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Myeongju Chae

Publications and source records attributed to Myeongju Chae.

18 recordsLinked to original sources
arXiv2026

Uniform-in-time propagation of chaos and bifurcation in two-type adhesion systems

We study a nonlocal adhesion model for two interacting tumor cell phenotypes, combining diffusion, pairwise interactions, and random phenotypic switching. The system admits a microscopic diffusion--jump particle description whose mean-field limit is a nonlinear McKean--Vlasov equation on a product space encoding position and internal state. We first establish uniform-in-time propagation of chaos in the weak-interaction regime using a coupling approach that combines reflection coupling for the diffusion with an optimal coupling of the spin-flip dynamics. As a byproduct, we obtain exponential long-time contraction for the nonlinear McKean--Vlasov equation in the first-order Wasserstein distance, implying uniqueness of the stationary distribution. We also investigate the complementary regime of strong interactions, where the homogeneous equilibrium may lose stability through a bifurcation mechanism.

math.AP↗
arXiv2025

Nonlinear instability of rolls in the 2-dimensional generalized Swift-Hohenberg equation

Within the framework developed in \cite{Gr, JLL, RT1}, we rigorously establish the nonlinear instability of roll solutions to the two-dimensional generalized Swift-Hohenberg equation (gSHE). Our analysis is based on spectral information near the maximally unstable Bloch mode, combined with precise semigroup estimates. We construct a certain class of small initial perturbations that grow in time and cause the solution to deviate from the underlying roll solution within a finite time. This result provides a clear transition from spectral to nonlinear instability in a genuinely two-dimensional setting, where the Bloch parameter $σ$ ranges over an unbounded domain.

math.AP↗
arXiv2023

The Stochastic Schwarz lemma on Kähler Manifolds by Couplings and Its Applications

We first provide a stochastic formula for the Carathéodory distance in terms of general Markovian couplings and prove a comparison result between the Carathéodory distance and the complete Kähler metric with a negative lower curvature bound using the Kendall-Cranston coupling. This probabilistic approach gives a version of the Schwarz lemma on complete non-compact Kähler manifolds with a further decomposition Ricci curvature into the orthogonal Ricci curvature and the holomorphic sectional curvature, which cannot be obtained by using Yau--Royden's Schwarz lemma. We also prove coupling estimates on quaternionic Kähler manifolds. As a byproduct, we obtain an improved gradient estimate of positive harmonic functions on Kähler manifolds and quaternionic Kähler manifolds under lower curvature bounds.

math.DG↗
arXiv2021

Nonlocal adhesion models for two cancer cell phenotypes in a multidimensional bounded domain

Cell-cell adhesion is an inherently nonlocal phenomenon. Numerous partial differential equation models with nonlocal term have been recently presented to describe this phenomenon, yet the mathematical properties of nonlocal adhesion model are not well understood. Here we consider a model with two kinds of nonlocal cell-cell adhesion, satisfying no-flux conditions in a multidimensional bounded domain. We show global-in-time well-posedness of the solution to this model and obtain the uniform boundedness of solution.

math.AP↗
arXiv2019

Global Well-posedness and Long Time Behaviors of Chemotaxis-Fluid System Modeling Coral Fertilization

We consider generalized models on coral broadcast spawning phenomena involving diffusion, advection, chemotaxis, and reactions when egg and sperm densities are different. We prove the global-in-time existence of the regular solutions of the models as well as their temporal decays in two and three dimensions. We also show that the total masses of egg and sperm density have positive lower bounds as time tends to infinity in three dimensions.

math.AP↗
arXiv2019

Nonlinear stability of planar traveling waves in a chemotaxis model of tumor angiogenesis with chemical diffusion

We consider a simplified chemotaxis model of tumor angiogenesis, described by a Keller-Segel system on the two dimensional infinite cylindrical domain $(x, y) \in \mathbb{R} \times {\mathbf S^λ}$, where $ \mathbf S^λ$ is the circle of perimeter $λ>0$. The domain models a virtual channel where newly generated blood vessels toward the vascular endothelial growth factor will be located. The system is known to allow planar traveling wave solutions of an invading type. In this paper, we establish the nonlinear stability of these traveling invading waves when chemical diffusion is present if $λ$ is sufficiently small. The same result for the corresponding system in one-dimension was obtained by Li-Li-Wang (2014) [16]. Our result solves the problem remained open in [3] at which only linear stability of the waves was obtained under certain artificial assumption.

math.AP↗
arXiv2016

Stability of planar traveling waves in a Keller-Segel equation on an infinite strip domain

