large finite cubic lattice. We prove that its solution converges to that of the NLS on the entire Euclidean space with simultaneous reduction in the lattice distance and expansion of the domain. Moreover, we obtain a precise global-in-time bound for the rate of convergence. Our proof heavily relies on Strichartz estimates on a finite lattice. A key observation is that, compared to the case of a lattice with a fixed size [Y. Hong, C. Kwak, S. Nakamura, and C. Yang, \emph{Finite difference scheme for two-dimensional periodic nonlinear {S}chrödinger equations}, Journal of Evolution Equations \textbf{21} (2021), no.~1, 391--418.], the loss of regularity in Strichartz estimates can be reduced as the domain expands, depending on the speed of expansion. This allows us to address the physically important three-dimensional case.

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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].

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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.

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arXiv2019

Critical well-posedness and scattering results for fractional Hartree-type equations

Scattering for the mass-critical fractional Schrödinger equation with a cubic Hartree-type nonlinearity for initial data in a small ball in the scale-invariant space of three-dimensional radial and square-integrable initial data is established. For this, we prove a bilinear estimate for free solutions and extend it to perturbations of bounded quadratic variation. This result is shown to be sharp by proving the unboundedness of a third order derivative of the flow map in the super-critical range.

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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.

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arXiv2018

Scattering results for Dirac Hartree-type equations with small initial data

We consider the Dirac equation with cubic Hartree-type nonlinearity derived by uncoupling the Dirac-Klein-Gordon systems. We prove small data scattering result in full subcritical range. Main ingredients of the proof are the localized Strichartz estimates, improved bilinear estimates thanks to null-structure hidden in Dirac operator and $Up,Vp$ function spaces. We apply the projection operator and get a system which of linear part is the Klein-Gordon type. It enables us to exploit the null-structures in equation. This result is shown to be almost optimal by showing that iteration method based on Duhamel's formula over supercritical range fails.

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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.

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