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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 (elliptic/parabolic PDEs, Fluid equations, and Kinetic Equations)
  • Harmonic Analysis (real-variable methods)

Activity

Publications / Preprints

1. Boundary mixed norm estimates for nonstationary Stokes equations on domains with curved boundary with H. Dong, arXiv:2408.17321 [math.AP]

Abstract. We consider nonstationary Stokes equations in nondivergence form with variable viscosity coefficients and Navier slip boundary conditions with slip coefficient $\alpha$ in a domain $\Omega$. On the one hand, if $\alpha$ is sufficiently smooth, then we obtain a priori local regularity estimates for solutions near a curved portion of the boundary of the domain. On the other hand, if $\alpha$ depends on the curvature of the boundary of the domain, then we obtain local boundary estimates of Hessians of solutions where the right-hand side does not contain the pressure. Our results are new even if the viscosity coefficients are constant.

2. Scattering of Vlasov-Riesz systems in the three dimensions, with W. Huang, arXiv:2407.16919 [math.AP]

Abstract. 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.

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

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

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.

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 $\mathbf{b}$:

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

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

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:

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

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

  • My footprints related to LaTeX (can be found on the Internet)
  • Seminars
  • Others
  • 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 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) image info