31 research outputs found
Well-Posedness of the Hele-Shaw-Cahn-Hilliard System
We study the well-posedness of the Hele-Shaw-Cahn-Hilliard system modeling binary fluid flow in porous media with arbitrary viscosity contrast but matched density between the components. for initial data in Hs, s\u3ed2+1, the existence and uniqueness of solution in C([0,T];Hs) ∪L2(0,T;Hs+2) that is global in time in the two-dimensional case (d=2) and local in time in the three-dimensional case (d=3) are established. Several blow-up criterions in the three-dimensional case are provided as well. One of the tools that we utilized is the Littlewood-Paley theory in order to establish certain key commutator estimates. © 2012 Elsevier Masson SAS
On the Cahn-Hilliard-Brinkman system
We consider a diffuse interface model for phase separation of an isothermal
incompressible binary fluid in a Brinkman porous medium. The coupled system
consists of a convective Cahn-Hilliard equation for the phase field ,
i.e., the difference of the (relative) concentrations of the two phases,
coupled with a modified Darcy equation proposed by H.C. Brinkman in 1947 for
the fluid velocity . This equation incorporates a diffuse interface
surface force proportional to , where is the so-called
chemical potential. We analyze the well-posedness of the resulting
Cahn-Hilliard-Brinkman (CHB) system for . Then we establish
the existence of a global attractor and the convergence of a given (weak)
solution to a single equilibrium via {\L}ojasiewicz-Simon inequality.
Furthermore, we study the behavior of the solutions as the viscosity goes to
zero, that is, when the CHB system approaches the Cahn-Hilliard-Hele-Shaw
(CHHS) system. We first prove the existence of a weak solution to the CHHS
system as limit of CHB solutions. Then, in dimension two, we estimate the
difference of the solutions to CHB and CHHS systems in terms of the viscosity
constant appearing in CHB
An unconditionally energy stable finite difference scheme for a stochastic Cahn-Hilliard equation
In this work, the MMC-TDGL equation, a stochastic Cahn-Hilliard equation is
solved numerically by using the finite difference method in combination with a
convex splitting technique of the energy functional. For the non-stochastic
case, we develop an unconditionally energy stable difference scheme which is
proved to be uniquely solvable. For the stochastic case, by adopting the same
splitting of the energy functional, we construct a similar and uniquely
solvable difference scheme with the discretized stochastic term. The resulted
schemes are nonlinear and solved by Newton iteration. For the long time
simulation, an adaptive time stepping strategy is developed based on both
first- and second-order derivatives of the energy. Numerical experiments are
carried out to verify the energy stability, the efficiency of the adaptive time
stepping and the effect of the stochastic term.Comment: This paper has been accepted for publication in SCIENCE CHINA
Mathematic