Kyungkeun Kang

We develop a conservative, second order accurate fully implicit discretization of ternary (three-phase) Cahn-Hilliard (CH) systems that has an associated discrete energy functional. This is an extension of our work for two-phase systems We analyze and prove convergence of the scheme. To efficiently solve the discrete system at the implicit time-level, we use a nonlinear multigrid method. The resulting scheme is efficient, robust and there is at most a 1 st order time step constraint for stabilit

Abstract We study boundary regularity of weak solutions of the Navier–Stokes equations in the half-space in dimension n ≥ 3. We prove that a weak solution u which is locally in the class L p, q with 2/p + n/q = 1, q > n near boundary is Hölder continuous up to the boundary. Our main tool is a pointwise estimate for the fundamental solution of the Stokes system, which is of independent interest.

Abstract For the Schrödinger flow from ℝ 2 × ℝ + to the 2‐sphere 𝕊 2 , it is not known if finite energy solutions can blow up in finite time. We study equivariant solutions whose energy is near the energy of the family of equivariant harmonic maps. We prove that such solutions remain close to the harmonic maps until the blowup time (if any), and that they blow up if and only if the length scale of the nearest harmonic map goes to 0. © 2006 Wiley Periodicals, Inc.

On a Quasilinear Parabolic Curl System Motivated by Time Evolution of Meissner States of Superconductors

Abstract We consider two dimensional chemotaxis equations coupled to the Navier–Stokes equations. We present a new localized regularity criterion that is localized in a neighborhood at each point. Secondly, we establish temporal decays of the regular solutions under the assumption that the initial mass of biological cell density is sufficiently small. Both results are improvements of previously known results given in Chae et al (2013 Discrete Continuous Dyn. Syst . A 33 2271–97) and Chae et al (

Abstract We prove short time regularity of suitable weak solutions of 3D incompressible Navier–Stokes equations near a point where the initial data is locally in $L^3$. The result is applied to the regularity problems of solutions with uniformly small local $L^3$ norms and of forward discretely self-similar solutions.