Research

I’m interested in or working on (or have done) these problems.

1. Elliptic and Parabolic equation with singular drift

Let \Omega be a bounded domain in \mathbb{R}^n. We consider Dirichlet problems for elliptic equations of second order:

(D)   \begin{equation*} \left\{\begin{alignedat}{2} -\triangle u +\Div(u\boldb) &= f &&\quad \text{in } \Omega,\\ u &=u_D && \quad \text{on } \partial\Omega, \end{alignedat}\right. \end{equation*}

and

(D’)   \begin{equation*} \left\{\begin{alignedat}{2} -\triangle v -\boldb\cdot \nabla v &= g &&\quad \text{in } \Omega,\\ v &=v_D & &\quad \text{on } \partial\Omega. \end{alignedat}\right. \end{equation*}

Here \boldb : \Omega \rightarrow \mathbb{R}^n is a given vector field. Similarly, we can also consider the Neumann boundary value problems. The \Sob{1}{p}(\Omega) solvability of these equations is well-known when \boldb \in \Leb{\infty}(\Omega) and \Omega is a bounded Lipschitz domain. What condition on \boldb gurantees the solvability in Sobolev space scale? If \boldb \in \Leb{n}(\Omega), \Sob{1}{2}-solvability of Dirichlet problem is shown by Droniou. \Sob{1}{p}-solvability is proved by Kim and Kim (2015, SIAM) when \Omega is a bounded C^1 domain. There are similar results for the Neumann boundary value problems. See Kang and Kim (2017, CPAA)

I’m working on an extension of these results in several directions (minimal assumption on domains, Bessel potential spaces, parabolic analog of the problem).

  • Dirichlet and Neumann problems for elliptic equations with singular drifts on Lipschitz domains (with Hyunseok Kim, preprint)

2. Equations originated from fluid mechanics and physics

I’m interested in the mathematical theory of fluid mechanics, especially the Navier-Stokes equation. Also, I have interested in the theory of dispersive equations.