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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.

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We provide a rigorous mathematical framework to establish the hydrodynamic limit of the Vlasov model introduced in Takata and Noguchi (J. Stat. Phys. 172:880-903, 2018) by Noguchi and Takata in order to describe phase transition of fluids by kinetic equations. We prove that, when the scale parameter tends to 0, this model converges to a nonlocal Cahn-Hilliard equation with degenerate mobility. For our analysis, we introduce apropriate forms of the short and long range potentials which allow us to derive Helmhotlz free energy estimates. Several compactness properties follow from the energy, the energy dissipation and kinetic averaging lemmas. In particular we prove a new weak compactness bound on the flux.