arXiv subjects

Junhee Ryu

Publications and source records attributed to Junhee Ryu.

13 recordsLinked to original sources

arXiv2026

Weighted mixed-norm estimates for fractional parabolic equations with space-time nonlocal operators

We establish weighted mixed-norm estimates for fractional parabolic equations \begin{equation*} \partial_t^\alpha u=Lu-\lambda u+f \text{ in } (0,T)\times\mathbb{R}^d, \end{equation*} with nonlocal operators in both time and space. Here, $\partial_t^\alpha$ is the Caputo derivative of order $\alpha\in(0,1)$, and $L$ is a spatially nonlocal operator of order $\sigma\in(0,2)$ whose kernel is merely measurable in time. We also obtain the corresponding estimates in the odd mixed-norm spaces, where the inner integration is taken in time and the outer one in space. The estimates are robust in the limit $\alpha\to1$ and $\sigma\to2$. We also establish unique solvability when either $T<\infty$ or $\lambda>0$. The proof is based on a direction-by-direction extension argument.

math.AP↗

arXiv2025

$L_p$-estimates for nonlocal equations with general L\'evy measures

We consider nonlocal operators of the form \begin{equation*} L_t u(x) = \int_{\mathbb{R}^d} \left( u(x+y)-u(x)-\nabla u(x)\cdot y^{(\sigma)} \right) \nu_t(dy), \end{equation*} where $\nu_t$ is a general L\'evy measure of order $\sigma \in(0,2)$. We allow this class of L\'evy measures to be very singular and impose no regularity assumptions in the time variable. Continuity of the operators and the unique strong solvability of the corresponding nonlocal parabolic equations in $L_p$ spaces are established. We also demonstrate that, depending on the ranges of $\sigma$ and $d$, the operator can or cannot be treated in weighted mixed-norm spaces.

math.AP↗

arXiv2025

A regularity theory for second-order parabolic partial differential equations in weighted mixed norm Sobolev-Zygmund spaces

We develop an optimal regularity theory for parabolic partial differential equations in weighted mixed norm Sobolev-Zygmund spaces. The results extend the classical Schauder estimates to coefficients that are merely measurable in time and to the critical case of integer-order regularity. In addition, nonzero initial data are treated in the optimal trace space via a sharp trace theorem.

math.AP↗

arXiv2025

On nondivergence form linear parabolic and elliptic equations with degenerate coefficients

We establish the unique solvability in weighted mixed-norm Sobolev spaces for a class of degenerate parabolic and elliptic equations in the upper half space. The operators are in nondivergence form, with the leading coefficients given by $x_d^2a_{ij}$, where $a_{ij}$ is bounded, uniformly nondegenerate, and measurable in $(t,x_d)$ except $a_{dd}$, which is measurable in $t$ or $x_d$. In the remaining spatial variables, they have weighted small mean oscillations. In addition, we investigate the optimality of the function spaces associated with our results.

math.AP↗

arXiv2025

The Dirichlet problem for stochastic partial differential equations with nonlocal operators in $C^{1,\sigma}$ open sets

This paper provides a comprehensive Sobolev regularity theory for the Dirichlet problem of stochastic partial differential equations in $C^{1,\sigma}$ open sets. We consider substantially large classes of nonlocal operators and generalized Gaussian noise. Our main results include the existence and uniqueness of strong solutions in weighted Sobolev spaces, along with maximal $L_p$-regularity estimates for the solutions.

math.PR↗

arXiv2024

Sobolev estimates for parabolic and elliptic equations in divergence form with degenerate coefficients

We study a class of degenerate parabolic and elliptic equations in divergence form in the upper half space $\{x_d>0\}$. The leading coefficients are of the form $x_d^2a_{ij}$, where $a_{ij}$ are bounded, uniformly elliptic, and measurable in $(t,x_d)$ except $a_{dd}$, which is measurable in $t$ or $x_d$. Additionally, they have small bounded mean oscillations in the other spatial variables. We obtain the well-posedness and regularity of solutions in weighted mixed-norm Sobolev spaces.

