Article ID Journal Published Year Pages File Type
4615461 Journal of Mathematical Analysis and Applications 2015 46 Pages PDF
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∂tu+∇l,uS(u)∋f where SS is a nonconvex functional, ∇l,uS(u)∇l,uS(u) is the strong-weak closure of the subgradient of SS 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] and [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.

Related Topics
Physical Sciences and Engineering Mathematics Analysis
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