arXiv subjects
Publications and source records attributed to Seongyeon Kim.
We obtain weighted $L^2$ estimates for solutions to the Dirac equation with weights in the Kerman-Sawyer class. This class has been introduced as a necessary and sufficient condition for the boundedness of fractional integral operators in weighted $L^2$ spaces.
We study the energy-critical inhomogeneous Hartree equation in space dimensions three and higher. Previous local well-posedness results left open the parameter regime where the inhomogeneity exponent is small and the Riesz potential exponent is either small or large. We establish local well-posedness for a new range of parameters, thereby substantially filling the remaining open parameter regime. In particular, our result completely resolves the remaining gap in dimensions $5$ and $6$.
We establish Morawetz-type estimates for solutions to the elastic wave equation with singular weights of the form $|x|^{-α}$ or $|(x,t)|^{-α}$. In particular, we show that space-time weights $|(x,t)|^{-α}$ admit stronger singularities and require weaker regularity assumptions on the initial data compared to purely spatial weights $|x|^{-α}$.
We investigate the persistence of spatial analyticity for solutions to the Majda-Biello and Hirota-Satsuma systems with analytic initial data. This result is the first to establish analyticity persistence in such coupled KdV systems.
Under asymmetrical faults, analyzing the transient stability of grid-forming voltage-source converters (GFM-VSCs) becomes essential because their behavior fundamentally differs from that under symmetrical faults. When current limiting is activated under asymmetrical faults, the point-of-common-coupling voltage of a GFM-VSC contains both positive- and negative-sequence components, and the interaction between these components generates a non-negligible negative-sequence-driven active power. However, the transient stability of GFM-VSCs under asymmetrical faults has not been sufficiently investigated, and the influence of negative-sequence-driven active power remains unclear. Accordingly, this letter derives the P-δ curve of a GFM-VSC with an elliptical current limiter under asymmetrical faults by explicitly accounting for negative-sequence effects. This enables a more accurate transient stability assessment when extending conventional symmetrical-fault analyses to asymmetrical conditions. The theoretical analysis is validated by the agreement between the derived P-δ curve and both simulation and experimental results.
We study the long time behaviour of solutions for the weakly damped forced Kawahara equation on the torus. More precisely, we prove the existence of a global attractor in $L^2$, to which as time passes all solutions draw closer. In fact, we show that the global attractor turns out to lie in a smoother space $H^2$ and be bounded therein. Further, we give an upper bound of the size of the attractor in $H^2$ that depends only on the damping parameter and the norm of the forcing term.
We obtain the local well-posedness for Dirac equations with a Hartree type nonlinearity derived by decoupling the Dirac-Klein-Gordon system. We extend the function space of initial data, enabling us to handle initial data that were not addressed in previous studies.
In this paper, we investigate the dichotomous behavior of solutions to the Kawahara equation with bounded variation initial data, analogous to the Talbot effect. Specifically, we observe that the solution is quantized at rational times, whereas at irrational times, it is a nowhere continuous differentiable function with a fractal profile. This phenomenon, however, has not been explored for the Kawahara equation, which is a fifth-order KdV type equation. To achieve this, we derive smoothing estimates for the nonlinear Duhamel solution, which, when combined with the known results on the linear solution, provides a mathematical description of the Talbot effect.
In this paper, we consider two types of damped wave equations: the weakly damped equation and the strongly damped equation. We recover the initial velocity from the trace of the solution on a space-time cylinder. This inverse problem is related to Photoacoustic Tomography (PAT), a hybrid medical imaging technique. PAT is based on generating acoustic waves inside of an object of interest and one of the mathematical problem in PAT is reconstructing the initial velocity from the solution of the wave equation measured on the outside of object. Using the spherical harmonics and spectral theorem, we demonstrate a way to recover the initial velocity.
We investigate the behavior of the Schrödinger equation under the influence of potentials, focusing on its relationship to quantum revivals and fractality. Our findings reveal that the solution displays fractal behavior at irrational times, while exhibiting regularity similar to the initial data at rational times. These extend the results of Oskolkov \cite{O} and Rodnianski \cite{R2} on the free Schrödinger evolution to the general case regarding potentials.
We study the Cauchy problem for the inhomogeneous Hartree equation in this paper. Although its well-posedness theory has been extensively studied in recent years, much less is known compared to the classical Hartree model of homogeneous type. In particular, the problem of Sobolev initial data with the Sobolev critical index remains unsolved. The main contribution of this paper is to establish the local existence of solutions to the inhomogeneous equation in the critical cases. To do so, we obtain all possible $L^p$ Strichartz estimates with singular weights.
