This note completely resolves the asymptotic development of order 2 by Γ-convergence of the mass-constrained Cahn–Hilliard functional, by showing that one of the critical assumptions of the authors’ previous work [10] is unnecessary.

....................................................................................................................................................7 Chapter One..............................................................................................................................................8 Introduction...................................................................................

We present an unconditionally energy stable and uniquely solvable finite difference scheme for the Cahn-Hilliard-Brinkman (CHB) system, which is comprised of a Cahn-Hilliard-type diffusion equation and a generalized Brinkman equation modeling fluid flow. The CHB system is a generalization of the Cahn-Hilliard-Stokes model and describes two phase very viscous flows in porous media. The scheme is...

In one spatial dimension, the metastable dynamics and coarsening process of an n-layer pattern of internal layers is studied for the Cahn-Hilliard equation, the viscous Cahn-Hilliard equation, and the constrained Allen-Cahn equation. These models from the continuum theory of phase transitions provide a caricature of the physical process of the phase separation of a binary alloy. A homotopy para...

Diblock copolymers are a class of materials formed by the reaction of two linear polymers. The different structures taken on by these polymers grant them special properties, which can prove useful in applications such as the development of new adhesives and asphalt additives. We consider a model for the formation of diblock copolymers first proposed by Ohta and Kawasaki [26]. Their model yields...

We use the idea in [33] to develop the energy law preserving method and compute the diffusive interface (phase-field) models of Allen-Cahn and Cahn-Hilliard type, respectively, governing the motion of two-phase incompressible flows. We discretize these two models using a C0 finite element in space and a modified midpoint scheme in time. To increase the stability in the pressure variable we trea...

A lattice Boltzmann equation method based on the Cahn-Hilliard diffuse interface theory is developed to investigate the bubble formation process in a microchannel with T-junction mixing geometry. The bubble formation process has different regimes, namely, squeezing, dripping and jetting regimes, which correspond to the primary forces acting on the system. Transition from regime to regime is gen...

We study an initial-boundary value problem (IBVP) for a coupled Cahn-Hilliard-Hele-Shaw system that models tumor growth. For large initial data with finite energy, we prove global (local resp.) existence, uniqueness, higher order spatial regularity and Gevrey spatial regularity of strong solutions to the IBVP in 2D (3D resp.). Asymptotically in time, we show that the solution converges to a con...

We study pattern-forming dissipative systems in growing domains. We characterize classes of boundary conditions that allow for defect-free growth and derive universal scaling laws for the wave number in the bulk of the domain. Scalings are based on a description of striped patterns in semibounded domains via strain-displacement relations. We compare predictions with direct simulations in the Sw...

Variational image segmentation based on theMumford and Shahmodel [31], together with implementation by the piecewise constant level-setmethod (PCLSM) [26], leads to fully nonlinear Total Variation (TV)-Allen-Cahn equations. The commonlyused numerical approaches usually suffer from the difficulties not only with the nondifferentiability of the TV-term, but alsowith directly evolving the disconti...