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Kyudong Choi

Publications and source records attributed to Kyudong Choi.

30 records · Page 2Linked to original sources
arXiv2019

Stability of Lamb dipoles

The Lamb dipole is a traveling wave solution to the two-dimensional Euler equations introduced by S. A. Chaplygin (1903) and H. Lamb (1906) at the early 20th century. We prove orbital stability of this solution based on a vorticity method initiated by V. I. Arnold. Our method is a minimization of a penalized energy with multiple constraints that deduces existence and orbital stability for a family of traveling waves. As a typical case, orbital stability of the Lamb dipole is deduced by characterizing a set of minimizers as an orbit of the dipole by a uniqueness theorem in the variational setting.

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arXiv2019

Global well-posedness of large perturbations of traveling waves in a hyperbolic-parabolic system arising from a chemotaxis model

We consider a one-dimensional system arising from a chemotaxis model in tumour angiogenesis, which is described by a Keller-Segel equation with singular sensitivity. This hyperbolic-parabolic system is known to allow viscous shocks (so-called traveling waves), and in literature, their nonlinear stabilities have been considered in the class of certain mean-zero small perturbations. We show the global existence of the solution without assuming the mean-zero condition for any initial data as arbitrarily large perturbations around traveling waves in the Sobolev space $H^1$ while the shock strength is assumed to be small enough. The main novelty of this paper is to develop the global well-posedness of any large $H^1$-perturbations of traveling wave connecting two different end states. The discrepancy of the end states is linked to the complexity of the corresponding flux, which requires a new type of an energy estimate. To overcome, we use the a priori contraction estimate of a weighted relative entropy functional up to a translation, which was proved by Choi-Kang-Kwon-Vasseur. The boundedness of the shift implies a priori bound of the relative entropy functional without a shift on any time interval of existence, which produces a $H^1$-estimate thanks to a De Giorgi type lemma. Moreover, to remove possibility of vacuum appearance, we use the lemma again.

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arXiv2019

Contraction for large perturbations of traveling waves in a hyperbolic-parabolic system arising from a chemotaxis model

We consider a hyperbolic-parabolic system arising from a chemotaxis model in angiogenesis, which is described by a Keller-Segel equation with singular sensitivity. It is known to allow viscous shocks (so-called traveling waves). We introduce a relative entropy of the system, which can capture how close a solution at a given time is to a given shock wave in almost $L^2$-sense. When the shock strength is small enough, we show the functional is non-increasing in time for any large initial perturbation. The contraction property holds independently of the strength of the diffusion.

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arXiv2019

On the estimate of distance traveled by a particle in a disk-like vortex patch

We consider the incompressible two-dimensional Euler equation in the plane in the case when its initial vorticity is the characteristic function of a bounded open set. We show that the travel distance grows linearly for most of fluid particles initially placed on the set when the area of the symmetric difference between the set and a disk is small enough.

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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.

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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.

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arXiv2015

Short-time stability of scalar viscous shocks in the inviscid limit by the relative entropy method

We consider inviscid limits to shocks for viscous scalar conservation laws in one space dimension, with strict convex fluxes. We show that we can obtain sharp estimates in $L^2$, for a class of large perturbations and for any bounded time interval. Those perturbations can be chosen big enough to destroy the viscous layer. This shows that the fast convergence to the shock does not depend on the fine structure of the viscous layers. This is the first application of the relative entropy method developed in [22], [23] to the study of an inviscid limit to a shock.

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arXiv2013

Finite time blow up for a 1D model of 2D Boussinesq system

The 2D conservative Boussinesq system describes inviscid, incompressible, buoyant fluid flow in gravity field. The possibility of finite time blow up for solutions of this system is a classical problem of mathematical hydrodynamics. We consider a 1D model of 2D Boussinesq system motivated by a particular finite time blow up scenario. We prove that finite time blow up is possible for the solutions to the model system.

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arXiv2011

Persistence of Hölder continuity for non-local integro-differential equations

In this paper, we consider non-local integro-differential equations under certain natural assumptions on the kernel, and obtain persistence of Hölder continuity for their solutions. In other words, we prove that a solution stays in $C^β$ for all time if its initial data lies in $C^β$. This result has an application for a fully non-linear problem, which is used in the field of image processing. The proof is in the spirit of the paper [18] of Kiselev and Nazarov where they established Hölder continuity of the critical surface quasi-geostrophic (SQG) equation.

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arXiv2011

Estimates on fractional higher derivatives of weak solutions for the Navier-Stokes equations

We study weak solutions of the 3D Navier-Stokes equations in whole space with $L^2$ initial data. It will be proved that $\nabla^αu $ is locally integrable in space-time for any real $α$ such that $1< α<3$, which says that almost third derivative is locally integrable. Up to now, only second derivative $\nabla^2 u$ has been known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in weak-$L_{loc}^{4/(α+1)}$. These estimates depend only on the $L^2$ norm of initial data and integrating domains. Moreover, they are valid even for $α\geq 3$ as long as $u$ is smooth. The proof uses a good approximation of Navier-Stokes and a blow-up technique, which let us to focusing on a local study. For the local study, we use De Giorgi method with a new pressure decomposition. To handle non-locality of the fractional Laplacian, we will adopt some properties of the Hardy space and Maximal functions.

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