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Sunghoon Kim

Publications and source records attributed to Sunghoon Kim.

36 records · Page 2Linked to original sources
arXiv2019

Local Continuity and Asymptotic Behaviour of Degenerate Parabolic Systems

We study the local Hölder continuity and the asymptotic behaviour of solution, $\mathbf{u}=(u^1,\cdots, u^k)$, of the degenerate system \begin{equation*} u^i_t=\nabla\cdot\left(m\,U^{m-1}\nabla u^i\right) \qquad \text{for $m>1$ and $i=1,\cdots,k$ } \end{equation*} which describes the populations density of $k$-species whose diffusion is determined by their total population density $U=u^1+\cdots+u^k$. For the local Hölder continuity, we adopt the intrinsic scaling and iteration arguments of DeGiorgi, Moser, and Dibenedetto. Under some regularity conditions, we also prove that the population density function of $i$-th species with the population $M_i$ converges in $C_s^{\infty}$ to $\frac{M_i}{M}\mathcal{B}_M(x,t)$ as $t\to \infty$ where $\mathcal{B}_M$ is the Barenblatt profile of the standard porous medium equation with $L^1$ mass $M=M_1+\cdots+M_k$. As a consequence of asymptotic behaviour, it is shown that each density function becomes a concave function after a finite time.

math.AP↗
arXiv2019

Vertex corrections to the dc conductivity in anisotropic multiband systems

For an isotropic single-band system, it is well known that the semiclassical Boltzmann transport theory within the relaxation time approximation and the Kubo formula with the vertex corrections provide the same result with the $(1-\cosθ)$ factor in the inverse transport relaxation time. In anisotropic multiband systems, the semiclassical Boltzmann transport equation is generalized to coupled integral equations relating transport relaxation times at different angles in different bands. Using the Kubo formula, we study the vertex corrections to the dc conductivity in anisotropic multiband systems and derive the relation satisfied by the transport relaxation time for both elastic and inelastic scatterings, verifying that the result is consistent with the semiclassical approach.

cond-mat.mes-hall↗
arXiv2018

A Regularized Spatial Market Segmentation Method with Dirichlet Process Gaussian Mixture Prior

Spatially referenced data are increasingly available thanks to the development of modern GPS technology. They also provide rich opportunities for spatial analytics in the field of marketing science. Our main interest is to propose a new efficient statistical framework to conduct spatial segmentation analysis for restaurants located in a metropolitan area in the U.S. The spatial segmentation problem poses important statistical challenges: selecting the optimal number of underlying structures of market segments, capturing complex and flexible spatial structures, and resolving any possible small n and large p issue which can be typical in latent class analysis. Existing approaches try to tackle these issues in heuristic ways or seem silent on them. To overcome these challenges, we propose a new statistical framework based on regularized Bayesian spatial mixture regressions with Dirichlet process integrating ridge or lasso regularization. Our simulation study demonstrates that the proposed models successfully recover the underlying spatial clustering structures and outperforms two existing benchmark models. In the empirical analysis using online customer satisfaction data from the Yelp, our models provides interesting insights on segment-level key drivers of customer satisfaction and interpretable relationships between regional demographics and restaurants' characteristics.

stat.AP↗
arXiv2018

Existence and large time behaviour of finite points blow-up solutions of the fast diffusion equation

Let $Ω\subset\R^n$ be a smooth bounded domain and let $a_1,a_2,\dots,a_{i_0}\inΩ$, $\widehatΩ=Ω\setminus\{a_1,a_2,\dots,a_{i_0}\}$ and $\widehat{R^n}=\R^n\setminus\{a_1,a_2,\dots,a_{i_0}\}$. We prove the existence of solution $u$ of the fast diffusion equation $u_t=Δu^m$, $u>0$, in $\widehatΩ\times (0,\infty)$ ($\widehat{R^n}\times (0,\infty)$ respectively) which satisfies $u(x,t)\to\infty$ as $x\to a_i$ for any $t>0$ and $i=1,\cdots,i_0$, when $0 \frac{n(1-m)}{2}$ and $u_0(x)\ge λ_i|x-a_i|^{-γ_i}$ for $x\approx a_i$ and some constants $γ_i>\frac{2}{1-m},λ_i>0$, for all $i=1,2,\dots,i_0$. We also find the blow-up rate of such solutions near the blow-up points $a_1,a_2,\dots,a_{i_0}$, and obtain the asymptotic large time behaviour of such singular solutions. More precisely we prove that if $u_0\geμ_0$ on $\widehatΩ$ ($\widehat{R^n}$, respectively) for some constant $μ_0>0$ and $γ_1>\frac{n-2}{m}$, then the singular solution $u$ converges locally uniformly on every compact subset of $\widehatΩ$ (or $\widehat{R^n}$ respectively) to infinity as $t\to\infty$. If $u_0\geμ_0$ on $\widehatΩ$ ($\widehat{R^n}$, respectively) for some constant $μ_0>0$ and satisfies $λ_i|x-a_i|^{-γ_i}\le u_0(x)\le λ_i'|x-a_i|^{-γ_i'}$ for $x\approx a_i$ and some constants $\frac{2}{1-m}<γ_i\leγ_i'<\frac{n-2}{m}$, $λ_i>0$, $λ_i'>0$, $i=1,2,\dots,i_0$, we prove that $u$ converges in $C^2(K)$ for any compact subset $K$ of $\2Ω\setminus\{a_1,a_2,\dots,a_{i_0}\}$ (or $\widehat{R^n}$ respectively) to a harmonic function as $t\to\infty$.

