arXiv subjects

Daehan Park

Publications and source records attributed to Daehan Park.

11 recordsLinked to original sources

arXiv2025

A regularity theory for evolution equations with space-time anisotropic non-local operators in mixed-norm Sobolev spaces

In this article, we study the regularity of solutions to inhomogeneous time-fractional evolution equations involving anisotropic non-local operators in mixed-norm Sobolev spaces of variable order, with non-trivial initial conditions. The primary focus is on space-time non-local equations where the spatial operator is the infinitesimal generator of a vector of independent subordinate Brownian motions, making it the sum of subdimensional non-local operators. A representative example of such an operator is $(\Delta_{x})^{\beta_{1}/2}+(\Delta_{y})^{\beta_{2}/2}$. We establish existence, uniqueness, and precise estimates for solutions in corresponding Sobolev spaces. Due to singularities arising in the Fourier transforms of our operators, traditional methods involving Fourier analysis are not directly applicable. Instead, we employ a probabilistic approach to derive solution estimates. Additionally, we identify the optimal initial data space using generalized real interpolation theory.

math.AP↗

arXiv2024

Sobolev regularity theory for stochastic reaction-diffusion-advection equations with spatially homogeneous colored noises and infinitesimal generators of subordinate Brownian motions

This article investigates the existence, uniqueness, and regularity of solutions to nonlinear stochastic reaction-diffusion-advection equations (SRDAEs) with spatially homogeneous colored noises and infinitesimal generators of subordinate Brownian motions in mixed norm $L_q(L_p)$-spaces. We introduce a new condition (strongly reinforced Dalang's condition) on colored noise, which facilitates a deeper understanding of the complicated relation between nonlinearities and stochastic forces. Additionally, we establish the space-time H\"older type regularity of solutions.

math.PR↗

arXiv2024

A regularity theory for parabolic equations with anisotropic non-local operators in $L_{q}(L_{p})$ spaces

In this paper, we present an $L_q(L_p)$-regularity theory for parabolic equations of the form: $$ \partial_t u(t,x)=\mathcal{L}^{\vec{a},\vec{b}}(t)u(t,x)+f(t,x),\quad u(0,x)=0. $$ Here, $\mathcal{L}^{\vec{a},\vec{b}}(t)$ represents anisotropic non-local operators encompassing the singular anisotropic fractional Laplacian with measurable coefficients: $$ \mathcal{L}^{\vec{a},\vec{0}}(t)u(x)=\sum_{i=1}^{d} \int_{\mathbb{R}}\left( u(x^{1},\dots,x^{i-1},x^{i}+y^{i},x^{i+1},\dots,x^{d}) - u(x) \right) \frac{a_{i}(t,y^{i})}{|y^{i}|^{1+α_{i}}} \mathrm{d}y^{i} . $$ To address the anisotropy of the operator, we employ a probabilistic representation of the solution and Calderón-Zygmund theory. As applications of our results, we demonstrate the solvability of elliptic equations with anisotropic non-local operators and parabolic equations with isotropic non-local operators.

math.AP↗

arXiv2023

An $L_{q}(L_{p})$-regularity theory for parabolic equations with integro-differential operators having low intensity kernels

In this article, we present the existence, uniqueness, and regularity of solutions to parabolic equations with non-local operators $$ \partial_{t}u(t,x) = \mathcal{L}^{a}u(t,x) + f(t,x), \quad t>0 $$ in $L_{q}(L_{p})$ spaces. Our spatial operator $\mathcal{L}^{a}$ is an integro-differential operator of the form $$ \int_{\mathbb{R}^{d}} \left( u(x+y)-u(x) -\nabla u(x) \cdot y \mathrm{1}_{|y|\leq 1} \right) a(t,y) j_{d}(|y|)dy. $$ Here, $a(t,y)$ is a merely bounded measurable coefficient, and we employed the theory of additive process to handle it. We investigate conditions on $j_{d}(r)$ which yield $L_{q}(L_{p})$-regularity of solutions. Our assumptions on $j_d$ are general so that $j_d(r)$ may be comparable to $r^{-d}\ell(r^{-1})$ for a function $\ell$ which is slowly varying at infinity. For example, we can take $\ell(r)=\log{(1+r^{\alpha})}$ or $\ell(r) = \min{\{r^{\alpha},1\}}$ ($\alpha\in(0,2)$). Indeed, our result covers the operators whose Fourier multiplier $\psi(\xi)$ does not have any scaling condition for $|\xi|\geq 1$. Furthermore, we give some examples of operators, which cannot be covered by previous results where smoothness or scaling conditions on $\psi$ are considered.