math.AP↗

arXiv2024

Tuning Fast Memory Size based on Modeling of Page Migration for Tiered Memory

Tiered memory, built upon a combination of fast memory and slow memory, provides a cost-effective solution to meet ever-increasing requirements from emerging applications for large memory capacity. Reducing the size of fast memory is valuable to improve memory utilization in production and reduce production costs because fast memory tends to be expensive. However, deciding the fast memory size is challenging because there is a complex interplay between application characterization and the overhead of page migration used to mitigate the impact of limited fast memory capacity. In this paper, we introduce a system, Tuna, to decide fast memory size based on modeling of page migration. Tuna uses micro-benchmarking to model the impact of page migration on application performance using three metrics. Tuna decides the fast memory size based on offline modeling results and limited information on workload telemetry. Evaluating with common big-memory applications and using 5% as the performance loss target, we show that Tuna in combination with a page management system (TPP) saves fast memory by 8.5% on average (up to 16%). This is in contrast to the 5% saving in fast memory reported by Microsoft Pond for the same workloads (BFS and SSSP) and the same performance loss target.

cs.PF↗

arXiv2023

Nonlocal elliptic and parabolic equations with general stable operators in weighted Sobolev spaces

We study nonlocal elliptic and parabolic equations on $C^{1,\tau}$ open sets in weighted Sobolev spaces, where $\tau\in (0,1)$. The operators we consider are infinitesimal generators of symmetric stable L\'evy processes, whose L\'evy measures are allowed to be very singular. Additionally, for parabolic equations, the measures are assumed to be merely measurable in the time variable.

math.AP↗

arXiv2023

Weighted Sobolev space theory for non-local elliptic and parabolic equations with non-zero exterior condition on $C^{1,1}$ open sets

We introduce a weighted Sobolev space theory for the non-local elliptic equation $$ \Delta^{\alpha/2}u=f, \quad x\in \mathcal{O}\,; \quad r_{\overline{\mathcal{O}}^c}u=g $$ as well as for the non-local parabolic equation $$ u_t=\Delta^{\alpha/2}u+f, \quad t>0,\, x\in \mathcal{O} \,; \quad r_{\mathcal{O}}u(0,\cdot)=u_0, \,r_{(0,T)\times \overline{\mathcal{O}}^c}u=g. $$ Here, $\alpha\in (0,2)$ and $\mathcal{O}$ is a $C^{1,1}$ open set. We prove uniqueness and existence results in weighted Sobolev spaces. We measure the Sobolev and H\"older regularities of arbitrary order derivatives of solutions using a system of weights consisting of appropriate powers of the distance to the boundary. One of the most interesting features of our results is that, unlike the classical results in Sobolev spaces without weights, the weighted regularities of solutions in $\mathcal{O}$ are less affected by those of exterior conditions on $\overline{\mathcal{O}}^c$. For instance, even if $g=\delta_{x_0}$, the dirac delta distribution concentrated at $x_0\in \overline{\mathcal{O}}^c $, the solution to the elliptic equation given with $f=0$ is infinitely differentiable in $\mathcal{O}$, and for any $k=0,1,2, 3,\cdots$, $\varepsilon>0$, and $\delta\in (0,1)$, it holds that $$ |d_x^{-\frac{\alpha}{2}+\varepsilon+k}D^k_xu|_{C_b(\mathcal{O})} +|d_x^{-\frac{\alpha}{2}+\varepsilon+k+\delta} D^k_xu|_{C^{\delta}(\mathcal{O})}<\infty, $$ where $d_x=dist(x, \partial \mathcal{O})$.

math.AP↗

arXiv2023

Rethinking Memory Profiling and Migration for Multi-Tiered Large Memory Systems

Multi-tiered large memory systems call for rethinking of memory profiling and migration because of the unique problems unseen in the traditional memory systems with smaller capacity and fewer tiers. We develop MTM, an application-transparent page management system based on three principles: (1) connecting the control of profiling overhead with the profiling mechanism for high-quality profiling; (2) building a universal page migration policy on the complex multi-tiered memory for high performance; and (3) introducing huge page awareness. We evaluate MTM using common big-data applications with realistic working sets (hundreds of GB to 1 TB). MTM outperforms seven state-of-the-art solutions by up to 42% (17% on average)

cs.PF↗

arXiv2022

Sobolev regularity theory for the non-local elliptic and parabolic equations on $C^{1,1}$ open sets

