Abstract We consider the Allen–Cahn equation $${\partial _t}{u}-\Delta u=u-u^3$$ ∂ t u - Δ = 3 with a rapidly mixing Gaussian field as init...

There has been recently an important interest in deriving rigorously the Cahn-Hilliard equation from nonlocal equation, also called aggregation equation. So far, only non-degenerate mobilities were treated. Since we are motivated by models for biomechanics of living tissues, it is useful to include degenerate mobilities. In this framework, present a new method show convergence local The include...

We study the effect of directional quenching on patterns formed in simple bistable systems such as the Allen-Cahn and the Cahn-Hilliard equation on the plane. We model directional quenching as an externally triggered change in system parameters, changing the system from monostable to bistable across an interface. We are then interested in patterns forming in the bistable region, in particular a...

We present a nonlinear model predictive framework for closedloop control of two-phase flows governed by the Cahn-Hilliard NavierStokes system. We adapt the concept for instantaneous control from [6, 12, 16] to construct distributed closed-loop control strategies for twophase flows. It is well known that distributed instantaneous control is able to stabilize the Burger’s equation [16] and also t...

This paper studies a phase field model for the mixture of two immiscible and incompressible fluids. The model is described by a nonlinear parabolic system consisting of the nonstationary Stokes equations coupled with the Allen-Cahn equation through an extra phase induced stress term in the Stokes equations and a fluid induced transport term in the Allen-Cahn equation. Both semi-discrete and ful...

In this paper, we reformulate the diffuse interface model of the tumor growth (S.M. Wise et al., Three-dimensional multispecies nonlinear tumor growth-I: model and numerical method, J. Theor. Biol. 253 (2008) 524--543). In the new proposed model, we use the conservative second-order Allen--Cahn equation with a space--time dependent Lagrange multiplier instead of using the fourth-order Cahn--Hil...

We prove a weak rate of convergence fully discrete scheme for stochastic Cahn--Hilliard equation with additive noise, where the spectral Galerkin method is used in space and backward Euler time. Compared Allen--Cahn type partial differential equation, error analysis here much more sophisticated due to presence unbounded operator front nonlinear term. To address such issues, novel direct approac...