math.AP↗

arXiv2022

Weighted maximal $L_{q}(L_{p})$-regularity theory for time-fractional diffusion-wave equations with variable coefficients

We present a maximal $L_{q}(L_{p})$-regularity theory with Muckenhoupt weights for the equation \begin{equation}\label{eqn 01.26.16:00} \partial^α_{t}u(t,x)=a^{ij}(t,x)u_{x^{i}x^{j}}(t,x)+f(t,x),\quad t>0,x\in\mathbb{R}^{d}. \end{equation} Here, $\partial^α_{t}$ is the Caputo fractional derivative of order $α\in(0,2)$ and $a^{ij}$ are functions of $(t,x)$. Precisely, we show that \begin{equation*} \begin{aligned} &\int_{0}^{T}\left(\int_{\mathbb{R}^{d}}|(1-Δ)^{γ/2}u_{xx}(t,x)|^{p}w_{1}(x)dx\right)^{q/p}w_{2}(t)dt \\ &\quad \leq N \int_{0}^{T}\left(\int_{\mathbb{R}^{d}}|(1-Δ)^{γ/2}f(t,x)|^{p}w_{1}(x)dx\right)^{q/p}w_{2}(t)dt, \end{aligned} \end{equation*} where $1<p,q<\infty$, $γ\in\mathbb{R}$, and $w_{1}$ and $w_{2}$ are Muckenhoupt weights. This implies that we prove maximal regularity theory, and sharp regularity of solution according to regularity of $f$. To prove our main result, we also proved the complex interpolation of weighted Sobolev spaces, $$ [H^{γ_{0}}_{p_{0}}(w_{0}), H^{γ_{1}}_{p_{1}}(w_{1})]_{[θ]} = H^γ_{p}(w), $$ where $θ\in (0,1)$, $γ_{0},γ_{1}\in\mathbb{R}$, $p_{0},p_{1}\in(1,\infty)$, $w_{i}$ ($i=0,1$) are arbitrary $A_{p_{i}}$ weight, and $$ γ=(1-θ)γ_{0}+θγ_{1}, \quad \frac{1}{p}=\frac{1-θ}{p_{0}} + \fracθ{p_{1}},\quad w^{1/p}=w^{\frac{(1-θ)}{p_{0}}}_{0}w^{\fracθ{p_{1}}}_{1}.

math.AP↗

arXiv2022

An $L_q(L_p)$-theory for time-fractional diffusion equations with nonlocal operators generated by Lévy processes with low intensity of small jumps

We investigate an $L_{q}(L_{p})$-regularity ($1<p,q<\infty$) theory for space-time nonlocal equations of the type $\partial^α_{t}u = \mathcal{L}u +f$. Here, $\partial^α_{t}$ is the Caputo fractional derivative of order $α\in(0,1)$ and $\mathcal{L}$ is an integro-differential operator $$ \mathcal{L}u(x) = \int_{\mathbb{R}^{d}} \left( u(x)-u(x+y) -\nabla u (x) \cdot y \mathbf{1}_{|y|\leq 1} \right) j_{d}(|y|)dy $$ which is the infinitesimal generator of an isotropic unimodal Lévy process. We assume that the jump kernel $j_{d}(r)$ is comparable to $r^{-d} \ell(r^{-1})$, where $\ell$ is a continuous function satisfying $$ C_{1}\left(\frac{R}{r}\right)^{δ_{1}} \leq \frac{\ell(R)}{\ell(r)} \leq C_{2} \left( \frac{R}{r} \right)^{δ_{2}} \quad \text{for}\;\; \,1\leq r\leq R<\infty, $$ where $0\leq δ_{1}\leq δ_{2}<2$. Hence, $\ell$ can be slowly varying at infinity. Our result covers $\mathcal{L}$ whose Fourier multiplier $Ψ(ξ)$ satisfies $Ψ(ξ)\asymp -\log{(1+|ξ|^β)}$ for $β\in (0,2]$ and $Ψ(ξ) \asymp-(\log(1+|ξ|^{β/4}))^{2}$ for $β\in(0,2)$ by taking $\ell(r) \asymp 1$ and $\ell(r) \asymp \log{(1+r^β)}$ for $r\geq1$ respectively. In this article, we use the Calderón-Zygmund approach and function space theory for operators having slowly varying symbols.

math.AP↗

arXiv2022

A Sobolev space theory for the Stochastic Partial Differential Equations with space-time non-local operators

