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Vanishing diffusion in a dynamic boundary condition for the Cahn–Hilliard equation

Abstract

The initial boundary value problem for a Cahn–Hilliard system subject to a dynamic boundary condition of Allen–Cahn type is treated. The vanishing of the surface diffusion on the dynamic boundary condition is the point of emphasis. By the asymptotic analysis as the diffusion coefficient tends to 0, one can expect that the solutions of the surface diffusion problem converge to the solution of the problem without the surface diffusion. This is actually the case, but the solution of the limiting problem naturally looses some regularity. Indeed, the system we investigate is rather complicate due to the presence of nonlinear terms including general maximal monotone graphs both in the bulk and on the boundary. The two graphs are related each to the other by a growth condition, with the boundary graph that dominates the other one. In general, at the asymptotic limit a weaker form of the boundary condition is obtained, but in the case when the two graphs exhibit the same growth the boundary condition still holds almost everywhere.

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Acknowledgements

The authors warmly thank Professor Ken Shirakawa for his valuable advice about Lemmas A.1 and A.2. P. Colli points out his affiliation as Research Associate to the IMATI – C.N.R. Pavia, Italy, and acknowledges support from the Italian Ministry of Education, University and Research (MIUR): Dipartimenti di Eccellenza Program (2018–2022) – Dept. of Mathematics “F. Casorati”, University of Pavia, and from the GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica). T. Fukao acknowledges the support from the JSPS KAKENHI Grant-in-Aid for Scientific Research(C), Japan, Grant Number 17K05321 and from the Grant Program of The Sumitomo Foundation, Grant Number 190367. Last but not least, the authors are very grateful to the reviewer for the careful reading of manuscript.

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Appendix A

Appendix A

We use the same setting as in the previous sections.

Lemma A.1

Assume (A2). Then (3.3) holds, that is,

$$\begin{aligned} \bigl |\beta _\varepsilon (r)\bigr | \le \varrho \bigl |\beta _{\Gamma ,\varepsilon } (r)\bigr |+c_0 \quad \text {for all } r \in \mathbb {R}, \end{aligned}$$

for all \(\varepsilon \in (0,1]\) with the same constants \(\varrho \ge 1 \) and \(c_0 >0 \).

Proof

Thanks to [6, Lemma 4.4], it is known that

$$\begin{aligned} \bigl |\beta _\varepsilon (r)\bigr | \le \varrho \bigl |\beta _{\Gamma ,\varepsilon \varrho } (r)\bigr |+c_0 \quad \text {for all } r \in \mathbb {R}, \end{aligned}$$

where \(\beta _{\Gamma ,\varepsilon \varrho }\) denotes the Yosida approximation of \(\beta _{\Gamma }\) with parameter \(\varepsilon \varrho \), i.e.,

$$\begin{aligned} \beta _{\Gamma , \varepsilon \varrho } (r) := \frac{1}{\varepsilon \varrho } \bigl ( r-(I+\varepsilon \varrho \beta _\Gamma )^{-1} (r) \bigr ). \end{aligned}$$
(5.10)

Then, recalling that \(\varrho \ge 1\), we may invoke the fundamental property [2, Proposition 2.6, p. 28] of Yosida approximations, which implies that

$$\begin{aligned} \bigl | \beta _{\Gamma , \varepsilon \varrho } (r) \bigr | \le |\beta _{\Gamma , \varepsilon } (r) \bigr | \quad \text {for all } r \in \mathbb {R}, \end{aligned}$$

because \(\varepsilon \le \varepsilon \varrho \). Thus we get the conclusion. \(\square \)

Lemma A.2

Assume (A2)\(^\prime \). Then (5.2) holds, that is,

$$\begin{aligned} \frac{1}{\varrho }\bigl | \beta _{\Gamma , \varepsilon } (r) \bigr |-c_0 \le \bigl |\beta _\varepsilon (r)\bigr | \le \varrho \bigl |\beta _{\Gamma ,\varepsilon } (r)\bigr |+c_0 \quad \text {for all } r \in \mathbb {R}, \end{aligned}$$

for all \(\varepsilon \in (0,1]\) with the same constants \(\varrho \ge 1 \) and \(c_0 >0\).

Proof

In view of Lemma A.1, is enough to prove that

$$\begin{aligned} \frac{1}{\varrho }\bigl | \beta _{\Gamma , \varepsilon } (r) \bigr |-c_0 \le \bigl |\beta _\varepsilon (r)\bigr | \quad \text {for all } r \in \mathbb {R}, \end{aligned}$$

which is the same as

$$\begin{aligned} \bigl |\beta _{\Gamma ,\varepsilon } (r)\bigr | \le \varrho \bigl |\beta _\varepsilon (r)\bigr | + \varrho \,c_0 \quad \text {for all } r \in \mathbb {R}. \end{aligned}$$

But this follows immediately from Lemma A.1 again. \(\square \)

Remark A.3

Comparing to previous works (see, e.g., [8, 9, 12]) in which the same kind of property (2.7) was assumed for the two maximal monotone graphs, the parameter of the Yosida regularizations is here the same for both graphs (see also [11, Section 2]) . Instead, in the approach devised in [6, Lemma 4.4] exactly the approximation \(\beta _{\Gamma , \varepsilon \varrho }\) defined by (5.10) was introduced and used for \(\beta _{\Gamma }\).

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Colli, P., Fukao, T. Vanishing diffusion in a dynamic boundary condition for the Cahn–Hilliard equation. Nonlinear Differ. Equ. Appl. 27, 53 (2020). https://doi.org/10.1007/s00030-020-00654-8

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Keywords

  • Cahn–Hilliard system
  • Dynamic boundary condition
  • Non-smooth potentials
  • Convergence
  • Well-posedness
  • Regularity

Mathematics Subject Classification

  • Primary 35K61
  • Secondary 35K25
  • 74N20
  • 80A22