Publications (41)
An aggregation equation with degenerate diffusion: qualitative property of solutions
Lincoln Chayes, Inwon Kim, Yao Yao
We study a nonlocal aggregation equation with degenerate diffusion, set in a periodic domain. This equation represents the generalization to of the McKean-Vlasov equation w…
Congested aggregation via Newtonian interaction
Katy Craig, Inwon Kim, Yao Yao
We consider a congested aggregation model that describes the evolution of a density through the competing effects of nonlocal Newtonian attraction and a hard height constraint. Thi…
On mean curvature flow with forcing
Inwon Kim, Dohyun Kwon
This paper investigates geometric properties and well-posedness of a mean curvature flow with volume-dependent forcing. With the class of forcing which bounds the volume of the evo…
A Hele-Shaw limit without monotonicity
Nestor Guillen, Inwon Kim, Antoine Mellet
We study the incompressible limit of the porous medium equation with a right hand side representing either a source or a sink term, and an injection boundary condition. This model…
The -contraction principle in optimal transport
Matt Jacobs, Inwon Kim, Jiajun Tong
In this work, we use the JKO scheme to approximate a general class of diffusion problems generated by Darcy's law. Although the scheme is now classical, if the energy density is sp…
Density-constrained Chemotaxis and Hele-Shaw flow
Inwon Kim, Antoine Mellet, Yijing Wu
We consider a model of congestion dynamics with chemotaxis, where the density of cells follows the chemical signal it generates, while observing an incompressibility constraint. We…
Global existence and uniqueness of solutions to a model of price formation
Lincoln Chayes, Maria del Mar Gonzalez, Maria Pia Gualdani +1
We study a model due to J.M. Lasry and P.L. Lions, describing the evolution of a scalar price which is realized as a free boundary in a 1-D diffusion equation with dynamically evol…
The supercooled Stefan problem: fractal freezing and the fine structure of maximal solutions
Raymond Chu, Inwon Kim, Sebastian Munoz
We study the supercooled Stefan problem in arbitrary dimensions. First, we study general solutions and their irregularities, showing generic fractal freezing and nucleation, based…
Incompressible limit of porous medium equation with bistable and monostable reaction terms
Inwon Kim, Antoine Mellet
We study the incompressible limit of the porous medium equation with a reaction term that is non-monotone with respect to the pressure variable. More specifically we consider react…
Darcy's Law with a Source term
Matt Jacobs, Inwon Kim, Jiajun Tong
We introduce a novel variant of the JKO scheme to approximate Darcy's law with a pressure dependent source term. By introducing a new variable that implicitly controls the source t…
The Nonlocal Stefan Problem via a Martingale Transport
Raymond Chu, Inwon Kim, Young-Heon Kim +1
We study the nonlocal Stefan problem, where the phase transition is described by a nonlocal diffusion as well as the change of enthalpy functions. By using a stochastic optimizatio…
Porous Medium Equation with A Drift: Free boundary Regularity
Inwon Kim, Yuming Paul Zhang
We study regularity properties of the free boundary for solutions of the porous medium equation with the presence of drift. We show the regularity of the free boundary,…
Liquid drops sliding down an inclined plane
Inwon Kim, Antoine Mellet
We investigate a one-dimensional model describing the motion of liquid drops sliding down an inclined plane (the so-called quasi-static approximation model). We prove existence and…
Mean Field Limit for Congestion Dynamics in One Dimension
Inwon Kim, Antoine Mellet, Jeremy Sheung-Him Wu
This paper addresses congested transport, which can be described, at macroscopic scales, by a continuity equation with a pressure variable generated from the hard-congestion constr…
On nonlinear cross-diffusion systems: an optimal transport approach
Inwon Kim, Alpár R. Mészáros
We study a nonlinear, degenerate cross-diffusion model which involves two densities with two different drift velocities. A general framework is introduced based on its gradient flo…
A drift approximation for parabolic PDEs with oblique boundary data
Damon Alexander, Inwon Kim
We consider solutions of a quasi-linear parabolic PDE with zero oblique boundary data in a bounded domain. Our main result states that the solutions can be approximated by solution…
Tumor Growth with Nutrients: Regularity and Stability
Matt Jacobs, Inwon Kim, Jiajun Tong
In this paper we study a tumor growth model with nutrients. The model presents dynamic patch solutions due to the contact inhibition among the tumor cells. We show that when the nu…
Well-posedness and Regularity for a Polyconvex Energy
Wilfrid Gangbo, Matt Jacobs, Inwon Kim
We prove the existence, uniqueness, and regularity of minimizers of a polyconvex functional in two and three dimensions, which corresponds to the projection of measure-preser…
Quantitative Homogenization of Elliptic PDE with Random Oscillatory Boundary Data
William M. Feldman, Inwon Kim, Panagiotis E. Souganidis
We study the averaging behavior of nonlinear uniformly elliptic partial differential equations with random Dirichlet or Neumann boundary data oscillating on a small scale. Under co…
Homogenization of oblique boundary value problems
Sunhi Choi, Inwon Kim
We consider a nonlinear Neumann problem, with periodic oscillation in the elliptic operator and on the boundary condition. Our focus is on problems posed in half-spaces, but with g…
Two phase Stefan-type problem: Regularization near initial data by phase dynamics
Sunhi Choi, Inwon Kim
