Optimal Control Problem for a Sixth-order Cahn-hilliard Equation with Nonlinear Diffusion
نویسندگان
چکیده
In this article, we study the initial-boundary-value problem for a sixth-order Cahn-Hilliard type equation ut = D μ, μ = γDu− a(u)Du− a′(u) 2 |Du| + f(u) + kut, which describes the separation properties of oil-water mixtures, when a substance enforcing the mixing of the phases is added. The optimal control of the sixth order Cahn-Hilliard type equation under boundary condition is given and the existence of optimal solution to the sixth order Cahn-Hilliard type equation is proved.
منابع مشابه
Numerical Analysis and Scientific Computing Preprint Seria C0 Interior Penalty Discontinuous Galerkin approximation of a sixth order Cahn-Hilliard equation modeling microemulsification processes
Microemulsions can be modeled by an initial-boundary value problem for a sixth order Cahn-Hilliard equation. Introducing the chemical potential as a dual variable, a Ciarlet-Raviart type mixed formulation yields a system consisting of a linear second order evolutionary equation and a nonlinear fourth order equation. The spatial discretization is done by a C0 Interior Penalty Discontinuous Galer...
متن کاملN ov 2 01 5 Nonlinear diffusion equations as asymptotic limits of Cahn – Hilliard systems Pierluigi Colli Dipartimento di Matematica , Università di Pavia and IMATI C . N . R . Pavia
An asymptotic limit of a class of Cahn–Hilliard systems is investigated to obtain a general nonlinear diffusion equation. The target diffusion equation may reproduce a number of well-known model equations: Stefan problem, porous media equation, Hele-Shaw profile, nonlinear diffusion of singular logarithmic type, nonlinear diffusion of Penrose–Fife type, fast diffusion equation and so on. Namely...
متن کاملThe Spectral Method for the Cahn-Hilliard Equation with Concentration-Dependent Mobility
In this paper, we apply the spectral method to approximate the solutions of Cahn-Hilliard equation, which is a typical class of nonlinear fourth-order diffusion equations. Diffusion phenomena is widespread in the nature. Therefore, the study of the diffusion equation caught wide concern. Cahn-Hilliard equation was proposed by Cahn and Hilliard in 1958 as a mathematical model describing the diff...
متن کاملSolving the regularized, strongly anisotropic Cahn-Hilliard equation by an adaptive nonlinear multigrid method
We present efficient, second-order accurate and adaptive finite-difference methods to solve the regularized, strongly anisotropic Cahn–Hilliard equation in 2D and 3D. When the surface energy anisotropy is sufficiently strong, there are missing orientations in the equilibrium level curves of the diffuse interface solutions, corresponding to those missing from the sharp interface Wulff shape, and...
متن کاملOptimal distributed control of a nonlocal convective Cahn–Hilliard equation by the velocity in 3D
In this paper we study a distributed optimal control problem for a nonlocal convective Cahn–Hilliard equation with degenerate mobility and singular potential in three dimensions of space. While the cost functional is of standard tracking type, the control problem under investigation cannot easily be treated via standard techniques for two reasons: the state system is a highly nonlinear system o...
متن کامل