Convergence to equilibrium for the Cahn–Hilliard equation with a logarithmic free energy

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Abstract

In this paper we investigate the asymptotic behavior of the nonlinear Cahn–Hilliard equation with a logarithmic free energy and similar singular free energies. We prove an existence and uniqueness result with the help of monotone operator methods, which differs from the known proofs based on approximation by smooth potentials. Moreover, we apply the Lojasiewicz–Simon inequality to show that each solution converges to a steady state as time tends to infinity.

Section snippets

Introduction and main results

The nonlinear Cahn–Hilliard equation tc=Δμ,μ=Δc+f(c), 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 c=c(t,x) 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, μ is the chemical potential, which accounts for mass transport in the system and f is the derivative of a function f, representing the

Preliminaries

For a set M the power set will be denoted by P(M). Moreover, we define R+n={xRn:xn>0} and R+=R+1. If X is a Banach space and X is its dual, then f,gf,gX,X=f(g),fX,gX, denotes the duality product. Moreover, if H is a Hilbert space, (,)H 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 H be a real-valued and separable Hilbert space. Recall that A:HP(H) is a monotone operator if (wz,xy)H0for all wA(x),zA(y). Moreover, D(A)={xH:A(x)}. Now let φ:HR{+} be a convex function. Then dom(φ)={xH:φ(x)<} and φ is called proper if dom(φ). Moreover, the subgradient φ:HP(H) is defined by wφ(x

Subgradients

In the following ϕ:[a,b]R denotes a continuous function and we set ϕ(x)=+ for x[a,b].

In this section we study the subgradient of the functional F(c)=12Ω|c(x)|2dx+Ωϕ(c(x))dx first defined on L(m)2(Ω), m(a,b), with domF={cH1(Ω)L(m)2(Ω):ϕ(c)L1(Ω)}. We denote by F(c):L(m)2(Ω)P(L(0)2(Ω)) the subgradient of F at cdomF in the sense that wF(c) if and only if (w,cc)L2F(c)F(c)for all cL(m)2(Ω). Note that L(m)2(Ω) is an affine subspace of L2(Ω) with tangent space L(0)2(Ω). Therefore

Existence of unique solutions

First of all, we can assume w.l.o.g. that m(c0)=1|Ω|Ωc0dx=0. 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 ddtΩc(x,t)dx=ΩΔμdx=0, we conclude m(c(t))=0 for almost all t>0.

We will consider (1.4), (1.5), (1.6), (1.7) as an evolution equation on H(0)1(Ω) in the following way: tc+A(c)+Bc=0,t>0,c|t=0=c0 where A(c),φH(0)1,H(0)1=(μ,φ)L2with μ=Δc+ϕ(c)Bc,φH(0)1,H(0)1=d(c,φ)L2,φH(0

Convergence to equilibrium

Again we assume w.l.o.g. (5.1).

From Theorem 1.2 it follows that cL(Jδ;H2(Ω)) and tcL2(Jδ;H1(Ω)), where Jδ[δ,T], T< and δ(0,T) may be arbitrarily small. Hence by Sobolev embedding we obtain cC12(Jδ;H1(Ω)) and interpolation yields cC(Jδ;H2r(Ω)) for all r[0,1). Observe that δ>0 does not depend on the initial value c0. We setZZ0={zH(0)1(Ω):E(z)<}, and define a family of operators S{S(t)}t>0 by S(t):ZZ,S(t)c0=c(t;c0),tR+. Here c(t;c0) 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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  • C.M. Elliot, S. Luckhaus, A generalized equation for phase separation of a multi-component mixture with interfacial...
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