We study the zero exterior problem for the elliptic equation $$ \Delta^{\alpha/2}u-\lambda u=f, \quad x\in D\,; \quad u|_{D^c}=0 $$ as well as for the parabolic equation $$ u_t=\Delta^{\alpha/2}u+f, \quad t>0,\, x\in D \,; \quad u(0,\cdot)|_D=u_0, \,u|_{[0,T]\times D^c}=0. $$ Here, $\alpha\in (0,2)$, $\lambda \geq 0$ and $D$ is a $C^{1,1}$ open set. We prove uniqueness and existence of solutions in weighted Sobolev spaces, and obtain global Sobolev and H\"older estimates of solutions and their arbitrary order derivatives. We measure the Sobolev and H\"older regularities of solutions and their arbitrary derivatives using a system of weights consisting of appropriate powers of the distance to the boundary. The range of admissible powers of the distance to the boundary is sharp.

math.AP↗

arXiv2021

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 $$ \partial_{t}^{\alpha}u=\left( \phi(\Delta) u +f(u) \right) + \partial_t^\beta \sum_{k=1}^\infty \int_0^t g^k(u)\,dw_s^k, \quad t>0, x\in \mathbb{R}^d; \,\,\, u(0,\cdot)=u_0 $$ as well as the SPDE driven by space-time white noise $$ \partial^{\alpha}_{t}u=\phi(\Delta)u + f(u) + \partial^{\beta-1}_{t}h(u) \dot{W}, \quad t>0,x\in \mathbb{R}^d; \quad u(0,\cdot)=u_{0}. $$ Here, $\alpha\in (0,1), \beta\in (-\infty, \alpha+1/2)$, $\{w_t^k : k=1,2,\cdots\}$ 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 $\partial_{t}^{\gamma}$ denotes the Caputo fractional derivative if $\gamma>0$ and the Riemann-Liouville fractional integral if $\gamma\leq0$. The the spatial non-local operator $\phi(\Delta)$ is a type of integro-differential operator whose symbol is $-\phi(|\xi|^2)$, where $\phi$ is a Bernstein function satisfying \begin{equation*} \kappa_0\left(\frac{R}{r}\right)^{\delta_{0}} \leq \frac{\phi(R)}{\phi(r)}, \qquad \forall\,\, 0 0$ and $\delta_0\in (0,1]$. We prove the uniqueness and existence results in Sobolev spaces, and obtain the maximal regularity results of solutions.

math.PR↗

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 $$ \partial_{t}^{\alpha}u=\phi(\Delta) u +f, \quad t>0,\, x\in \mathbb{R}^d \quad\, ;\, u(0,\cdot)=u_0. $$ Here $p,q>1$, $\alpha\in (0,1)$, $\partial_{t}^{\alpha}$ is the Caputo fractional derivative of order $\alpha$, and $\phi$ is a Bernstein function satisfying the following: $\exists \delta_0\in (0,1]$ and $c>0$ such that \begin{equation} \label{eqn 8.17.1} c \left(\frac{R}{r}\right)^{\delta_0}\leq \frac{\phi(R)}{\phi(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^{\alpha}_t u|+|u|+|\phi(\Delta)u|\|_{L_q([0,T];L_p)}\leq N(\|f\|_{L_q([0,T];L_p)}+ \|u_0\|_{B_{p,q}^{\phi,2-2/ \alpha q}}), \end{align*} where $B_{p,q}^{\phi,2-2/\alpha q}$ is a modified Besov space on $\mathbb{R}^d$ related to $\phi$. Our approach is based on BMO estimate for $p=q$ and vector-valued Calder\'on-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↗