We deal with the Sobolev space theory for the stochastic partial differential equation (SPDE) driven by Wiener processes $$ \partial_{t}^αu=\left( ϕ(Δ) u +f(u) \right) + \partial_t^β\sum_{k=1}^\infty \int_0^t g^k(u)\,dw_s^k, \quad t>0, x\in \mathbb{R}^d; \,\,\, u(0,\cdot)=u_0 $$ as well as the SPDE driven by space-time white noise $$ \partial^α_{t}u=ϕ(Δ)u + f(u) + \partial^{β-1}_{t}h(u) \dot{W}, \quad t>0,x\in \mathbb{R}^d; \quad u(0,\cdot)=u_{0}. $$ Here, $α\in (0,1), β\in (-\infty, α+1/2)$, $\{w_t^k : k=1,2,\cdots\}$ is a family of independent one-dimensional Wiener processes, and $\dot{W}$ is a space-time white noise defined on $[0,\infty)\times \mathbb{R}^d$. The time non-local operator $\partial_{t}^γ$ denotes the Caputo fractional derivative if $γ>0$ and the Riemann-Liouville fractional integral if $γ\leq0$. The the spatial non-local operator $ϕ(Δ)$ is a type of integro-differential operator whose symbol is $-ϕ(|ξ|^2)$, where $ϕ$ is a Bernstein function satisfying \begin{equation*} κ_0\left(\frac{R}{r}\right)^{δ_{0}} \leq \frac{ϕ(R)}{ϕ(r)}, \qquad \forall\,\, 0 0$ and $δ_0\in (0,1]$. We prove the uniqueness and existence results in Sobolev spaces, and obtain the maximal regularity results of solutions.

math.PR↗

arXiv2022

A Sobolev space theory for the time-fractional stochastic partial differential equations driven by Levy processes

We present an $L_{p}$-theory ($p\geq 2$) for time-fractional stochastic partial differential equations driven by Lévy processes of the type $$ \partial^α_{t}u=\sum_{i,j=1}^d a^{ij}u_{x^{i}x^{j}} +f+\sum_{k=1}^{\infty}\partial^β_{t}\int_{0}^{t} (\sum_{i=1}^dμ^{ik} u_{x^i} +g^k) dZ^k_{s} $$ given with nonzero intial data. Here $\partial^α_t$ and $\partial^β_t$ are the Caputo fractional derivatives, $α\in (0,2), β\in (0,α+1/p)$, and $\{Z^k_t:k=1,2,\cdots\}$ is a sequence of independent Lévy processes. The coefficients are random functions depending on $(t,x)$. We prove the uniqueness and existence results in Sobolev spaces, and obtain the maximal regularity of the solution.

math.AP↗

arXiv2021

Electronic properties of bilayer graphene with magnetic quantum structures studied using the Dirac equation

The electronic properties of bilayer graphene with a magnetic quantum dot and a magnetic quantum ring are investigated. The eigenenergies and wavefunctions of quasiparticle states are calculated analytically by solving decoupled fourth-order differential equations. For the magnetic quantum dot, in the case of a negative inner magnetic field, two peculiar characteristics of the eigenenergy evolution are found: (i) the energy eigenstates change in a stepwise manner owing to energy anticrossing and (ii) the quantum states approach zero energy. For the magnetic quantum ring, there is an angular momentum transition of eigenenergy as the inner radius of the ring varies, and the Aharonov--Bohm effect is observed in the eigenenergy spectra for both positive and negative magnetic fields inside the inner radius.

cond-mat.mes-hall↗

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

Weighted $L_q(L_p)$-estimate with Muckenhoupt weights for the diffusion-wave equations with time-fractional derivatives

We present a weighted $L_{q}(L_{p})$-theory ($p,q\in(1,\infty)$) with Muckenhoupt weights for the equation $$ \partial_{t}^αu(t,x)=Δu(t,x) +f(t,x), \quad t>0, x\in \mathbb{R}^d. $$ Here, $α\in (0,2)$ and $\partial_{t}^α$ is the Caputo fractional derivative of order $α$. In particular we prove that for any $p,q\in (1,\infty)$, $w_{1}(x)\in A_p$ and $w_{2}(t)\in A_q$, $$ \int^{\infty}_0\left(\int_{\mathbb{R}^d} |u_{xx}|^p \,w_{1} dx \right)^{q/p}\,w_{2}dt \leq N \int^{\infty}_0\left(\int_{\mathbb{R}^d} |f|^p \,w_{1} dx \right)^{q/p}\,w_{2}dt, $$ where $A_p$ is the class of Muckenhoupt $A_p$ weights. Our approach is based on the sharp function estimates of the derivatives of solutions.

math.AP↗