Searcharxiv⌕ Search

arXiv subjects

Ki-Ahm Lee

Publications and source records attributed to Ki-Ahm Lee.

53 records · Page 3Linked to original sources
arXiv2015

Multi-D Fast Diffusion Equation via Diffusive Scaling of Generalized Carleman Kinetic Equation

In this paper, we investigate generalized Carleman kinetic equation for n$\ge$2 and prove convergence towards the solution of equation with fast diffusion or porous medium type, $u_t=Δu^m$ ($0\le m\le2$), in its diffusive hydrodynamic limit. Using comparison principle of system combined with fixed speed propagation property of transport equation, we create a new barrier argument for this hyperbolic system. It is crucial to construct explicit local sub and solution of system and this is done by employing an ansatz from second order asymptotic expansion. This allow us prove diffusive limit toward subcritical FDE, which is thought to be difficult with previous method due to the lack of mass conservation. Moreover, we can also prove convergence with growing initial data in slow diffusion range (including $n=1$), which was also unknown before.

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↗
arXiv2015

Hölder regularity and Uniqueness theorem on weak solutions to the degenerate Keller-Segel system

In this paper, we present local Hölder estimates for the degenerate Keller-Segel system \eqref{eq-cases-aligned-main-problem-of-Keller-Segel-System} below in the range of $m>1$ and $q>1$ before a blow-up of solutions. To deal with difficulties caused by the degeneracy of the operator, we find uniform estimates depending sup-norm of the density function and modified the energy estimates and intrinsic scales considered in Porous Medium Equation. As its application, the uniqueness of weak solution to \eqref{eq-cases-aligned-main-problem-of-Keller-Segel-System} is also showed in the class of Hölder continuous functions by proving $L^1$-contraction in this class.

math.AP↗
arXiv2014

Qualitative properties of multi-bubble solutions for nonlinear elliptic equations involving critical exponents

The objective of this paper is to obtain qualitative characteristics of multi-bubble solutions to the Lane-Emden-Fowler equations with slightly subcritical exponents given any dimension $n \ge 3$. By examining the linearized problem at each $m$-bubble solution, we provide a number of estimates on the first $(n+2)m$-eigenvalues and their corresponding eigenfunctions. Specifically, we present a new proof of the classical theorem due to Bahri-Li-Rey (Calc. Var. Partial Differential Equations 3 (1995) 67-93) which states that if $n \ge 4$, then the Morse index of a multi-bubble solution is governed by a certain symmetric matrix whose component consists of a combination of Green's function, the Robin function, and their first and second derivatives. Our proof also allows us to handle the intricate case $n = 3$.

math.AP↗
arXiv2014

Regularity for fully nonlinear integro-differential operators with regularly varying kernels

In this paper, the regularity results for the integro-differential operators of the fractional Laplacian type by Caffarelli and Silvestre \cite{CS1} are extended to those for the integro-differential operators associated with symmetric, regularly varying kernels at zero. In particular, we obtain the uniform Harnack inequality and Hölder estimate of viscosity solutions to the nonlinear integro-differential equations associated with the kernels $K_{σ, β}$ satisfying $$ K_{σ,β}(y)\asymp \frac{ 2-σ}{|y|^{n+σ}}\left( \log\frac{2}{|y|^2}\right)^{β(2-σ)}\quad \mbox{near zero} $$ with respect to $σ\in(0,2)$ close to $2$ (for a given $β\in\mathbb R$), where the regularity estimates do not blow up as the order $ σ\in(0,2)$ tends to $2.$

math.AP↗
arXiv2014

Parabolic Harnack inequality of viscosity solutions on Riemannian manifolds

We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold $M$, with the sectional curvature bounded from below by $-κ$ for $κ\geq 0$. In the elliptic case, Wang and Zhang \cite{WZ} recently extended the results of \cite{Ca} to nonlinear elliptic equations in nondivergence form on such $M$, where they obtained the Harnack inequality for classical solutions. We establish the Harnack inequality for nonnegative {\it viscosity solutions} to nonlinear uniformly {\it parabolic equations} in nondivergence form on $M$. The Harnack inequality of nonnegative viscosity solutions to the elliptic equations is also proved.

math.AP↗
arXiv2013

Hölder Estimates for Singular Non-local Parabolic Equations

In this paper, we establish local Hölder estimate for non-negative solutions of the singular equation \eqref{eq-nlocal-PME-1} below, for $m$ in the range of exponents $(\frac{n-2σ}{n+2σ},1)$. Since we have trouble in finding the local energy inequality of $v$ directly. we use the fact that the operator $(-\La)^σ$ can be thought as the normal derivative of some extension $v^{\ast}$ of $v$ to the upper half space, \cite{CS}, i.e., $v$ is regarded as boundary value of $v^{\ast}$ the solution of some local extension problem. Therefore, the local Hölder estimate of $v$ can be obtained by the same regularity of $v^{\ast}$. In addition, it enables us to describe the behaviour of solution of non-local fast diffusion equation near their extinction time.

math.AP↗
arXiv2013

Viscosity method for Homogenization of Parabolic Nonlinear Equations in Perforated Domains