math.AP↗
arXiv2016

Singular limits and properties of solutions of some degenerate elliptic and parabolic equations

Let $n\geq 3$, $0\le m<\frac{n-2}{n}$, $ρ_1>0$, $β>β_0^{(m)}=\frac{mρ_1}{n-2-nm}$, $α_m=\frac{2β+ρ_1}{1-m}$ and $α=2β+ρ_1$. For any $λ>0$, we prove the uniqueness of radially symmetric solution $v^{(m)}$ of $\La(v^m/m)+α_m v+βx\cdot\nabla v=0$, $v>0$, in $\R^n\setminus\{0\}$ which satisfies $\lim_{|x|\to 0}|x|^{\frac{α_m}β}v^{(m)}(x)=λ^{-\frac{ρ_1}{(1-m)β}}$ and obtain higher order estimates of $v^{(m)}$ near the blow-up point $x=0$. We prove that as $m\to 0^+$, $v^{(m)}$ converges uniformly in $C^2(K)$ for any compact subset $K$ of $\R^n\setminus\{0\}$ to the solution $v$ of $\La\log v+αv+βx\cdot\nabla v=0$, $v>0$, in $\R^n\bs\{0\}$, which satisfies $\lim_{|x|\to 0}|x|^{\fracαβ}v(x)=λ^{-\frac{ρ_1}β}$. We also prove that if the solution $u^{(m)}$ of $u_t=Δ(u^m/m)$, $u>0$, in $(\R^n\setminus\{0\})\times (0,T)$ which blows up near $\{0\}\times (0,T)$ at the rate $|x|^{-\frac{α_m}β}$ satisfies some mild growth condition on $(\R^n\setminus\{0\})\times (0,T)$, then as $m\to 0^+$, $u^{(m)}$ converges uniformly in $C^{2+θ,1+\fracθ{2}}(K)$ for some constant $θ\in (0,1)$ and any compact subset $K$ of $(\R^n\setminus\{0\})\times (0,T)$ to the solution of $u_t=\La\log u$, $u>0$, in $(\R^n\setminus\{0\})\times (0,T)$. As a consequence of the proof we obtain existence of a unique radially symmetric solution $v^{(0)}$ of $\La \log v+αv+βx\cdot\nabla v=0$, $v>0$, in $\R^n\setminus\{0\}$, which satisfies $\lim_{|x|\to 0}|x|^{\fracαβ}v(x)=λ^{-\frac{ρ_1}β}$.

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↗
arXiv2015

Singular limit of the generalized Burgers equation with absorption

We prove the convergence of the solutions $u_{m,p}$ of the equation $u_t+(u^m)_x=-u^p$ in $\R\times (0,\infty)$, $u(x,0)=u_0(x)\ge 0$ in $\R$, as $m\to\infty$ for any $p>1$ and $u_0\in L^1(\R)\cap L^{\infty}(\R)$ or as $p\to\infty$ for any $m>1$ and $u_0\in L^{\infty}(\R)$ . We also show that in general $\underset{p\to\infty}\lim\underset{m\to\infty}\lim u_{m,p}\ne\underset{m\to\infty}\lim\underset{p\to\infty}\lim u_{m,p}$.

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

Performance of SSE and AVX Instruction Sets

SSE (streaming SIMD extensions) and AVX (advanced vector extensions) are SIMD (single instruction multiple data streams) instruction sets supported by recent CPUs manufactured in Intel and AMD. This SIMD programming allows parallel processing by multiple cores in a single CPU. Basic arithmetic and data transfer operations such as sum, multiplication and square root can be processed simultaneously. Although popular compilers such as GNU compilers and Intel compilers provide automatic SIMD optimization options, one can obtain better performance by a manual SIMD programming with proper optimization: data packing, data reuse and asynchronous data transfer. In particular, linear algebraic operations of vectors and matrices can be easily optimized by the SIMD programming. Typical calculations in lattice gauge theory are composed of linear algebraic operations of gauge link matrices and fermion vectors, and so can adopt the manual SIMD programming to improve the performance.

