The 3D strict separation property for the nonlocal Cahn-Hilliard equation with singular potential

Abstract

We consider the nonlocal Cahn-Hilliard equation with singular (logarithmic) potential and constant mobility in three-dimensional bounded domains and we establish the validity of the instantaneous strict separation property. This means that any weak solution, which is not a pure phase initially, stays uniformly away from the pure phases $\pm 1$ from any positive time on. This work extends the result in dimension two for the same equation and gives a positive answer to the long standing open problem of the validity of the strict separation property in dimensions higher than two. In conclusion, we show how this property plays an essential role to achieve higher-order regularity for the solutions and to prove that any weak solution converges to a single equilibrium.

References in corpus (8)

• Phase Segregation Dynamics in Particle Systems with Long Range Interactions I: Macroscopic Limits
• Local and Global Well-Posedness for Aggregation Equations and Patlak-Keller-Segel Models with Degenerate Diffusion
• Global and trajectory attractors for a nonlocal Cahn-Hilliard-Navier-Stokes system
• Nonlocal-to-local convergence of Cahn-Hilliard equations: Neumann boundary conditions and viscosity terms
• On the nonlocal Cahn--Hilliard equation with nonlocal dynamic boundary condition and boundary penalization
• Global Well-posedness of a Navier-Stokes-Cahn-Hilliard System with Chemotaxis and Singular Potential in 2D
• On the Mass-Conserving Allen-Cahn Approximation for Incompressible Binary Fluids
• On a phase field model for RNA-Protein dynamics

Cited by in corpus (1)

• Strong well-posedness and separation properties for a bulk-surface convective Cahn--Hilliard system with singular potentials