I’m interested in or working on (or have done) these problems.
1. Elliptic and Parabolic equation with singular drift
Let be a bounded domain in . We consider Dirichlet problems for elliptic equations of second order:
Here is a given vector field. Similarly, we can also consider the Neumann boundary value problems. The solvability of these equations is well-known when and is a bounded Lipschitz domain. What condition on gurantees the solvability in Sobolev space scale? If , -solvability of Dirichlet problem is shown by Droniou. -solvability is proved by Kim and Kim (2015, SIAM) when is a bounded 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.