Nonlinear Analysis: Theory, Methods & Applications
Convergence to equilibrium for the Cahn–Hilliard equation with a logarithmic free energy
Section snippets
Introduction and main results
The nonlinear Cahn–Hilliard equation has been studied by several authors over the last few years. This system describes the dynamics of phase separation of a two-component mixture. Here is proportional to the concentration difference of the two components and is often called the order parameter of the system, e.g. the mass density, is the chemical potential, which accounts for mass transport in the system and is the derivative of a function , representing the
Preliminaries
For a set the power set will be denoted by . Moreover, we define and . If is a Banach space and is its dual, then denotes the duality product. Moreover, if is a Hilbert space, will denote its inner product. In the following all Hilbert spaces will be real-valued and separable.
Evolution equations for monotone operators
We refer the reader to Brézis [2] and Showalter [20] for basic results in the theory of monotone operators. In the following we just summarize some basic facts and definitions. Let be a real-valued and separable Hilbert space. Recall that is a monotone operator if Moreover, . Now let be a convex function. Then and is called proper if . Moreover, the subgradient is defined by
Subgradients
In the following denotes a continuous function and we set for .
In this section we study the subgradient of the functional first defined on , , with We denote by the subgradient of at in the sense that if and only if Note that is an affine subspace of with tangent space . Therefore
Existence of unique solutions
First of all, we can assume w.l.o.g. that As in the previous section we can reduce to this case by a simple shift. Since (5.1) implies that any solution of (1.4), (1.5) as in Theorem 1.2 satisfies we conclude for almost all .
We will consider (1.4), (1.5), (1.6), (1.7) as an evolution equation on in the following way: where
Convergence to equilibrium
Again we assume w.l.o.g. (5.1).
From Theorem 1.2 it follows that and , where , and may be arbitrarily small. Hence by Sobolev embedding we obtain and interpolation yields for all . Observe that does not depend on the initial value . We set and define a family of operators by Here denotes the solution due to Assumption 1.1 with
Acknowledgement
The authors are grateful to the anonymous referee for valuable comments on the literature and careful reading of the manuscript.
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