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Kyeong-hun Kim

Publications and source records attributed to Kyeong-hun Kim.

5 recordsLinked to original sources
arXiv2020

An $L_q(L_p)$-theory for diffusion equations with space-time nonlocal operators

We present an $L_q(L_{p})$-theory for the equation $$ \partial_{t}^αu=ϕ(Δ) u +f, \quad t>0,\, x\in \mathbb{R}^d \quad\, ;\, u(0,\cdot)=u_0. $$ Here $p,q>1$, $α\in (0,1)$, $\partial_{t}^α$ is the Caputo fractional derivative of order $α$, and $ϕ$ is a Bernstein function satisfying the following: $\exists δ_0\in (0,1]$ and $c>0$ such that \begin{equation} \label{eqn 8.17.1} c \left(\frac{R}{r}\right)^{δ_0}\leq \frac{ϕ(R)}{ϕ(r)}, \qquad 0<r<R<\infty. \end{equation} We prove uniqueness and existence results in Sobolev spaces, and obtain maximal regularity results of the solution. In particular, we prove \begin{align*} \| |\partial^α_t u|+|u|+|ϕ(Δ)u|\|_{L_q([0,T];L_p)}\leq N(\|f\|_{L_q([0,T];L_p)}+ \|u_0\|_{B_{p,q}^{ϕ,2-2/ αq}}), \end{align*} where $B_{p,q}^{ϕ,2-2/αq}$ is a modified Besov space on $\mathbb{R}^d$ related to $ϕ$. Our approach is based on BMO estimate for $p=q$ and vector-valued Calderón-Zygmund theorem for $p\neq q$. The Littlewood-Paley theory is also used to treat the non-zero initial data problem. Our proofs rely on the derivative estimates of the fundamental solution, which are obtained in this article based on the probability theory.

math.AP↗
arXiv2019

Boundary behavior and interior Hölder regularity of solution to nonlinear stochastic partial differential equations driven by space-time white noise

We present uniqueness and existence in weighted Sobolev spaces of the equation $$ u_t=(au_{xx}+bu_x+cu)+ ξ|u|^{1+λ} {\dot{B}}, \quad\,\, t>0, \, x\in (0,1) $$ with initial data $u(0,\cdot)=u_0$ and zero boundary data. Here $λ\in [0,1/2)$, $\dot{B}$ is a space-time white noise, and the coefficients $a,b,c$ and the function $ξ$ depend on $(ω,t,x)$ and the initial data $u_0$ depends on $(ω,x)$. More importantly, we obtain various interior Hölder regularities and boundary behaviors of the solution. For instance, if the initial data is in appropriate $L_p$ spaces, then for any small $\varepsilon>0$ and $T<\infty$, almost surely $$ ρ^{-1/2-κ}u \in C^{\frac{1}{4}-\fracκ{2}-\varepsilon, \frac{1}{2}-κ-\varepsilon}_{t,x}([0,T]\times (0,1)), \quad \forall\, κ\in (λ, 1/2), $$ where $ρ(x)$ is the distance from $x$ to the boundary. Taking $κ\downarrow λ$, one gets the the maximal Hölder exponents in time and space, which are $1/4-λ/2-\varepsilon$ and $1/2-λ-\varepsilon $ respectively. Also, letting $κ\uparrow 1/2$, one gets better decay or behavior near the boundary.

math.PR↗
arXiv2019

A sharp $L_p$-regularity result for second-order stochastic partial differential equations with unbounded and fully degenerate leading coefficients