hep-lat↗
arXiv2012

Extinction profile of the logarithmic diffusion equation

Let $u$ be the solution of $u_t=Δ\log u$ in $\R^N\times (0,T)$, N=3 or $N\ge 5$, with initial value $u_0$ satisfying $B_{k_1}(x,0)\le u_0\le B_{k_2}(x,0)$ for some constants $k_1>k_2>0$ where $B_k(x,t) =2(N-2)(T-t)_+^{N/(N-2)}/(k+(T-t)_+^{2/(N-2)}|x|^2)$ is the Barenblatt solution for the equation. We prove that the rescaled function $\4{u}(x,s)=(T-t)^{-N/(N-2)}u(x/(T-t)^{-1/(N-2)},t)$, $s=-\log (T-t)$, converges uniformly on $\R^N$ to the rescaled Barenblatt solution $\4{B}_{k_0}(x)=2(N-2)/(k_0+|x|^2)$ for some $k_0>0$ as $s\to\infty$. We also obtain convergence of the rescaled solution $\4{u}(x,s)$ as $s\to\infty$ when the initial data satisfies $0\le u_0(x)\le B_{k_0}(x,0)$ in $\R^N$ and $|u_0(x)-B_{k_0}(x,0)|\le f(|x|)\in L^1(\R^N)$ for some constant $k_0>0$ and some radially symmetric function $f$.

math.AP↗
arXiv2012

Long time existence of Smooth solution for the porous medium equation in a bounded domain

In this paper, we are going to show the long time existence of the smooth solution for the porous medium equations in a smooth bounded domain: {equation} {cases} u_t=\La u^m\quad\text{in $Ω\times [0,\infty)$} u(x,0)=u_0>0\quad\text{in $Ω$} u(x,t)=0\quad\text{for $x\in\partialΩ$} {cases} {equation} where $m>1$ is the permeability. The proof is based on the short time existence of $C^{2,\barγ}_{s}$-smooth solution, the global $C^{1}_{s}$-estimate, the Hölder estimate of divergence type degenerate equation with measurable coefficients and $C^{1,\barγ}_s$-estimate of mixed type equation with Lipschitz coefficients.

math.FA↗
arXiv2012

Decay rate and radial symmetry of the exponential elliptic equation

Let $n\geq 3$, $α$, $β\in\mathbb{R}$, and let $v$ be a solution $Δv+αe^v+βx\cdot\nabla e^v=0$ in $\mathbb{R}^n$, which satisfies the conditions $\lim_{R\to\infty}\frac{1}{\log R}\int_{1}^{R}ρ^{1-n} (\int_{B_ρ}e^v\,dx)dρ\in (0,\infty)$ and $|x|^2e^{v(x)}\le A_1$ in $\R^n$. We prove that $\frac{v(x)}{\log |x|}\to -2$ as $|x|\to\infty$ and $α>2β$. As a consequence under a mild condition on $v$ we prove that the solution is radially symmetric about the origin.

math.AP↗
arXiv2011

Large time behaviour of higher dimensional logarithmic diffusion equation

Let $n\ge 3$ and $ψ_{λ_0}$ be the radially symmetric solution of $Δ\logψ+2βψ+βx\cdot\nablaψ=0$ in $R^n$, $ψ(0)=λ_0$, for some constants $λ_0>0$, $β>0$. Suppose $u_0\ge 0$ satisfies $u_0-ψ_{λ_0}\in L^1(R^n)$ and $u_0(x)\approx\frac{2(n-2)}β\frac{\log |x|}{|x|^2}$ as $|x|\to\infty$. We prove that the rescaled solution $\widetilde{u}(x,t)=e^{2βt}u(e^{βt}x,t)$ of the maximal global solution $u$ of the equation $u_t=Δ\log u$ in $R^n\times (0,\infty)$, $u(x,0)=u_0(x)$ in $R^n$, converges uniformly on every compact subset of $R^n$ and in $L^1(R^n)$ to $ψ_{λ_0}$ as $t\to\infty$. Moreover $\|\widetilde{u}(\cdot,t)-ψ_{λ_0}\|_{L^1(R^n)} \le e^{-(n-2)βt}\|u_0-ψ_{λ_0}\|_{L^1(R^n)}$ for all $t\ge 0$.

math.AP↗
arXiv2010

Convergence of the Dirichlet solutions of the very fast diffusion equation

For any $-1 0$, $0\le u_0\in L^{\infty}(R)$ such that $u_0(x)\le (μ_0 |m||x|)^{\frac{1}{m}}$ for any $|x|\ge R_0$ and some constants $R_0>1$ and $0<μ_0\leq μ$, and $f,\,g \in C([0,\infty))$ such that $f(t),\, g(t) \geq μ_0$ on $[0,\infty)$ we prove that as $R\to\infty$ the solution $u^R$ of the Dirichlet problem $u_t=(u^m/m)_{xx}$ in $(-R,R)\times (0,\infty)$, $u(R,t)=(f(t)|m|R)^{1/m}$, $u(-R,t)=(g(t)|m|R)^{1/m}$ for all $t>0$, $u(x,0)=u_0(x)$ in $(-R,R)$, converges uniformly on every compact subsets of $R\times (0,T)$ to the solution of the equation $u_t=(u^m/m)_{xx}$ in $R\times (0,\infty)$, $u(x,0)=u_0(x)$ in $R$, which satisfies $\int_Ru(x,t)\,dx=\int_Ru_0dx-\int_0^t(f(s)+g(s))\,ds$ for all $0<t<T$ where $\int_0^T(f+g)\,ds=\int_Ru_0dx$. We also prove that the solution constructed is equal to the solution constructed in [Hu3] using approximation by solutions of the corresponding Neumann problem in bounded cylindrical domains.

math.AP↗