In this paper, we develop a viscosity method for Homogenization of Nonlinear Parabolic Equations constrained by highly oscillating obstacles or Dirichlet data in perforated domains. The Dirichlet data on the perforated domain can be considered as a constraint or an obstacle. Homogenization of nonlinear eigen value problems has been also considered to control the degeneracy of the Porous medium equation in perforated domains. For the simplicity, we consider obstacles that consist of cylindrical columns distributed periodically and perforated domains with punctured balls. If the decay rate of the capac- ity of columns or the capacity of punctured ball is too high or too small, the limit of uǫ will converge to trivial solutions. The critical decay rates of having nontrivial solution are obtained with the construction of barriers. We also show the limit of uǫ satisfies a homogenized equation with a term showing the effect of the highly oscillating obstacles or perforated domain in viscosity sense.

math.AP↗
arXiv2013

Geometric Properties of Gelfand's Problems with Parabolic Approach

We consider the asymptotic profiles of the nonlinear parabolic flows $$(e^{u})_{t}= \La u+λe^u$$ to show the geometric properties of the following elliptic nonlinear eigenvalue problems known as a Gelfand's problem: \begin{equation*} \begin{split} \La \vp &+ λe^{\vp}=0, \quad \vp>0\quad\text{in $Ω$}\\ \vp&=0\quad\text{on $Ω$} \end{split} \end{equation*} posed in a strictly convex domain $Ω\subset\re^n$. In this work, we show that there is a strictly increasing function $f(s)$ such that $f^{-1}(\vp(x))$ is convex for $0<λ\leqλ^{\ast}$, i.e., we prove that level set of $\vp$ is convex. Moreover, we also present the boundary condition of $\vp$ which guarantee the $f$-convexity of solution $\vp$.

math.AP↗
arXiv2013

Geometric property of the Ground State Eigenfunction for Cauchy Process

We consider the asymptotic behavior of nonlinear nonlocal flows $u_t+(-\La)^{1/2}u=0$ to find the geometric property of the solutions in nonlinear eigenvalue problem: (-\La)^{1/2}\vp=λ\vp posed in a strictly convex domain $Ω\subset\R^n$ with $\vp>0$ in $Ω$ and $\vp=0$ on $\R^n\bsΩ$. This is corresponding to an eigenvalue problem for Cauchy process. The concavity of $\vp$ is well known for the dimension $n=1$. In this paper, we will show $\vp^{-\frac{2}{n+1}}$ is convex. Moreover, the eventual power-convexity of the parabolic flows is also proved. In the final section, We extend geometric results to Cauchy problem for the fractional Heat operator.

math.AP↗
arXiv2012

Harnack inequality for nondivergent parabolic operators on Riemannian manifolds

We consider second-order linear parabolic operators in non-divergence form that are intrinsically defined on Riemannian manifolds. In the elliptic case, Cabré proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature of the underlying manifold is nonnegative. Later, Kim improved Cabré's result by replacing the curvature condition by a certain condition on the distance function. Assuming essentially the same condition introduced by Kim, we establish Krylov-Safonov Harnack inequality for nonnegative solutions of the non-divergent parabolic equation. This, in particular, gives a new proof for Li-Yau Harnack inequality for positive solutions to the heat equation in a manifold with nonnegative Ricci curvature.

math.AP↗
arXiv2011

Regularity results for fully nonlinear parabolic integro-differential operators

In this paper, we consider the regularity theory for fully nonlinear parabolic integro-differential equations with symmetric kernels. We are able to find parabolic versions of Alexandrov-Backelman-Pucci estimate with 0<σ<2. And we show a Harnack inequality, Hölder regularity, and C^{1,α}-regularity of the solutions by obtaining decay estimates of their level sets.

math.AP↗
arXiv2011

Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels

In this paper, we consider fully nonlinear integro-differential equations with possibly nonsymmetric kernels. We are able to find different versions of Alexandroff-Backelman-Pucci estimate corresponding to the full class $\cS^{\fL_0}$ of uniformly elliptic nonlinear equations with $1<σ<2$ (subcritical case) and to their subclass $\cS^{\fL_0}_η$ with $0<σ\leq 1$. We show that $\cS^{\fL_0}_η$ still includes a large number of nonlinear operators as well as linear operators. And we show a Harnack inequality, Hölder regularity, and $C^{1,α}$-regularity of the solutions by obtaining decay estimates of their level sets in each cases.

math.AP↗
arXiv2010

Regularity results for fully nonlinear integro-differential operators with nonsymmetric positive kernels : Subcritical Case

We introduce a new class of fully nonlinear integro-differential operators with possible nonsymmetric kernels, which includes the ones that arise from stochastic control problems with purely jump Lèvy processes. If the index of the operator $σ$ is in $ (1,2)$ (subcritical case), then we obtain a comparison principle, a nonlocal version of the Alexandroff-Backelman-Pucci estimate, a Harnack inequality, a Hölder regularity, and an interior $\rm C^{1,α}$-regularity for fully nonlinear integro-differential equations associated with such a class. Moreover, our estimates remain uniform as the index $σ$ of the operator is getting close to two, so that they can be regarded as a natural extension of regularity results for elliptic partial differential equations.

math.CA↗