arXiv subjects

Hyunwoo Kwon

Publications and source records attributed to Hyunwoo Kwon.

12 recordsLinked to original sources

arXiv2026

GELATO: Multi-Material Topology Optimization of Programmable Gel-Elastomer Structures

Gel-elastomer composites, comprising an active swellable hydrogel and a passive elastomer, are a compelling class of programmable material systems (PMS) capable of shape morphing under multiphysics actuation. The precise design of the topology and material distribution unlocks complex programmability instrumental in wearable electronics, soft robots, and drug delivery; however, the structure-function relationship is highly non-intuitive, rendering both trial-and-error and conventional design approaches largely intractable. To address this, we present a topology optimization (TO) framework for the automated design of such structures, enabling systematic exploration of the design space for target functionalities realized via programmable shape morphing. In particular, we propose a multi-material TO framework that concurrently optimizes the structural topology and the spatial distribution of the gel-elastomer phases. The design is represented via a coordinate-based neural network, and the mechanical response of both phases is described within a unified constitutive framework based on the Flory-Rehner theory. Furthermore, we present an end-to-end differentiable design framework with implicit differentiation that accommodates various objective functions, constraints, and discretizations. We demonstrate the framework on shape-programming structures and soft actuators. The framework is further validated through the design of organogel-hydrogel composites for multi-stimuli responsiveness across chemically distinct solvent environments, and of anisotropic hydrogels wherein the local fiber orientation is optimized concurrently with the topology. The codebase implemented in JAX is publicly shared to support benchmarking and reproducibility.

cs.CE↗

arXiv2026

Global well-posedness of the one-phase Muskat problem with surface tension

In this paper, we establish the global well-posedness of the one-phase Muskat problem with surface tension for small initial data. This problem describes the motion of the interface separating a wet region from a dry region within a porous medium, a process governed by Darcy's law. Although physically essential, the inclusion of surface tension introduces an additional challenge. We prove that if the initial free boundary is sufficiently small in $H^s$, $s>d/2+1$, then the problem admits a unique global strong solution. Moreover, the solution converges to zero in Lipschitz norm as $t\rightarrow\infty$. To the best of our knowledge, this work constitutes the first global well-posedness result for the one-phase Muskat problem with surface tension.

math.AP↗

arXiv2026

Scattering map for the Vlasov--Poisson system with a repulsive harmonic potential

We consider the Vlasov--Poisson system with a repulsive harmonic potential and prove the (modified) scattering of solutions, as well as the existence of wave operators, in any spatial dimension $d\geq 2$. The main novelty of this work is the construction of the wave operators and the introduction of the lens transform for the Vlasov--Poisson system. In addition, we provide a new and simpler proof that relaxes the assumptions on the initial data compared with those in Bigorgne, Velozo Ruiz, and Velozo Ruiz (2025) and Velozo Ruiz, and Velozo Ruiz (2024).

math.AP↗

arXiv2024

Spatial $C^1$, $C^2$, and Schauder estimates for nonstationary Stokes equations with Dini mean oscillation coefficients

We establish the spatial differentiability of weak solutions to nonstationary Stokes equations in divergence form with variable viscosity coefficients having $L_2$-Dini mean oscillations. As a corollary, we derive local spatial Schauder estimates for such equations if the viscosity coefficient belongs to $C^\alpha_x$. Similar results also hold for strong solutions to nonstationary Stokes equations in nondivergence form.

math.AP↗

arXiv2024

Nonstationary Stokes equations on a domain with curved boundary under slip boundary conditions

We consider nonstationary Stokes equations in nondivergence form with variable viscosity coefficients and generalized Navier slip boundary conditions with slip tensor $\mathcal{A}$ in a domain $\Omega$ in $\mathbb{R}^d$. First, under the assumption that slip matrix $\mathcal{A}$ is sufficiently smooth, we establish a priori local regularity estimates for solutions near a curved portion of the domain boundary. Second, when $\mathcal{A}$ is the shape operator, we derive local boundary estimates for the Hessians of the solutions, where the right-hand side does not involve the pressure. Notably, our results are new even if the viscosity coefficients are constant.

math.AP↗

arXiv2024

Scattering of the Vlasov-Riesz system in the three dimensions

We consider an asymptotic behavior of solutions to the Vlasov-Riesz system of order $\alpha$ in $\mathbb{R}^3$ which is a kinetic model induced by Riesz interactions. We prove small data scattering when $1/2<\alpha<1$ and modified scattering when $1<\alpha<1+\delta$ for some $\delta>0$. Moreover, we show the existence of (modified) wave operators for such a regime. To the best of our knowledge, this is the first result on the existence of modified scattering with polynomial correction in kinetic models.

math.AP↗

arXiv2023

Global solutions to Stokes-Magneto equations with fractional dissipations

In this paper, we investigate a Stokes-Magneto system with fractional diffusions. We first deal with the non-resistive case in $\mathbb{T}^{d}$ and establish the local and global well-posedness with initial magnetic field $\mathbf{b}_0\in H^{s}(\mathbb{T}^d)$. We also show the existence of a unique mild solution of the resistive case with initial data $\mathbf{b}_0$ in the critical $L^{p}(\mathbb{R}^d)$ space. Moreover, we show that $\|\mathbf{b}(t)\|_{L^{p}}$ converges to zero as $t\rightarrow\infty$ when the initial data is sufficiently small.

