New results for the Cahn-Hilliard equation with non-degenerate mobility: well-posedness and longtime behavior

2024/10/29 by Monica Conti, Conti, Monica, Pietro Galimberti +5 · 5 citations
Computer Science · Materials Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Solidification and crystal growth phenomena

paper · pdf · doi:10.48550/arxiv.2410.22234

openalex publication_date 2024/10/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Cahn-Hilliard equation with non-degenerate concentration-dependent mobility and logarithmic potential in two dimensions. We show that any weak solution is unique, exhibits propagation of uniform-in-time regularity, and stabilizes towards an equilibrium state of the Ginzburg-Landau free energy for large times. These results improve the state of the art dating back to a work by Barrett and Blowey. Our analysis relies on the combination of enhanced energy estimates, elliptic regularity theory and tools in critical Sobolev spaces.

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