I am an assistant professor at the Department of Mathematics, the University of Wisconsin-Madison. My research area is the field of partial differential equations in the kinetic theory and fluid dynamics.
(new) I will give a lecture series for a summer school at USC. For more information link.
Contact: ckim (dot) pde (at) gmail (dot) com
- Formation and Propagation of Discontinuity for Boltzmann Equation in Non-Convex Domains, Communications in Mathematical Physics, Volume 308, Number 3, 641-701 (2011) link
- The Boltzmann Equation near a Rotational Local Maxwellian, (S. Yun), SIAM Journal on mathematical Analysis. 44, no.4 (2012), 2560-2598, link
- Boltzmann Equation with a Large Potential in a Periodic Box, Comm. PDE, 39, Issue 8, (2014) 1393-1423 link
- The viscous surface-internal wave problem: global well-posedness and decay (I.Tice, Y. Wang), Arch. Rational Mech. Anal., 212, Issue 1, pp 1-92 (2014) link
- Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law (R.Esposito, Y. Guo, R.Marra), Communications in Mathematical Physics, Volume 323, Number 177-239 (2013) link
- BV-regularity of the Boltzmann equation in Non-convex Domains, (Y.Guo, D.Tonon, A.Trescases), Arch. Rational Mech. Anal., 220, Issue 3, pp 1045-1093 (2016) link
- Regularity of the Boltzmann Equation in Convex Domains, (Y.Guo, D.Tonon, A.Trescases), Invent. Math, 207, Issue 1, pp 115–290 (2017) link, Presentation
Abstract. A basic question about regularity of Boltzmann solutions in the presence of physical boundary conditions has been open due to characteristic nature of the boundary as well as the non-local mixing of the collision operator. Consider the Boltzmann equation in a strictly convex domain with the specular, bounce-back and diffuse boundary condition. With the aid of a distance function toward the grazing set, we construct weighted classical C1 solutions away from the grazing set for all boundary conditions. For the diffuse boundary condition, we construct W1,p solutions for 1 < p < 2 and weighted W1,p solutions for 2 ≤ p ≤ ∞ as well.
- Stationary solutions to the Boltzmann equation in the Hydrodynamic limit, (R.Esposito, Y. Guo, R.Marra), submitted, link, Presentation1, Presentation2
Abstract. Despite its conceptual and practical importance, the rigorous derivation of the steady incompressible Navier-Stokes-Fourier system from the Boltzmann theory has been an outstanding open problem for general domains in 3D. We settle this open question in the affirmative, in the presence of a small external field and a small boundary temperature variation for the diffuse boundary condition. We employ a recent quantitative L2−L∞ approach with new L6 estimates for the hydrodynamic part Pf of the distribution function. Our results also imply the validity of Fourier law in the hydrodynamical limit, and our method leads to asymptotical stability of steady Boltzmann solutions as well as the derivation of the unsteady Navier-Stokes Fourier system.
- The Boltzmann equation with specular boundary condition in convex domains, (D. Lee), accepted in Comm. Pure Appl. Math., link
Abstract. We establish the global-wellposedness and stability of the Boltzmann equation with the specular reflection boundary condition in general smooth convex domains when an initial datum is close to the Maxwellian with or without a small external potential. In particular, we have completely solved the long standing open problem after an announcement of  in 1977.
- Dynamics and stability of surfactant-driven surface waves (I.Tice), submitted, link