math.AP↗

arXiv2023

Interior and boundary mixed norm derivative estimates for nonstationary Stokes equations

We obtain weighted mixed norm Sobolev estimates in the whole space for nonstationary Stokes equations in divergence and nondivergence form with variable viscosity coefficients that are merely measurable in time variable and have small mean oscillation in spatial variables in small cylinders. As an application, we prove interior mixed norm derivative estimates for solutions to both equations. We also discuss boundary mixed norm Hessian estimates for solutions to equations in nondivergence form under the Lions boundary conditions.

math.AP↗

arXiv2023

Global existence and uniqueness of weak solutions of a Stokes-Magneto system with fractional diffusions

We consider a Stokes-Magneto system in $\mathbb{R}^d$ ($d\geq 2$) with fractional diffusions $Λ^{2α}\boldsymbol{u}$ and $Λ^{2β}\boldsymbol{b}$ for the velocity $\boldsymbol{u}$ and the magnetic field $\boldsymbol{b}$, respectively. Here $α,β$ are positive constants and $Λ^s = (-Δ)^{s/2}$ is the fractional Laplacian of order $s$. We establish global existence of weak solutions of the Stokes-Magneto system for any initial data in $L_{2}$ when $α$, $β$ satisfy $1/2<α<(d+1)/2$, $β>0$, and $\min\{α+β,2α+β-1\}>d/2$. It is also shown that weak solutions are unique if $β\geq 1$ and $\min \{α+β,2α+β-1\}\geq d/2+1$, in addition.

math.AP↗

arXiv2021

Dirichlet and Neumann problems for elliptic equations with singular drifts on Lipschitz domains

We consider the Dirichlet and Neumann problems for second-order linear elliptic equations: \[ -\triangle u +\mathrm{div}(u\mathbf{b}) =f \quad\text{ and }\quad -\triangle v -\mathbf{b} \cdot \nabla v =g \] in a bounded Lipschitz domain $Ω$ in $\mathbb{R}^n$ $(n\geq 3)$, where $\mathbf{b}:Ω\rightarrow \mathbb{R}^n$ is a given vector field. Under the assumption that $\mathbf{b} \in L^{n}(Ω)^n$, we first establish existence and uniqueness of solutions in $L_α^{p}(Ω)$ for the Dirichlet and Neumann problems. Here $L_α^{p}(Ω)$ denotes the Sobolev space (or Bessel potential space) with the pair $(α,p)$ satisfying certain conditions. These results extend the classical works of Jerison-Kenig [17] and Fabes-Mendez-Mitrea [12] for the Poisson equation. We also prove existence and uniqueness of solutions of the Dirichlet problem with boundary data in $L^{2}(\partialΩ)$. Our results for the Dirichlet problems hold even for the case $n=2$.

math.AP↗

arXiv2021

Elliptic equations in divergence form with drifts in $L^2$

We consider the Dirichlet problem for second-order linear elliptic equations in divergence form \begin{equation*} -\mathrm{div }(A\nabla u)+\mathbf{b} \cdot \nabla u+λu=f+\mathrm{div } \mathbf{F}\quad \text{in } Ω\quad\text{and}\quad u=0\quad \text{on } \partialΩ, \end{equation*} in bounded Lipschitz domain $Ω$ in $\mathbb{R}^2$, where $A:\mathbb{R}^2\rightarrow \mathbb{R}^{2^2}$, $\mathbf{b} : Ω\rightarrow \mathbb{R}^2$, and $λ\geq 0$ are given. If $2<p<\infty$ and $A$ has a small mean oscillation in small balls, $Ω$ has small Lipschitz constant, and $\mathrm{div } A,\,\mathbf{b} \in L^{2}(Ω;\mathbb{R}^2)$, then we prove existence and uniqueness of weak solutions in $W^{1,p}_0(Ω)$ of the problem. Similar result also holds for the dual problem.

math.AP↗

arXiv2021

Existence and uniqueness of weak solution in $W^{1,2+\varepsilon}$ for elliptic equation with drifts in weak-$L^{n}$ spaces

We consider the following Dirichlet problems for elliptic equations with singular drift $\mathbf{b}$: \[ \text{(a) } -\operatorname{div}(A \nabla u)+\operatorname{div}(u\mathbf{b})=f,\quad \text{(b) } -\operatorname{div}(A^T \nabla v)-\mathbf{b} \cdot \nabla v =g \quad \text{in } Ω, \] where $Ω$ is a bounded Lipschitz domain in $\mathbb{R}^n$, $n\geq 2$. Assuming that $\mathbf{b}\in L^{n,\infty}(Ω)^n$ has non-negative weak divergence in $Ω$, we establish existence and uniqueness of weak solution in $W^{1,2+\varepsilon}_0(Ω)$ of the problem (b) when $A$ is bounded and uniformly elliptic. As an application, we prove unique solvability of weak solution $u$ in $\bigcap_{q<2} W^{1,q}_0(Ω)$ for the problem (a) for every $f\in \bigcap_{q<2} W^{-1,q}(Ω)$.

math.AP↗