Adversarial training has been considered an imperative component for safely deploying neural network-based applications to the real world. To achieve stronger robustness, existing methods primarily focus on how to generate strong attacks by increasing the number of update steps, regularizing the models with the smoothed loss function, and injecting the randomness into the attack. Instead, we analyze the behavior of adversarial training through the lens of response frequency. We empirically discover that adversarial training causes neural networks to have low convergence to high-frequency information, resulting in highly oscillated predictions near each data. To learn high-frequency contents efficiently and effectively, we first prove that a universal phenomenon of frequency principle, i.e., \textit{lower frequencies are learned first}, still holds in adversarial training. Based on that, we propose phase-shifted adversarial training (PhaseAT) in which the model learns high-frequency components by shifting these frequencies to the low-frequency range where the fast convergence occurs. For evaluations, we conduct the experiments on CIFAR-10 and ImageNet with the adaptive attack carefully designed for reliable evaluation. Comprehensive results show that PhaseAT significantly improves the convergence for high-frequency information. This results in improved adversarial robustness by enabling the model to have smoothed predictions near each data.
This work studies the Cauchy problem for the energy-critical inhomogeneous Hartree equation with inverse square potential $$i\partial_t u-\mathcal K_λu=\pm |x|^{-τ}|u|^{p-2}(I_α*|\cdot|^{-τ}|u|^p)u, \quad \mathcal K_λ=-Δ+\fracλ{|x|^2}$$ in the energy space $H_λ^1:=\{f\in L^2,\quad\sqrt{\mathcal{K}_λ}f\in L^2\}$. In this paper, we develop a well-posedness theory and investigate the blow-up of solutions in $H_λ^1$. Furthermore we present a dichotomy between energy bounded and non-global existence of solutions under the ground state threshold. To this end, we use Caffarelli-Kohn-Nirenberg weighted interpolation inequalities and some equivalent norms considering $\mathcal K_λ$, which make it possible to control the non-linearity involving the singularity $|x|^{-τ}$ as well as the inverse square potential. The novelty here is the investigation of the energy critical regime which remains still open and the challenge is to deal with three technical problems: a non-local source term, an inhomogeneous singular term $|\cdot|^{-τ}$, and the presence of an inverse square potential.
We discuss the Lugiato-Lefever equation and its variant with third-order dispersion, which are mathematical models used to describe how a light beam forms patterns within an optical cavity. It is mathematically demonstrated that the solutions of these equations follow the Talbot effect, which is a phenomenon of periodic self-imaging of an object under certain conditions of diffraction. The Talbot effect is regarded as the underlying cause of pattern formation in optical cavities.
We study the well-posedness for the inhomogeneous Hartree equation $i\partial_t u + Δu = λ(I_α\ast |\cdot|^{-b}|u|^p)|x|^{-b}|u|^{p-2}u$ in $H^s$, $s\ge0$. Until recently, its well-posedness theory has been intensively studied, focusing on solving the problem for the critical index $p=1+\frac{2-2b+α}{n-2s}$ with $0\le s \le 1$, but the case $1/2\leq s \leq 1$ is still an open problem. In this paper, we develop the well-posedness theory in this case, especially including the energy-critical case. To this end, we approach to the matter based on the Sobolev-Lorentz space which can lead us to perform a finer analysis for this equation. This is because it makes it possible to control the nonlinearity involving the singularity $|x|^{-b}$ as well as the Riesz potential $I_α$ more effectively.
In this paper we obtain some new Strichartz estimates for the Dirac flow in the context of Wiener amalgam spaces which control the local regularity of a function and its decay at infinity separately unlike $L^p$ spaces. While it is well understood recently for some flows such as the Schrödinger and wave flows that work in the non-relativistic regime, nothing is known about the Dirac flow which governs a physical system in the case of relativistic fields.
We prove dispersive estimate for the elastic wave equation by which we extend the known Strichartz estimates for the classical wave equation to those for the elastic wave equation. In particular, the endpoint Strichartz estimates are deduced. For the purpose we diagonalize the symbols of the Lamé operator and its semigroup, which also gives an alternative and simpler proofs of the previous results on perturbed elastic wave equations. Furthermore, we obtain uniform Sobolev inequalities for the elastic wave operator.
We study pointwise convergence of the solution to the elastic wave equation to the initial data which lies in the Sobolev spaces. We prove that the solution converges along every lines to the initial data almost everywhere whenever the initial regularity is greater than one half. We show this is almost optimal.