On systems of Cahn–Hilliard and Allen–Cahn equations considered as gradient flows in Hilbert spaces

Abstract

We study systems of Allen-Cahn and Cahn-Hilliard equations with the mobility coefficients depending on c and ∇c. We interpret these systems of equations as gradient flows in Hilbert spaces with a densely defined Riemannian metric. In particular, we study gradient flows (curves of maximal slope) of the formis the strong-weak closure of the subgradient of S and f is a time dependent right hand side. The article generalizes the results by Rossi and Savaré [36] to this setting and applies for systems of multiple phases derived by Heida, Málek and Rajagopal [20,19] in a simplified form. More generally, we will show that a certain class of reaction-diffusion equations coming from a modeling approach by Rajagopal and Srinivasa [32] or by Mielke [27], are automatically subject to the presented theory of curves of maximal slope.

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

“…, [17]). We say that for any u ∈ H, ξ ∈ H is an element of the limiting subdifferential d l S(u) of S in u if there are u n ∈ H with u n → u strongly and ξ n ∈ dS(u n ) such that ξ n ⇀ ξ weakly in H. The limiting subgradient is defined through…”

“…Short sketch of the mathematical approach. The proofs of Theorems 1.1 and 1.2 are based on a recent result by the author [17]. The basic idea is to consider (1.2) as a gradient flow in H −1 (0) (Ω) of the functional S given in (1.4) and with respect to local scalar products g u (·, ·).…”

“…We will furthermore introduce basic notation for the work with boundary derivatives. In Section 3 we will introduce some functional analytical tools, in particular the theory of Young measures, whereas in Section 4, we will introduce the theory of gradient flows in the way it is presented in [17].…”

“…This work deals with the existence of solutions to a variety of Cahn-Hilliard models generalizing the applications in [17]. In what follows, we will introduce three types of equations that will be discussed in this paper, where we use some notation and Hilbert spaces as they are introduced below in Section 2.…”

“…The first problem in most parts was treated in [17] and we will not spend too much effort discussing it; we rather consider it as an introductory exercise for the other two problems, as it will help to improve understanding of the method. In the aforementioned paper, the author developed and applied a generalized concept of gradient flows to the following problem:…”