Asymptotic stabilization of weak solutions to phase-field equations with non-degenerate mobility and singular potential

TL;DR

This paper advances the understanding of phase-field models by relaxing assumptions for proving the convergence of weak solutions to equilibrium, enabling analysis of complex models with singular potentials in three dimensions.

Contribution

It generalizes the method for proving convergence to equilibrium, handling all main phase-field models with singular potentials in 3D, including nonlinear and nonlocal variants.

Findings

Established convergence to equilibrium for complex phase-field models.

Proved asymptotic regularization and strict separation properties.

Extended applicability to coupled hydrodynamic models.

Abstract

A common paradigm in phase-field models with singular potentials is that global-in-time weak solutions converge to a single equilibrium only after undergoing asymptotic regularization. However, in arXiv:2510.17296 we introduced a novel method to establish the convergence to a single equilibrium for solutions to Cahn--Hilliard equations, and some related coupled systems, with non-degenerate mobility and singular potentials, under very general assumptions: we only require the existence of a global weak solution satisfying an energy inequality and then we make use of a Lojasiewicz--Simon inequality. Here we take a non-trivial step further. We relax the assumptions needed to prove the precompactness of trajectories, which is an essential ingredient of the complete proof. Thanks to this generalization, we can handle all the main phase-field models, with fully general singular potentials, in a three-dimensional domain, whose asymptotic behavior has so far remained an open problem. Namely, we apply the new method to the Cahn--Hilliard equation with nonlinear diffusion, the conserved Allen--Cahn equation, and the nonlocal Cahn--Hilliard equation. In the case of second-order equations, De Giorgi's iteration argument is crucial to show that weak solutions stay asymptotically uniformly away from pure phases (strict separation property), which is the key ingredient to apply the Lojasiewicz--Simon inequality. We expect this new technique to have a wider range of applications to coupled problems, including hydrodynamic models like the conserved Allen--Cahn--Navier--Stokes system or the nonlocal Abels--Garcke--Gr\"un system with non-degenerate mobility. Further applications to multi-component models are also possible.