We present existence, uniqueness, and sharp regularity results of solution to the stochastic partial differential equation (SPDE) \begin{align} \label{abs eqn} du=(a^{ij}(ω,t)u_{x^ix^j}+f)dt + (σ^{ik}(ω,t)u_{x^i}+g^k)dw^k_t, \quad u(0,x)=u_0, \end{align} where $\{w^k_t:k=1,2,\cdots\}$ is a sequence of independent Brownian motions. The coefficients are merely measurable in $(ω,t)$ and can be unbounded and fully degenerate, that is, coefficients $a^{ij}$, $σ^{ik}$ merely satisfy \begin{align} \label{abs only} \left(α^{ij}(ω,t)\right)_{d\times d}:= \left(a^{ij}(ω,t)-\frac{1}{2}\sum_{k=1}^{\infty} σ^{ik}(ω,t)σ^{jk}(ω,t)\right) \geq 0. \end{align} In this article, we prove that there exists a unique solution $u$ to \eqref{abs eqn}, and \begin{align} \notag \|u_{xx}\|_{\mathbb{H}^γ_p(τ,δ)} &\leq N(d,p) \bigg( \|u_0\|_{\mathbb{B}_p^{γ+2 \left(1-1/ p \right)}} + \| f\|_{\mathbb{H}^γ_p( τ,δ^{1-p} )} \label{abs est} &\qquad \qquad+\|g_x\|^p_{\mathbb{H}^γ_p( τ, |σ|^p δ^{1-p},l_2)}+ \| g_x\|_{\mathbb{H}^γ_p( τ,δ^{1-p/2},l_2)} \bigg), \end{align} where $p\geq 2$, $γ\in \mathbf{R}$, $τ$ is an arbitrary stopping time, $δ(ω, t)$ is the smallest eigenvalue of $α^{ij}(ω, t)$, $\mathbb{H}_p^γ(τ, δ)$ is a weighted stochastic Sobolev space, and $\mathbb{B}_p^{γ+2 \left(1-1/ p \right)}$ is a stochastic Besov space.

math.PR↗
arXiv2018

On the second order derivative estimates for degenerate parabolic equations

We study the parabolic equation \begin{align} \notag &u_t(t,x)=a^{ij}(t)u_{x^ix^j}(t,x)+f(t,x), \quad (t,x) \in [0,T] \times \mathbf{R}^d \\ &u(0,x)=u_0(x) \label{main eqn} \end{align} with the full degeneracy of the leading coefficients, that is, \begin{align} (a^{ij}(t)) \geq δ(t)I_{d\times d} \geq 0. \end{align} It is well known that if $f$ and $u_0$ are not smooth enough, say $f\in \mathbb{L}_p(T):=L_p([0,T] ; L_p(\mathbf{R}^d))$ and $u_0\in L_p(\mathbf{R}^d)$, then in general the solution is only in $C([0,T];L_p(\mathbf{R}^d))$, and thus derivative estimates are not possible. In this article we prove that $u_{xx}(t,\cdot)\in L_p(\mathbf{R}^d)$ on the set $\{t: δ(t)>0 \}$ and \begin{align*} \int^T_0 \|u_{xx}(t)\|^p_{L_p} δ(t)dt\leq N(d,p) \left(\int^T_0 \|f(t)\|^p_{L_p}δ^{1-p}(t)dt + \|u_0\|^p_{B^{2-2/ p}_p} \right), \end{align*} where $B^{2-2/ p}_p$ is the Besov space of order $2-2/p$. We also prove that $u_{xx}(t,\cdot)\in L_p(\mathbf{R}^d)$ for all $t>0$ and \begin{equation} \label{10.13.3} \int^T_0 \|u_{xx}\|^p_{L_p(\mathbf{R}^d)}\,dt \leq N \|u_0\|^p_{B^{2-2/(βp)}_p}, \end{equation} if $f=0$, $\int^t_0 δ(s)ds>0$ for each $t>0$, and a certain asymptotic behavior of $δ(t)$ holds near $t=0$ (see (1.3)). Here $β>0$ is the constant related to the asymptotic behavior in (1.3). For instance, if $d=1$ and $a^{11}(t)=δ(t)=1+\sin(1/t)$, then the estimate holds with $β=1$, which actually equals the maximal regularity of the heat equation $u_t=Δu$.

math.AP↗