A simplified model of the tumor angiogenesis can be described by a Keller-Segel equation \cite{FrTe,Le,Pe}. The stability of traveling waves for the one dimensional system has recently been known by \cite{JinLiWa,LiWa}. In this paper we consider the equation on the two dimensional domain $ (x, y) \in \mathbf R \times {\mathbf S^λ}$ for a small parameter $λ>0$ where $ \mathbf S^λ$ is the circle of perimeter $λ$. Then the equation allows a planar traveling wave solution of invading types. We establish the nonlinear stability of the traveling wave solution if the initial perturbation is sufficiently small in a weighted Sobolev space without a chemical diffusion. When the diffusion is present, we show a linear stability. Lastly, we prove that any solution with our front conditions eventually becomes planar under certain regularity conditions. The key ideas are to use the Cole-Hopf transformation and to apply the Poincaré inequality to handle with the two dimensional structure.

math.AP↗
arXiv2015

Small data global existence and decay for relativistic Chern-Simons equations

We establish a general small data global existence and decay theorem for Chern-Simons theories with a general gauge group, coupled with a massive relativistic field of spin 0 or 1/2. Our result applies to a wide range of relativistic Chern-Simons theories considered in the literature, including the abelian/non-abelian self-dual Chern-Simons-Higgs equation and the Chern-Simons-Dirac equation. A key idea is to develop and employ a gauge invariant vector field method for relativistic Chern-Simons theories, which allows us to avoid the long range effect of charge.

math.AP↗
arXiv2015

Global Well-posedness of the Chemotaxis-Navier-Stokes Equations in two dimensions

We consider two dimensional Keller-Segel equations coupled with the Navier-Stokes equations modelled by Tuval et al.[32]. Assuming that the chemotactic sensitivity and oxygen consumption rate are nondecreasing and differentiable, we prove that there is no blow-up in a finite time for solutions with large initial data to chemotaxis-Navier-Stokes equations in two dimensions. In addition, temporal decays of solutions are shown, as time tends to infinity.

math.AP↗
arXiv2014

The stability of nonlinear Schrödinger equations with a potential in high Sobolev norms revisited

We consider the nonlinear Schrödinger equations with a potential on $\mathbb T^d$. For almost all potentials, we show the almost global stability in very high Sobolev norms. We apply an iteration of the Birkhoff normal form, as in the formulation introduced by Bourgain \cite{Bo00}. This result reprove a dynamical consequence of the infinite dimensional Birkhoff normal form theorem by Bambusi and Grebert \cite{BG}

math.AP↗
arXiv2013

Global existence and temporal decay in Keller-Segel models coupled to fluid equations

We consider a Keller-Segel model coupled to the incompressible Navier-Stokes equations in spatial dimensions two and three. We establish the local existence of regular solutions and present some blow-up criteria for both cases that equations of oxygen concentration is of parabolic or hyperbolic type. We also prove global existence and decay estimate in time under the some smallness conditions of initial data.

math.AP↗
arXiv2012

Existence of Smooth Solutions to Coupled Chemotaxis-Fluid Equations

We consider a system coupling the parabolic-parabolic Keller-Segel equations to the in- compressible Navier-Stokes equations in spatial dimensions two and three. We establish the local existence of regular solutions and present some blow-up criteria. For two dimensional Navier-Stokes-Keller-Segel equations, regular solutions constructed locally in time are, in reality, extended globally under some assumptions pertinent to experimental observation in [20] on the consumption rate and chemotactic sensitivity. We also show the existence of global weak solutions in spatially three dimensions with rather restrictive consumption rate and chemotactic sensitivity.

math.AP↗
arXiv2009

Mass concentration for the $L^2$-critical Nonlinear Schrödinger equations of higher orders

We consider the mass concentration phenomenon for the $L^2$-critical nonlinear Schrödinger equations of higher orders. We show that any solution $u$ to $iu_{t} + (-Δ)^{\fracα2} u =\pm |u|^\frac{2α}{d}u$, $u(0,\cdot)\in L^2$ for $α>2$, which blows up in a finite time, satisfies a mass concentration phenomenon near the blow-up time. We verify that as $α$ increases, the size of region capturing a mass concentration gets wider due to the stronger dispersive effect.

math.AP↗
arXiv2003

On the dynamics of Gowdy space times

We study the behavior near the singularity t=0 of Gowdy metrics. We prove existence of an open dense set of boundary points near which the solution is smoothly "asymptotically velocity term dominated" (AVTD). We show that the set of AVTD solutions satisfying a uniformity condition is open in the set of all solutions. We analyse in detail the asymptotic behavior of "power law" solutions at the (hitherto unchartered) points at which the asymptotic velocity equals zero or one. Several other related results are established.

gr-qc↗