Convergence to equilibrium of weak solutions to the Cahn--Hilliard equation with non-degenerate mobility and singular potential

We consider the classical initial and boundary value problem for the Cahn–Hilliard equation with non-degenerate mobility and singular (e.g., logarithmic) potential. We prove that any weak solution converges to a single equilibrium using only minimal assumptions, that is, the existence of a global weak solution which satisfies an energy inequality. This result appears to be new in the literature and also holds in the three-dimensional case, which was an open problem due to the lack of regularity results, especially when the mobility is just a continuous function. We then prove the same result for a Cahn–Hilliard-Navier–Stokes type system with unmatched densities and viscosities proposed by Abels, Garcke, and Grün (Math. Models Methods Appl. Sci. 22, 2012), always assuming a non-degenerate mobility. We expect that this novel method can be used to analyze the same issue for other models where the regularization properties of the solutions are unknown or unlikely.

💡 Research Summary

The paper addresses the long‑time behavior of the classical Cahn–Hilliard equation with a non‑degenerate (strictly positive and continuous) mobility function and a singular logarithmic (Flory–Huggins) potential. While the existence of global weak solutions and an associated energy inequality are well‑known, proving convergence to a single equilibrium in three dimensions has remained open because standard techniques rely on finite‑time regularization (e.g., H²‑regularity) and an instantaneous strict separation from the pure phases ±1. Such regularity is unavailable for non‑degenerate mobility, especially in 3D.

The authors propose a novel framework that requires only two minimal assumptions: (i) the existence of a global weak solution, and (ii) the validity of the energy inequality
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