# Research

From Newton’s mechanics, many scientists and mathematicians have devoted to studying differential equations to explain several phenomena in the nature. It is natural to ask whether we can solve boundary value problems of given partial differential equations. If we know an existence of solution of the problem, one can also ask some properties of solutions that have.

## Elliptic equations with singular drift terms

For several years, I have devoted to studying unique solvability of linear elliptic equations with singular drift terms in Sobolev spaces. To explain the results, let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$, $n\geq 2$. We consider the following Dirichlet problems of elliptic equations of second-order: \begin{equation}\tag{$D$} -\mathrm{div}(A \nabla u)+\mathrm{div}(u\mathbf{b})=f\quad \text{in } \Omega\qquad u=0\quad \text{on } \partial\Omega \end{equation} and \begin{equation}\tag{$D’$} -\mathrm{div}(A \nabla v)-\mathbf{b}\cdot \nabla v =g\quad \text{in } \Omega\qquad v=0\quad \text{on } \partial\Omega \end{equation} Here $A=(a^{ij}):\mathbb{R}^n\rightarrow \mathbb{R}^{n\times n}$ denotes an $n\times n$ real matrix-valued function which is uniformly elliptic, that is, there exists $0<\delta<1$ such that \begin{equation} \delta|\xi|^2 \leq a^{ij}(x)\xi_i \xi_j\quad \text{and}\quad |a^{ij}(x)|\leq \delta^{-1}\quad \text{for all } x,\xi \in \mathbb{R}^n, \end{equation} $\mathbf{b}=(b^1,b^2,\dots,b^n) : \Omega\rightarrow \mathbb{R}^n$ is a given vector field. Here we follow the usual summation convention for repeated indices.

When $\mathbf{b}$ is zero or a bounded vector field, $L^p$-theory for these equations is well-known by several authors. One might ask $L^p$-theory for unbounded vector field, $\mathbf{b}\in L^q(\Omega;\mathbb{R}^n)$.

During the master course, I wrote a research article [KK18] with my academic advisor Prof. Hyunseok Kim. At the Republic of Korea Air Force Academy, I wrote two research articles [K20] and [K21].

- In a joint work with Hyunseok Kim [KK18], when $A=I$ and $\mathbf{b} \in L^{n}(\Omega;\mathbb{R}^n)$, we proved $L^{\alpha,p}$–solvability for the problems $(D)$ on arbitrary bounded Lipschitz domains $\Omega$ in $\mathbb{R}^n$, $n\geq 3$ for appropriate $(\alpha,p)\in \mathscr{A}\cap \mathscr{B}$. Here $L^{\alpha,p}(\Omega)$ denotes the Bessel potential spaces over $\Omega$, which generalizes the classical Sobolev spaces $W^{k,p}(\Omega)$. Here $\mathscr{A}$ is the set of admissible pairs $(\alpha,p)$ such that the Dirichlet problem for the Poisson equation has a unique solution in $L^{\alpha,p}_0(\Omega) ={ u\in L^{\alpha,p}(\Omega) : u=0\text{ on } \partial\Omega}$. Also, $\mathscr{B}$ denotes the set of some pairs $(\alpha,p)$ for which $\mathbf{b} \cdot \nabla u \in L^{\alpha-2,p}(\Omega)$ for all $u\in L_0^{\alpha,p}(\Omega)$. Similar result also proved for the problem $(D’)$. This result extends the classical result of Jerison-Kenig [JK95]. In the same paper, Neumann problems were also considered.
- When $A$ satisfies (1) and $\mathbf{b} \in L^{n,\infty}(\Omega;\mathbb{R}^n)$, $n\geq 2$, $\mathrm{div} \mathbf{b} \geq 0$ in $\Omega$, I proved that if $q>2$, then there exists $\varepsilon>0$ such that if $g\in W^{-1,q}(\Omega)$, then there exists a unique $v\in W_0^{1,2+\varepsilon}(\Omega)$ of the problem $(D’)$. As an application, I proved that if $f\in \bigcap_{q<2} W^{-1,q}(\Omega)$, there exists a unique $u\in \bigcap_{q<2} W_0^{1,q}(\Omega)$ satisfying the problem $(D)$. The result can be found in [K20]. When $n=2$, this result extends Kim-Tsai [KT20] and Chernobai-Shilkin [CS19].
- In [K21], I proved that if $2<p<\infty$, $A$ satisfies mean oscillation in small balls and $\mathrm{div} A \in L^{2}(\Omega;\mathbb{R}^2)$, $\Omega$ is a bounded Lipschitz domain in $\mathbb{R}^2$ which has a small Lipschitz constant, and $\mathbf{b} \in L^{2}(\Omega;\mathbb{R}^2)$, then the problem $(D’)$ has a unique solution in $W_0^{1,p}(\Omega)$. A similar result was also proved for the problem $(D)$. Results are new even if $A=I$. The results complement Kim-Kim [KK15] and Kang-Kim [KK17] when $n=2$. Further research should be done to remove an additional assumption on $\mathrm{div} A$.

