**Hyunwoo Kwon (권현우, 權賢宇) / Will Kwon**

## Contact

Division of Applied Mathematics,

Brown University,

182 George Street, Providence, RI 02912, USA

Email. hyunwoo_kwon[at]brown[dot]edu

## About me

I am Ph. D. candidate in the Division of Applied Mathematics at Brown University. My thesis advisor is Prof. Hongjie Dong. Previously, I worked as a researcher in the Department of Mathematics at Sogang University. I was an assistant professor and lecturer (with the rank Lieutenant) in the Department of Mathematics at Republic of Korea Air Force Academy. I did my Bachelor and Master in mathematics at Sogang University. My master thesis advisor was Prof. Hyunseok Kim.

You can pronounce my Korean name as “Hyeo-nu” (there is a linking sound between un and woo)

Here is my CV.

The drawing was given by one of my students at the Republic of Korea Air Force Academy.

## Research Interests

- Partial Differential Equations
- Harmonic Analysis (real-variable methods)

## Activity

- In Fall 2024, I will organize learning seminar on the Cauchy problem on free surface flows.
- Starting from Spring 2023, I am organizing seminar Graduate Lecture Series in Analysis and PDEs with Tainara Gobetti Borges.
- In Fall 2022, I organized learning seminar on Harmonic analysis related to PDEs.

## Publications / Preprints

## 1. Working progress on kinetic equations with W. Huang

**Abstract.** TBA

## 2. Working progress on Stokes equations with H. Dong

**Abstract.** TBA

## 3. Global solutions to Stokes-Magneto equations with fractional dissipations, with H. Bae and J. Shin, arXiv:2310.03255 [math.AP]

**Abstract.** 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 $\boldsymbol{b}_0\in H^s(\mathbb{T}^d)$. We also show the existence of a unique mild solution of the resisitive case with initial data $\boldsymbol{b}_0$ in the critical $L^p(\mathbb{R}^d)$ space. Moreover, we show that the $L^p$-norm of $\boldsymbol{b}(t)$ converges to zero as $t\rightarrow\infty$ if the initial data is sufficiently small.

## 4. Interior and boundary mixed norm derivative estimates for nonstationary Stokes equations, with H. Dong, arXiv:2308.09220 [math.AP]

**Abstract.** 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.

## 5. Global existence and uniqueness of weak solutions of a Stokes-Magneto system with fractional diffusions, with H. Kim, J. Differential Equations (2023), vol. 374, 497--547, [Journal] arXiv:2302.02046 [math.AP]

**Abstract.** We consider a Stokes-Magneto system in $\mathbb{R}^d$ ($d\geq 2)$ with fractional diffusions $\Lambda^{2\alpha}\boldsymbol{u}$ and $\Lambda^{2\beta}\boldsymbol{b}$ for the velocity $\boldsymbol{u}$ and the magnetic field $\boldsymbol{b}$, respectively. Here $\alpha,\beta$ are positive constants and $\Lambda^s = (-\Delta)^{s/2}$ 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 $\alpha$, $\beta$ satisfy $1/2<\alpha<(d+1)/2$, $\beta >0$,
and $\min(\alpha+\beta,2\alpha+\beta-1)>d/2$. It is also shown that weak solutions are unique if $\beta \geq 1$ and $\min (\alpha+\beta,2\alpha+\beta-1)\geq d/2+1$, in addition.

## 6. Elliptic equations in divergence form with drifts in $L^2$, Proc. Amer. Math. Soc. 150(2022), no. 8, 3415–-3429, 2022 [Journal], arXiv:2104.01300 [math.AP]

**Abstract.** We consider the Dirichlet problem for second-order linear elliptic equations in divergence form

in bounded Lipschitz domain $\Omega$ in $\mathbb{R}^2$, where $A:\mathbb{R}^2\rightarrow \mathbb{R}^{2^2}$, $\mathbf{b}: \Omega\rightarrow \mathbb{R}^2$, and $\lambda \geq 0$ are given. If $2<p<\infty$ and $A$ has a small mean oscillation in small balls, $\Omega$ has small Lipschitz constant, and $\mathrm{div } A, \mathbf{b}\in L^2(\Omega;\mathbb{R}^2)$, then we prove existence and uniqueness of weak solutions in $W_0^{1,p}(\Omega)$ of the problem. Similar result also holds for the dual problem.

