Geodesic
On the Cahn–Hilliard equation with kinetic rate dependent dynamic boundary conditions and non-smooth potentials: Well-posedness and asymptotic limits
Maoyin Lv1 orcid ; Hao Wu1 orcid
1Fudan University, Shanghai, P. R. China
Interfaces and free boundaries, Tome 27 (2025) no. 2, pp. 177-234

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DOI

We analyze a class of Cahn–Hilliard equations with kinetic rate dependent dynamic boundary conditions that describe possible short-range interactions between the binary mixture and the solid boundary. In the presence of surface diffusion on the boundary, the initial boundary value problem can be viewed as a transmission problem consisting of Cahn–Hilliard-type equations both in the bulk and on the boundary. We first establish the existence, uniqueness, and continuous dependence of global weak solutions. In the construction of weak solutions, an explicit convergence rate in terms of the parameter for the Yosida approximation is obtained. Under some additional assumptions, we further prove the existence and uniqueness of global strong solutions. Next, we study the asymptotic limit as the coefficient of the boundary diffusion goes to zero and show that the limit problem with a forward-backward dynamic boundary condition is well posed in a suitable weak formulation. At last, we investigate the asymptotic limits as the kinetic rate tends to zero and infinity, respectively. Our results are valid for a general class of bulk and boundary potentials with double-well structure, including the physically relevant logarithmic potential and the non-smooth double obstacle potential.
DOI : 10.4171/ifb/532
Classification : 35K35, 35K61, 35B20, 35B40, 80A22
Mots-clés : Cahn–Hilliard equation, dynamic boundary condition, bulk-boundary interaction, non-smooth potential, well-posedness, asymptotic limit

Maoyin Lv  1   ; Hao Wu  1

1 Fudan University, Shanghai, P. R. China 
Maoyin Lv; Hao Wu. On the Cahn–Hilliard equation with kinetic rate dependent dynamic boundary conditions and non-smooth potentials: Well-posedness and asymptotic limits. Interfaces and free boundaries, Tome 27 (2025) no. 2, pp. 177-234. doi: 10.4171/ifb/532
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     author = {Maoyin Lv and Hao Wu},
     title = {On the {Cahn{\textendash}Hilliard} equation with kinetic rate dependent dynamic boundary conditions and non-smooth potentials: {Well-posedness} and asymptotic limits},
     journal = {Interfaces and free boundaries},
     pages = {177--234},
     year = {2025},
     volume = {27},
     number = {2},
     doi = {10.4171/ifb/532},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/532/}
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