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Inwon Kim

Publications and source records attributed to Inwon Kim.

42 records · Page 3Linked to original sources
arXiv2013

Homogenization for nonlinear PDEs in general domains with oscillatory Neumann boundary data

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 both in the operator and in the Neumann data. Our main result states that, when the domain does not have flat boundary parts and when the homogenized operator is rotation invariant, the solutions uniformly converge to the homogenized solution solving a Neumann boundary problem. Furthermore we show that the homogenized Neumann data is continuous with respect to the normal direction of the boundary.

math.AP↗
arXiv2012

An aggregation equation with degenerate diffusion: qualitative property of solutions

We study a nonlocal aggregation equation with degenerate diffusion, set in a periodic domain. This equation represents the generalization to $m > 1$ of the McKean-Vlasov equation where here the "diffusive" portion of the dynamics are governed by Porous medium self-interactions. We focus primarily on $m\in(1,2]$ with particular emphasis on $m = 2$. In general, we establish regularity properties and, for small interaction, exponential decay to the uniform stationary solution. For $m=2$, we obtain essentially sharp results on the rate of decay for the entire regime up to the (sharp) transitional value of the interaction parameter.

math.AP↗
arXiv2012

Liquid drops sliding down an inclined plane

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 uniqueness of a solution and investigate its long time behavior for both homogeneous and inhomogeneous medium (i.e. constant and non-constant contact angle). We also obtain some homogenization results.

math.AP↗
arXiv2011

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.

math.AP↗
arXiv2010

Two phase Stefan-type problem: Regularization near initial data by phase dynamics

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 is very close to a Lipschitz profile in space-time. We introduce a new decomposition argument to generalize the preceding ones by Choi, Jerion and Kim([CJK1], [CJK2]) and by Choi and Kim([CK]) on one-phase problems.

math.AP↗