Inwon C. Kim

The Patlak–Keller–Segel Model and Its Variations: Properties of Solutions via Maximum Principle

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 concern radial solutions. The challenge in the analysis consists of the nonlocal aggregation term as well as the degeneracy of the diffusion term which generates compactly supported solutions. The key tools used in the paper are maximum-principle type arguments as well as estimates on mass concentration of solutions.

Viscosity Solutions for the Two-Phase Stefan Problem

We introduce a notion of viscosity solutions for the two-phase Stefan problem, which incorporates possible existence of a mushy region generated by the initial data. We show that a comparison principle holds between viscosity solutions, and investigate the coincidence of the viscosity solutions and the weak solutions defined via integration by parts. In particular, in the absence of initial mushy region, viscosity solution is the unique weak solution with the same boundary data.

Global Existence and Uniqueness of Solutions to a Model of Price Formation

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 one-dimensional diffusion equation with dynamically evolving, nonstandard sources. We establish global existence and uniqueness.

Regularity for the one-phase Hele-Shaw problem from a Lipschitz initial surface

In this paper we show that if the Lipschitz constant of the initial free boundary is small, then for small positive time the solution is smooth and satisfies the Hele-Shaw equation in the classical sense. A key ingredient in the proof which is of independent interest is an estimate up to order of magnitude of the speed of the free boundary in terms of initial data.

Quasi-static evolution and congested crowd transport

We consider the relationship between Hele-Shaw evolution with drift, the porous medium equation with drift, and a congested crowd motion model originally proposed by Maury et al (2010 Math. Models Methods Appl. Sci. 20 1787–821). We first use viscosity solutions to show that the porous medium equation solutions converge to the Hele-Shaw solution as m → ∞ provided the drift potential is strictly subharmonic. Next, using the gradient-flow structure of both the porous medium equation and the crowd motion model, we prove that the porous medium equation solutions also converge to the congested crowd motion as m → ∞. Combining these results lets us deduce that in the case where the initial data to the crowd motion model is given by a patch, or characteristic function, the solution evolves as a patch that is the unique solution to the Hele-Shaw problem. While proving our main results we also obtain a comparison principle for solutions with the minimizing movement scheme based on the Wasserstein metric, of independent interest.

Viscosity solutions for a model of contact line motion

This paper considers a free boundary problem that describes the motion of contact lines of a liquid droplet on a flat surface. The elliptic nature of the equation for droplet shape and the monotonic dependence of contact line velocity on contact angle allows us to introduce a notion of “viscosity” solutions for this problem. Unlike similar free boundary problems, a comparison principle is only available for a modified short-time approximation because of the constraint that conserves volume. We use this modified problem to construct viscosity solutions to the original problem under a weak geometric restriction on the free boundary shape. We also prove uniqueness provided there is an upper bound on front velocity.

A free boundary problem arising in flame propagation

Abstract We study a free boundary problem describing the propagation of laminar flames. The problem arises as the limit of a singular perturbation problem. We introduce the notion of viscosity solutions for the problem to show the maximum principle-type property of the solutions. Using this property we show the uniform convergence of the approximating solutions and the uniqueness of the viscosity solution under several geometric conditions on the initial data.

Global Existence and Finite Time Blow-Up for Critical Patlak--Keller--Segel Models with Inhomogeneous Diffusion

The

Homogenization of one-phase Stefan-type problems in periodic and random media

L^1

Homogenization of a Model Problem on Contact Angle Dynamics

-critical parabolic-elliptic Patlak--Keller--Segel system is a classical model of chemotactic aggregation in microorganisms well known to have critical mass phenomena [Blanchet, Dolbeault, and Perthame, Electron. J. Differential Equations, 2006 (2006), pp. 1--23; Blanchet, Carrillo, and Laurencot, Calc. Var., 35 (2009), pp. 133--168]. In this paper we study this critical mass phenomenon in the context of Patlak--Keller--Segel models with spatially varying diffusivity of the chemoattractant. The primary issue is how, if possible, one localizes the presence of the inhomogeneity in the nonlocal term. Our methods also provide new blow-up results for the homogeneous problem with nonlinear diffusion, showing that there exist blow-up solutions with arbitrarily large (positive) initial free energy.