Separation property and convergence to equilibrium for the equation and dynamic boundary condition of Cahn-Hilliard type with singular potential

Abstract

We consider a class of Cahn-Hilliard equation that models phase separation process of binary mixtures involving nontrivial boundary interactions in a bounded domain with non-permeable wall. The system is characterized by certain dynamic type boundary conditions and the total mass, in the bulk and on the boundary, is conserved for all time. For the case with physically relevant singular (e.g., logarithmic) potential, global regularity of weak solutions is established. In particular, when the spatial dimension is two, we show the instantaneous strict separation property such that for arbitrary positive time any weak solution stays away from the pure phases +1 and -1, while in the three dimensional case, an eventual separation property for large time is obtained. As a consequence, we prove that every global weak solution converges to a single equilibrium as the time goes to infinity, by the usage of an extended Lojasiewicz-Simon inequality.

34 pages

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Cited by in corpus (6)

• Phase-field dynamics with transfer of materials: The Cahn--Hilliard equation with reaction rate dependent dynamic boundary conditions
• A Review on the Cahn-Hilliard Equation: Classical Results and Recent Advances in Dynamic Boundary Conditions
• On the nonlocal Cahn--Hilliard equation with nonlocal dynamic boundary condition and boundary penalization
• Stability of the semi-implicit method for the Cahn-Hilliard equation with logarithmic potentials
• Long-time dynamics of the Cahn--Hilliard equation with kinetic rate dependent dynamic boundary conditions
• Strong well-posedness and separation properties for a bulk-surface convective Cahn--Hilliard system with singular potentials