Existence of Global Attractor for Cahn-Hilliard Perturbed Phase-Field System with Dirichlet Boundary Condition and Regular Potentiel
Mangoubi Jean De Dieu *
Faculte des Sciences et Techniques, Universit Marien Ngouabi, B.P.69, Brazzaville, Congo.
Daniel Moukoko
Faculte des Sciences et Techniques, Universit Marien Ngouabi, B.P.69, Brazzaville, Congo.
Dieudonne Ampini
Faculte des Sciences et Techniques, Universit Marien Ngouabi, B.P.69, Brazzaville, Congo.
Fidele Moukamba
Faculte des Sciences et Techniques, Universit Marien Ngouabi, B.P.69, Brazzaville, Congo.
*Author to whom correspondence should be addressed.
Abstract
Aims/ objectives: We are interested in a hyperbolic phase field system of Cahn-Hilliard type, parameterized by ϵ for which the solution is a function defined on (0; T) × Ω . We show the existence and uniqueness of the solutoin, existence of the global attractor for a hyperbolic phase field system of Cahn-Hilliard type, with homogenous conditions Dirichlet on the boundary, this system is governed by a regular potential, in a bounded and smooth domain. the hyperbolic phase field system of Cahn-Hilliard type is based on a thermomecanical theory of deformable continu.
Note that the global attractor is the smallest compact set in the phase space, which is invariant by the semigroup and attracts all bounded sets of initial data, as time goes to infinity. So the global attractor allows to make description of asymptotic behaviour about dynamic system.
Study Design: Propagation study of waves.
Place and Duration of Study: Departement of mathematics (group of research called G.R.A.F.E.D.P), Sciences Faculty and Technical of Marien NGOUABI University PO Box 69, between October 2015 and July 2016.
Methodology: To prove the existence of the global attractor to based of the classic methode about the perturbed hyperbolic system, with initial conditions and homogenous conditions Dirichlet on the boundary, we proceed by proving the dissipativity and regularity of the semigroup associated to the system, and we then split the semigroup such that we have the sum of two continuous operators, where the first tends uniformly to zero when the time goes to infinity, and the second is regularizing.
Results: We show the existence of global attractor, about a hyperbolic phase field system of Cahn-Hilliard type, governed by regular potential.
Conclusion: All the procedures explained in the methodology being demonstrated , we can assert the existence of the smallest compact set of the phase space, invariant by the semigroup and which attracts all the bounded sets of initial data from a some time.
Keywords: Cahn-Hilliard phase- eld system, dissipativity, global attractor, dirichlet boundary conditions.