The Cahn-Hilliard Equation and the Allen-Cahn Equation on Manifolds with Conical Singularities

Abstract

We consider the Cahn-Hilliard equation on a manifold with conical singularities. We first show the existence of bounded imaginary powers for suitable closed extensions of the bilaplacian. Combining results and methods from singular analysis with a theorem of Clément and Li we then prove the short time solvability of the Cahn-Hilliard equation in Lp-Mellin-Sobolev spaces and obtain the asymptotics of the solution near the conical points.We deduce, in particular, that regularity is preserved on the smooth part of the manifold and singularities remain confined to the conical points.We finally show how the Allen-Cahn equation can be treated by simpler considerations. Again we obtain short time solvability and the behavior near the conical points.2000 Mathematics Subject Classification. 35J70,35R05,58J40.

Existence and maximalLp-regularity of solutions for the porous medium equation on manifolds with conical singularities

“…By returning to the conic manifold B, the closed extension ∆ 0 from Theorem 3.3 satisfies the assumptions of [43,Theorem 5.6] and the conditions (i), (ii) and (iii) of [43,Theorem 5.7]. Therefore, according to [43,Theorem 5.6 and Theorem 5.7], see also [40,Theorem 2.9 and Remark 2.10], the closed extension ∆ ∧ satisfies the condition (E3) from [43, Section 3.2], i.e. its spectrum is contained in (−∞, 0].…”

“…We regard ∆ as a cone differential operator or a Fuchs type operator and recall some basic facts and results from the related underlined pseudodifferential theory, which is called cone calculus, towards the direction of the study of nonlinear partial differential equations. For more details we refer to [6], [13], [14], [25], [28], [36], [37], [38], [39], [40], [41], [42], [43], [44] and [45].…”

Existence of solutions for two types of generalized versions of the Cahn-Hilliard equation

“…We suppose that s has the properties as above and A : R × T Γ → R n×n is Lipschitz continuous, bounded and uniformly elliptic, which means there is a constant C > 0 such that C −1 |ξ| This problem is of particular interest for numerical simulations in vesicles formation in biological membranes, see Lowengrub, Rätz, Voigt [22], as well as Mercker and coworkers [23], [25], [24]. A former mathematical study of the Cahn-Hilliard and the Allen-Cahn equations on manifolds can be found in [36]. The aforementioned publication has its focus on singularities of the manifolds and assumes A ≡ const.…”

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“…However, these two did not issue any commentsonthespinodal line; this may be due to the fact that the spinodal is notreallypartofthephase diagram. Figures 2.8a and b also contain results of a study by Dubieland Inden (Roidos & Schrohe, 2013) which was aimed at further characterization of the miscibilitygaplineaswellas eutectoid temperature of the reaction σ→α + α′. They performedlongterm annealing (2 to 11 years at 460, 500 and 510oC) on Fe-Cr binaryalloysof15,20,48 and 70 at.% Cr and determined the chemical compositionofαorα′phases applying Mössbauer Spectroscopy (MS).…”

“…We shall consider here the version As usual, we model a manifold with conical singularities by a manifold with boundary B of dimension n + 1, n ≥ 1, endowed with a conically degenerate Riemannian metric. On one hand, working on a manifold with boundary simplifies the analysis; on the other hand, the degeneracy of the Riemannian metric entails that geometric operators such as the Laplacian show the typical degeneracy they have on spaces with conic points in Euclidean space (Roidos & Schrohe, 2013;Vertman, 2016;Bahuaud & Helliwell, 2011). Permanent magnet is a ferromagnetic material that has a magnetization even if there is no external applied field.…”