Global Weak Solutions to a Sixth Order Cahn--Hilliard Type Equation

Abstract

In this paper we study a sixth order Cahn-Hilliard type equation that arises as a model for the faceting of a growing surface. We show global in time existence of weak solutions and uniform in time a priori estimates in the H 3 norm. These bounds enable us to show the uniqueness of weak solutions.

Global Attractors of Sixth Order PDEs Describing the Faceting of Growing Surfaces

“…3.2. On the other hand, strong continuity of S in the two-dimensional case of (1) has been already established in [10].…”

“…(4) and (8) in the terms of the semigroup theory. Continuity of S(t), t > 0 follows from results in [10,11]. The uniqueness theorems imply that the family {S(t)} t≥0 has the semigroup property.…”

“…Im fact, we treat in [10,11] existence of solutions in cases d = 1 and d = 2 differently. For the one-dimensional problem we switch to a new variable, the slope u = h x , i.e.…”

“…In [10,11] we showed only exponential bounds on the growth of solutions which is not particularly suitable for studying long time behavior. We will find the remedy here and we will show existence of a global attractor of (1) for d = 1, 2.…”

“…In the next section we recall the notion of weak solutions and the necessary facts from [10,11]. In addition we state the main results.…”