## Mathematical Fluid Dynamics: equations from Magnetohydrodynamics

I also worked on the global well-posedness of fluid equations that have fractional dissipations. To explain my result, let us first consider the generalized magnetohydrodynamics equations:

Here $u:\mathbb{R}^d\times (0,\infty)\rightarrow \mathbb{R}^d$ denotes the velocity field and the magnetic field, respectively. For $s>0$, $\Lambda^s$ denotes the fractional Laplacian which is defined by the Fourier transform:

When $\alpha=\beta=1$, the system is reduced to the classical MHD equation which was first introduced by Hannes Alfvén to describe the fluid of motions that were induced by magnetic fields.

Many researchers have devoted themselves to establishing well-posedness theory and regularity properties of solutions to the equation. Duvaut-Lions [DL72] proved the existence of weak solutions to the equation. Later, the result was extended by Wu [W03] for general $\alpha$ and $\beta$.

Starting from Arnol’d pioneering work, many researchers studied differential equations via dynamical points of view. It is natural to ask a equilibrium point of the corresponding dynamical systems. Arnol’d considered incompressible Euler equations that are regarded as a geodesic equations to find a fixed point of the equations, which is a steady Euler equation.

Later, Moffatt considered similar situations by working on magnetic fields. He constructed magnetohydrodyanmics equilibria from ideal MHD equations by letting $t\rightarrow \infty$. This procedure is called a *magnetic relaxation*. However, his construction has a limitation because of the problem on discontinuity in time variables. Later, Moffatt suggested a model that could lead to a desirable magnetic relaxation:

Moffatt introduced this model when $\alpha=1$ and $\eta=0$. When $\nu,\eta>0$, $\alpha=\beta=1$, and $d=2,3$, McCormick-Robinson-Rodrigo [MRR14] proved global existence of weak solutions of (4). For the two-dimensional case, they also proved uniqueness and regularity of weak solutions. Recently, Ji-Tan [JT21] proved global existence of strong solutions of (4) when $d=3$, $\alpha=1$, and $\beta\geq 3/2$. On the other hand, when $\nu>0$, $\eta=0$, and $\alpha=0$, Brenier [B14] proved global existence of dissipative weak solutions on the two-dimensional torus $\mathbb{T}^2$. When $\nu>0$, $\eta=0$, and $\alpha=1$, Fefferman et.al. [FMRR14] proved local existence and uniqueness of strong solutions of (4) on $\mathbb{R}^d$, $d=2,3$. Recently, when $\nu>0$, $\eta=0$, and $\alpha>d/2+1$, Beekie-Friedlander-Vicol [BFV21] established global existence of strong solutions of (4) on the torus $\mathbb{T}^d$, $d=2,3$. Moreover, they also investigated 2D stability and 3D instability as well as the long-time behavior of solutions. One natural question is whether we can relax these regularity assumptions on $\alpha$ and $s$ to guarantee the global solution to this Stokes-Magneto system.

In a joint work with Hyunseok Kim [HK22], we proved the existence of global weak solutions of the problem for general $\alpha,\beta$ on whole space $\mathbb{R}^d$, $d\geq 2$. More precisely, if $\alpha$, $\beta$ satisfy $1/2<\alpha<(d+1)/2$, $\beta >0$, and $\min(\alpha+\beta,2\alpha+\beta-1)>d/2$, then we proved the existence of global weak solutions of (4). Moreover, we also proved that weak solutions are unique if $\beta \geq 1$ and $\min (\alpha+\beta,2\alpha+\beta-1)\geq d/2+1$, in addition.

## Mathematical Fluid Dynamics: local regularity theory for Stokes equations

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