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Study of a coupled Cahn-Hilliard/Allen-Cahn system in phase separation

Abstract : This thesis is a theoretical study of a coupled system of equations of Cahn-Hilliard and Allen-Cahn that represents phase separation of binary alloys. The main goal of this study is to investigate the asymptotic behavior of the solution in terms of exponential/global attractors. For this reason, the existence and unicity of the solution are first studied. One of the most important applications of this proposed model of equations is crystallography. In the first part of the thesis, the system is studied with boundary conditions of Dirichlet type and a regular nonlinearity (a polynomial). There, we prove the existence of an exponential attractor that leads to the existence of a global attractor of finite dimension. Then, a singular nonlinearity (a logarithmic potential) is considered in the second part. This function is approximated by a sequence of regular ones and a global attractor is found.At the end, the system of equations is coupled with temperature: with the Fourrier law in the first case, then with the type III law (in the context of thermoelasticity) in the second case. The dynamics of the equations are studied and the existence of an exponential attractor is obtained.
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Wafa Saoud. Study of a coupled Cahn-Hilliard/Allen-Cahn system in phase separation. Analysis of PDEs [math.AP]. Université de Poitiers, 2018. English. ⟨NNT : 2018POIT2285⟩. ⟨tel-02087788⟩

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