Time optimal controls of the Cahn-Hilliard equation with internal control

2014 ◽  
Vol 36 (4) ◽  
pp. 566-582 ◽  
Author(s):  
Jiashan Zheng
Keyword(s):  
Internal Control ◽  
Optimal Controls ◽  
Hilliard Equation ◽  
Time Optimal ◽  
Cahn Hilliard Equation

scholarly journals Analysis and Optimal Velocity Control of a Stochastic Convective Cahn–Hilliard Equation

2021 ◽  
Vol 31 (2) ◽  
Author(s):  
Luca Scarpa
Keyword(s):  
Phase Separation ◽  
Random Component ◽  
Optimal Controls ◽  
Convection Term ◽  
Velocity Control ◽  
Optimal Velocity ◽  
Magnetic Devices ◽  
Hilliard Equation ◽  
Backward Problem ◽  
Cahn Hilliard Equation

AbstractA Cahn–Hilliard equation with stochastic multiplicative noise and a random convection term is considered. The model describes isothermal phase-separation occurring in a moving fluid, and accounts for the randomness appearing at the microscopic level both in the phase-separation itself and in the flow-inducing process. The call for a random component in the convection term stems naturally from applications, as the fluid’s stirring procedure is usually caused by mechanical or magnetic devices. Well-posedness of the state system is addressed, and optimisation of a standard tracking type cost with respect to the velocity control is then studied. Existence of optimal controls is proved, and the Gâteaux–Fréchet differentiability of the control-to-state map is shown. Lastly, the corresponding adjoint backward problem is analysed, and the first-order necessary conditions for optimality are derived in terms of a variational inequality involving the intrinsic adjoint variables.

scholarly journals On the sharp interface limit of a model for phase separation on biological membranes

Analysis ◽  
2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Helmut Abels ◽  
Johannes Kampmann
Keyword(s):  
Lipid Raft ◽  
Cell Membranes ◽  
System Modeling ◽  
Bulk Diffusion ◽  
Type Equation ◽  
Sharp Interface ◽  
Interface Problem ◽  
Hilliard Equation ◽  
System A ◽  
Cahn Hilliard Equation

AbstractWe rigorously prove the convergence of weak solutions to a model for lipid raft formation in cell membranes which was recently proposed in [H. Garcke, J. Kampmann, A. Rätz and M. Röger, A coupled surface-Cahn–Hilliard bulk-diffusion system modeling lipid raft formation in cell membranes, Math. Models Methods Appl. Sci. 26 2016, 6, 1149–1189] to weak (varifold) solutions of the corresponding sharp-interface problem for a suitable subsequence. In the system a Cahn–Hilliard type equation on the boundary of a domain is coupled to a diffusion equation inside the domain. The proof builds on techniques developed in [X. Chen, Global asymptotic limit of solutions of the Cahn–Hilliard equation, J. Differential Geom. 44 1996, 2, 262–311] for the corresponding result for the Cahn–Hilliard equation.

scholarly journals On global dynamics of 2D convective Cahn–Hilliard equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Dynamics ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Initial Value ◽  
Hilliard Equation ◽  
Long Time ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in $H^{4}(\Omega )$ H 4 ( Ω ) when the initial value belongs to $H^{1}(\Omega )$ H 1 ( Ω ) .

scholarly journals A free–energy stable p–adaptive nodal discontinuous Galerkin for the Cahn–Hilliard equation

2021 ◽  
pp. 110409
Author(s):  
Gerasimos Ntoukas ◽  
Juan Manzanero ◽  
Gonzalo Rubio ◽  
Eusebio Valero ◽  
Esteban Ferrer
Keyword(s):  
Free Energy ◽  
Discontinuous Galerkin ◽  
Hilliard Equation ◽  
Energy Stable ◽  
Cahn Hilliard Equation
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