## 7. Existence and uniqueness of weak solution in $W^{1,2+\varepsilon}$ for elliptic equations with drifts in weak-$L^{n}$ spaces, J. Math. Anal. Appl. 500(1) (2021) 125165, [Journal], arXiv:2011.07524 [math.AP]

**Abstract.** We consider the following Dirichlet problems for elliptic equations with singular drift $\mathbf{b}$:

where $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^n$, $n\geq 2$. Assuming that $\mathbf{b}\in L^{n,\infty}(\Omega)^n$ has non-negative weak divergence in $\Omega$, we establish existence and uniqueness of weak solution in $W^{1,2+\varepsilon}_0(\Omega)$ of the problem (b) when $A$ is bounded and uniformly elliptic. As an application, we prove unique solvability of weak solution $u$ in $W^{1,2-}_0(\Omega)$ for the problem (a) for every $f\in W^{-1,2-}(\Omega)$.

## 8. Dirichlet and Neumann problems for elliptic equations with singular drifts on Lipschitz domain, with H. Kim, Trans. Amer. Math. Soc. 375(9), 2022, [Journal], arXiv:1811.12619 [math.AP]

**Abstract.** We consider the Dirichlet and Neumann problems for second-order linear elliptic equations:

in a bounded Lipschitz domain $\Omega$ in $\mathbb{R}^n$ ($n\geq 3$), where $\mathbf{b}:\Omega\rightarrow \mathbb{R}^n$ is a given vector field. Under the assumption that $\mathbf{b}\in L^n(\Omega)^n$, we first establish existence and uniqueness of solutions in $L^p_{\alpha}(\Omega)$ for the Dirichlet and Neumann problems. Here $L^p_{\alpha}(\Omega)$ denotes the Sobolev space (or Bessel potential space) with the pair $(\alpha,p)$ satisfying certain conditions. These results extend the classical works of Jerison-Kenig (1995) and Fabes-Mendez-Mitrea (1998) for the Poisson equation. We also prove existence and uniqueness of solutions of the Dirichlet problem with boundary data in $L^2(\partial\Omega)$. Our results for the Dirichlet problems hold even for the case $n=2$.

## Previous events

- 2023 Workshop for admission to graduate school for Sogang alumni, co-organized with various alumni in Sogang, July 2023.
- Workshop for young mathematicians in Korea, co-organized with D. Ko and B. Sohn, January 2022.
- Workshop for students in Analysis and PDE, co-organized with S. Ryoo, H. Kim, and Y. Lee, July 2018.

## Miscellaneous

- My footprints related to LaTeX (can be found on the Internet)
- I was a technical editor of ICM Proceedings 2014 which can be found in pdf files in IMU homepage.
- My name was written in the acknowledgement of Sobolev and Viscosity Solutions for Fully Nonlinear Elliptic and Parabolic Equations written by N. V. Krylov.
- I gave several online lectures on Youtube (in Korean) link. Lecture notes can be found in ‘Notes’ page.

- Seminars
- Others
- CVGMT
- MathSciNet
- Julia Julia textbook NeuralPDE Julia Calculus
- Brown University SIAM Student Chapter
- Stupid Mathematics Seminar in Korea (I am a founder of this seminar)

- My favorite quotes
- 不爲也, 非不能也 (孟子) / A lack of effort, not a lack of ability. (Mencius) translation source
- It is not only courage to boldly jump into a dangerous place. It is also courage to ignore the strong temptation to jump in and go on my own path silently. It is foolish to react immediately when a reverse flow occurs in a forward flow. Maintaining your current flow when your opponent has reversed flow becomes countercurrent from the opponent’s point of view. Therefore, the attitude of maintaining my flow without wavering is the best defense and offensive means. (Changho Lee, one of the best “baduk(Go)” player) [translated by Google]

- Some random footprint in the internet
- When I was young, I made a train simulator addon (Seoul Metro Line 4 with Teciyoo, released in 2007) based on the program Bve Transim 4.

- Some random quote made by me? (Usually I do this joke when I teach proof by a contradiction)