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Yoonjung Lee

Publications and source records attributed to Yoonjung Lee.

12 recordsLinked to original sources
arXiv2025

The 3D energy-critical inhomogeneous nonlinear Schrodinger equation with strong singularity

In this paper, we study the Cauchy problem for the 3D energy-critical inhomogeneous nonlinear Schrödinger equation(INLS) $$i\partial_{t}u+Δu=\pm|x|^{-α}|u|^{4-2α}u$$ with strong singularity $3/2\leq α<2$. The well-posedness problem is well-understood for $0<α<3/2$, but the case $3/2\leq α<2$ has remained open so far. We address the local/small data global well-posedness result for $3/2\leq α<11/6$ by improving the inhomogeneous Strichartz estimates on the weighted space.

math.AP↗
arXiv2024

Global smooth solutions to the irrotational Euler-Riesz system in three dimensions

This paper investigates the global dynamics of the Euler--Riesz system in three dimensions, focusing on the well-posedness and large-time behavior of solutions near equilibrium. The system generalizes classical interactions by incorporating the Riesz interactions $\nabla (-Δ)^{-σ/2}(ρ- 1)$. We show that the system admits a global smooth solution for small irrotational initial perturbations. Specifically, we establish that if the initial data is sufficiently small, the solution remains regular globally in time and decays over time at a rate dependent on $σ$.

math.AP↗
arXiv2024

The global Cauchy problem for the Euler-Riesz equations

We completely resolve the global Cauchy problem for the multi-dimensional Euler-Riesz equations, where the interaction forcing is given by $\nabla (-Δ)^{-σ/2}ρ$ for some $σ\in (0,2)$. We construct the global-in-time unique solution to the Euler-Riesz system in a $H^s$ Sobolev space under a smallness assumption on the initial density and a dispersive spectral condition on the initial velocity. Moreover, we investigate the algebraic time decay of convergences for the constructed solutions. Our results cover the both attractive and repulsive cases as well as the whole regime $σ\in (0,2)$.

math.AP↗
arXiv2023

Sharp weighted Strichartz estimates and critical inhomogeneous Hartree equations

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.

math.AP↗
arXiv2023

Damped Euler system with attractive Riesz interaction forces

We consider the barotropic Euler equations with pairwise attractive Riesz interactions and linear velocity damping in the periodic domain. We establish the global-in-time well-posedness theory for the system near an equilibrium state. We also analyze the large-time behavior of solutions showing the exponential rate of convergence toward the equilibrium state as time goes to infinity.

math.AP↗
arXiv2023

On local well-posedness of nonlinear dispersive equations with partially regular data

We revisit the local well-posedness theory of nonlinear Schrödinger and wave equations in Sobolev spaces $H^s$ and $\dot{H}^s$, $0< s\leq 1$. The theory has been well established over the past few decades under Sobolev initial data regular with respect to all spatial variables. But here, we reveal that the initial data do not need to have complete regularity like Sobolev spaces, but only partially regularity with respect to some variables is sufficient. To develop such a new theory, we suggest a refined Strichartz estimate which has a different norm for each spatial variable. This makes it possible to extract a different integrability/regularity of the data from each variable.

math.AP↗
arXiv2022

Endpoint Strichartz estimates with angular integrability and some applications

The endpoint Strichartz estimate $\|e^{itΔ} f\|_{L_t^2 L_x^\infty} \lesssim \|f\|_{L^2}$ is known to be false in two space dimensions. Taking averages spherically on the polar coordinates $x=ρω$, $ρ>0$, $ω\in\mathbb{S}^1$, Tao showed a substitute of the form $\|e^{itΔ} f\|_{L_t^2L_ρ^\infty L_ω^2} \lesssim \|f\|_{L^2}$. Here we address a weighted version of such spherically averaged estimates. As an application, the existence of solutions for the inhomogeneous nonlinear Schrödinger equation is shown for $L^2$ data.

math.AP↗
arXiv2021

The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation

In this paper we study the Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation $i\partial_{t}u+Δu=λ|x|^{-α}|u|^βu$ in $H^1$. The well-posedness theory in $H^1$ has been intensively studied in recent years, but the currently known approaches do not work for the critical case $β=(4-2α)/(n-2)$. It is still an open problem. The main contribution of this paper is to develop the theory in this case.

math.AP↗
arXiv2021

On the integrability of the wave propagator arising from the Liouville-von Neumann equation

The Liouville-von Neumann equation describes the change in the density matrix with time. Interestingly, this equation was recently regarded as a wave equation for wave functions but not a equation for density functions. This setting leads to an extended form of the Schrödinger wave equation governing the motion of a quantum particle. In this paper we obtain the integrability of the wave propagator arising from the Liouville-von Neumann equation in this setting.

math.AP↗
arXiv2021

On well-posedness for the inhomogeneous nonlinear Schrödinger equation in the critical case

In this paper we study the well-posedness for the inhomogeneous nonlinear Schrödinger equation $i\partial_{t}u+Δu=λ|x|^{-α}|u|^βu$ in Sobolev spaces $H^s$, $s\geq0$. The well-posedness theory for this model has been intensively studied in recent years, but much less is understood compared to the classical NLS model where $α=0$. The conventional approach does not work particularly for the critical cases $β=\frac{4-2α}{d-2s}$. It is still an open problem. The main contribution of this paper is to develop the well-posedness theory in this critical case (as well as non-critical cases). To this end, we approach to the matter in a new way based on a weighted $L^p$ setting which seems to be more suitable to perform a finer analysis for this model. This is because it makes it possible to handle the singularity $|x|^{-α}$ in the nonlinearity more effectively. This observation is a core of our approach that covers the critical case successfully.

math.AP↗