We investigate the regularizing behavior of two-phase Stefan problem near initial data. The main step in the analysis is to establish that in any given scale, the scaled solution i…
Singular limit of the porous medium equation with a drift
Inwon Kim, Norbert Požár, Brent Woodhouse
We study the "stiff pressure limit" of a nonlinear drift-diffusion equation, where the density is constrained to stay below the maximal value one. The challenge lies in the presenc…
Quasi-static evolution and congested crowd transport
Damon Alexander, Inwon Kim, Yao Yao
We consider the relationship between Hele-Shaw evolution with drift, the porous medium equation with superharmonic drift, and a congested crowd motion model originally proposed by…
Regularity of Hele-Shaw Flow with source and drift
Inwon Kim, Yuming Paul Zhang
In this paper we study the regularity property of Hele-Shaw flow, where source and drift are present in the evolution. More specifically we consider Hölder continuous source and L…
Tumor growth with nutrients: stability of the tumor patches
Inwon Kim, Jona Lelmi
In this paper, we study a tumor growth model with nutrients. The contact inhibition for the tumor cells, presented in the model, results in the evolution of a congested tumor patch…
The Patlak-Keller-Segel model and its variations: properties of solutions via maximum principle
Inwon Kim, Yao Yao
In this paper we investigate qualitative and asymptotic behavior of solutions for a class of diffusion-aggregation equations. Most results except the ones in section 3 and 6 concer…
On volume-preserving crystalline mean curvature flow
Inwon Kim, Dohyun Kwon, Norbert Požár
In this work we consider the global existence of volume-preserving crystalline curvature flow in a non-convex setting. We show that a natural geometric property, associated with re…
Porous medium equation to Hele-Shaw flow with general initial density
Inwon Kim, Norbert Pozar
In this paper we study the "stiff pressure limit" of the porous medium equation, where the initial density is a bounded, integrable function with a sufficient decay at infinity. Ou…
Interface Dynamics in a Two-phase Tumor Growth Model
Inwon Kim, Jiajun Tong
We study a tumor growth model in two space dimensions, where proliferation of the tumor cells leads to expansion of the tumor domain and migration of surrounding normal tissues int…
Volume preserving mean curvature flow for star-shaped sets
Inwon Kim, Dohyun Kwon
We study the evolution of star-shaped sets in volume preserving mean curvature flow. Constructed by approximate minimizing movements, our solutions preserve a strong version of sta…
Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data
Sunhi Choi, Inwon Kim
In this article we investigate averaging properties of fully nonlinear PDEs in bounded domains with oscillatory Neumann boundary data. The oscillation is periodic and is present bo…
Quasistatic Droplet percolation
Nestor Guillen, Inwon Kim
We consider the Hele-Shaw problem in a randomly perforated domain with zero Neumann boundary conditions. A homogenization limit is obtained as the characteristic scale of the domai…
Regularity of the free boundary for the supercooled Stefan problem in arbitrary dimensions
Max Engelstein, Inwon Kim, Sebastian Munoz
We study the free boundary in the supercooled Stefan problem, a classical model for the solidification of water below its freezing temperature. In contrast with the melting problem…
Head and tail speeds of mean curvature flow with forcing
Hongwei Gao, Inwon Kim
In this paper, we investigate the large time behavior of interfaces moving with motion law , where is positive, Lipschitz and -periodic. It turns o…
Uniform convergence for the incompressible limit of a tumor growth model
Inwon Kim, Olga Turanova
We study a model introduced by Perthame and Vauchelet that describes the growth of a tumor governed by Brinkman's Law, which takes into account friction between the tumor cells. We…
Free boundary regularity for tumor growth with nutrients and diffusion
Carson Collins, Matt Jacobs, Inwon Kim
In this paper, we study a tumor growth model where the growth is driven by nutrient availability and the tumor expands according to Darcy's law with a mechanical pressure resulting…
Aggregation-diffusion phenomena: from microscopic models to free boundary problems
Inwon Kim, Antoine Mellet, Jeremy Sheung-Him Wu
This paper reviews (and expands) some recent results on the modeling of aggregation-diffusion phenomena at various scales, focusing on the emergence of collective dynamics as a res…
A density-constrained model for Chemotaxis
Inwon Kim, Antoine Mellet, Yijing Wu
We consider a model of congestion dynamics with chemotaxis: The density of cells follows a chemical signal it generates, while subject to an incompressibility constraint. The incom…
On the singular limit of Brinkman's law to Darcy's law
Noemi David, Matt Jacobs, Inwon Kim
In this paper we study singular limits of congestion-averse growth models, connecting different models describing the effect of congestion. These models arise in particular in the…
Regularity Properties of Degenerate Diffusion Equations with Drifts
Inwon Kim, Yuming Zhang
This paper considers a class of nonlinear, degenerate drift- diffusion equations. We study well-posedness and regularity properties of the solutions, with the goal to achieve unifo…
Weak solutions to the Muskat problem with surface tension via optimal transport
Matt Jacobs, Inwon Kim, Alpár R. Mészáros
Inspired by recent works on the threshold dynamics scheme for multi-phase mean curvature flow (by Esedoglu-Otto and Laux-Otto), we introduce a novel framework to approximate soluti…