Degenerate Cahn-Hilliard equation: From nonlocal to local

From Vlasov Equation to Degenerate Nonlocal Cahn-Hilliard Equation

“…Compactness in time follows from (E). An Aubin-Lions argument yields the result (see [25,Lemma D.1] for the precise statement).…”

“…Concerning the degenerate mobility, in [25], we investigated the case m(u) = u with a smooth potential, motivated by biological applications [23,24]. This result has been extended to the case of systems in [10] which appear in modeling of cell-cell adhesion [13,28].…”

“…This question was recently addressed by several authors [18][19][20]50] but only for the case of constant mobility. Our recent article [25] was the first one to establish rigorously the convergence for the degenerate mobility m(u) = u and smooth potential as a consequence of nonlocal compactness results developed by Bourgain-Brezis-Mironcescu [3] and Ponce [55]. In the current paper we conclude our analysis by showing that the reasoning from [25] can be extended to the mobility m(u) = u (1 − u) and the logarithmic Flory-Huggins potential as in the original works of Giacomin-Lebowitz [40] and Elliot-Garcke [27].…”

“…With classical arguments, we are able to prove that the latter admits a unique positive classical solution. Let us finally remark that the nonlocal Cahn-Hilliard equation is nowadays a topic of intense research activity, see for instance [19,28,45,46,67].…”

“…The main contribution of Theorem 1.2 is the possibility to consider more general, nonlinear mobilities m(u). In [25], where we considered the particular mobility m(u) = u, we exploited the formula…”