Degenerate Cahn-Hilliard equation: From nonlocal to local

Abstract

We consider the nonlocal Cahn-Hilliard equation with degenerate mobility and smooth potential. As the scaling parameter related to nonlocality tends to zero, we prove that the equation converges to a local Cahn-Hilliard equation. The proof relies on compactness properties and an adapted result from Bourgain-Brezis-Mironescu and Ponce. 2020 Mathematics Subject Classification. 35K25. Key words and phrases. Degenerate Cahn-Hilliard equation; Nonlocal Cahn-Hilliard equation; Aggregation-Diffusion; Singular limit. J.S. was supported by the National Agency of Academic Exchange project "Singular limits in parabolic equations" no. BPN/BEK/2021/1/00044. Authors are grateful to Benoît Perthame for fruitful discussions and helpful suggestions that greatly improved this paper. (C) F 2 has bounded second derivative i.e. F ′′ 2 ∞ < ∞ and F 2 (u) ≥ −C 3 − C 4 u 2 where C 4 is sufficiently small: more precisely 4 C 4 < C p with C p is the constant in Lemma C.1.Moreover, we require one of the following: (D1) F 1 = 0 or (D2) F 1 has a k-growth, i.e. for some constants C 5 , C 6 , C 7 and C 8 we haveExample 1.2. The following potentials satisfy Assumption 1.1.(1) power-type potential F (u) = u γ used in the context of tumour growth models [9,15,17,34],(2) double-well potential F (u) = u 2 (u − 1) 2 which is an approximation of logarithmic doublewell potential often used in Cahn-Hilliard equation, see [32, Chapter 1],(3) any F ∈ C 2 such that for some interval I ⊂ R we have F ′′ (u) > a > 0 for u ∈ R \ I andfor all u ∈ R \ I, see Lemma A.3 for details.Note that (3) is a more general version of (2). Notation 1.3 (exponents s and k).In what follows we writeWe also define s = 2k k−1 and s ′ its conjugate exponent. Now, we define weak solutions of the nonlocal and local degenerate Cahn-Hilliard equation.

From Vlasov Equation to Degenerate Nonlocal Cahn-Hilliard Equation

“…We proved that macroscopic densities { ε } formed from solutions of the Vlasov-Cahn-Hilliard equation (1) converge to the solutions of non-local degenerate Cahn-Hilliard (11). It is an open question whether one can obtain a local version of this equation by sending short-range interaction kernel ω S to the Dirac mass δ 0 .…”

“…where = −δ . One can try to perform this limit either on equation (11) or directly on (1), by sending ω L α * δ 0 , ω S * δ 0 together, see Fig. in the nondegenerate Cahn-Hilliard.…”

“…The nonlocality comes from the convolution of the Laplace operator with a smooth kernel ω S concentrated around the origin. There is a different possibility to approximate this operator nonlocally, we refer for instance to [5,25], where the authors prove the convergence of a nonlocal Cahn-Hilliard equation with constant mobility to a local Cahn-Hilliard equation and to [11] for a similar result with degenerate mobility. In our case, because of the degenerate mobility, it is not clear that we can pass from the nonlocal Equation ( 11) to a local one by sending ω S to a Dirac mass.…”

Degenerate Cahn-Hilliard equation: From nonlocal to local

“…Passage to the limit ε → 0 and nonlocal compactness results. We use the strategy developed by the second and third author in [32] for the single Cahn-Hilliard equation. The main tool is the compactness result due to Bourgain-Brezis-Mironescu [11] and Ponce [48] which reads as follows:…”