website picture

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 a full-time 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

Publications / Preprints

1. Working progress with H. Dong

Abstract. TBA

2. 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.

3. 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.

4. Global existence and uniqueness of weak solutions of a Stokes-Magneto system with fractional diffusions, with H. Kim, to appear in J. Differential Equations, 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.

5. 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

$-\mathrm{div} (A\nabla u) + \mathbf{b}\cdot \nabla u +\lambda u = f+\mathrm{div } \mathbf{F}\quad \text{in } \Omega\quad \text{and}\quad u=0\quad \text{on } \partial\Omega$,

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.

6. 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 $\mathbf{b}$:

$\text{(a)}\, −\mathrm{div}(A\nabla u)+\mathrm{div}(u\mathbf{b})=f,\qquad\text{(b)}\, −\mathrm{div}(A^T\nabla v)−\mathbf{b}\cdot\nabla v=g\quad \text{in } \Omega,$

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 $f\in W^{-1,2-}(\Omega)$.

7. 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:

$−\triangle u+\mathrm{div}(u\mathbf{b})=f\quad \text{and}\quad −\triangle v−\mathbf{b}\cdot\nabla v=g$

in a bounded Lipschitz domain $\Omega$ 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

Miscellaneous