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 form∂tu+∇l,uS(u)∋f where S is a nonconvex functional, ∇l,uS(u) is 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.