Stability of the Semi-Implicit Method for the Cahn-Hilliard Equation with Logarithmic Potentials

Dong Liand Tao Tang

1 Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong

2 Division of Science and Technology, BNU-HKBU United International College, Zhuhai 519087, Guangdong, China; and SUSTech International Center for Mathematics, Southern University of Science and Technology, Shenzhen 518055, Guangdong, China

Abstract. We consider the two-dimensional Cahn-Hilliard equation with logarithmic potentials and periodic boundary conditions. We employ the standard semi-implicit numerical scheme, which treats the linear fourth-order dissipation term implicitly and the nonlinear term explicitly. Under natural constraints on the time step we prove strict phase separation and energy stability of the semiimplicit scheme. This appears to be the first rigorous result for the semi-implicit discretization of the Cahn-Hilliard equation with singular potentials.

Key words: Cahn-Hilliard equation, logarithmic kernel, semi-implicit scheme, energy stability.

1 Introduction

Consider the 2D Cahn-Hilliard equation on ?=T=[?π,π):

where u:?→(?1,1) is the order parameter of a two-phase system such as a binary alloy, and the term μ denotes the chemical potential. The two end-points u=±1 correspond to pure states. The coefficient ν>0 denotes mobility. In this paper we take it to be a constant parameter. The thermodynamic potential F:(?1,1)→R is given by

Note that for u ∈(?1,1), the term F(u) is bounded by an absolute constant, and the only coercive quantity in E(u) is the gradient term.

Remark 1.1. We note that the usual quartic polynomial approximation of the free energy F(u) is given by (below the series converges for u∈[?1,1])

The standard double-well potential const·(u?1)corresponds to the specific choice θ/θ=3/4. However, this approximation introduces a nontrivial shift of the location of the minimum. Namely for the original free energy F(u), its two equal minima occur at±u,where u>0 is the positive root of the equation f(u)=0(see(1.2b)). In particular,0

The strict phase separation turns out to play an important role in the rigorous analysis of (1.1).

Mathematically speaking, the system (1.1) can be recast as a gradient flow of a Ginzburg-Landau (GL) type energy functional ψ(u) in H, i.e.,

Here the gradient term in the GL energy accounts for surface tension effects,or more generally, short range interactions in the material. This particular form of energy functional can be derived from an approximation of a nonlocal term representing long range interactions [2]. In [10,11] Giacomin and Lebowitz considered a lattice gas model with certain long range Kac potentials, and gave a rigorous derivation of the nonlocal Cahn-Hilliard equation. Further results such as regularity and traveling waves on these and similar models can be found in [12,13,16] and the references therein.

The main point is to derive ?-independent estimates on the regularized problem and extract the desired solution in the vanishing ?-limit. In [6], Debussche and Dettori adopted a different regularization of F(u):

where

There are some subtle technical difficulties associated with the numerical discretization of(1.1). We now point out two most pronounced issues. Denote u≈u(t)as the numerical solution at time step t=nτ, where τ>0 is the time step.

1. How do we guarantee that u∈(?1,1) for all n≥0?

2. How to ensure the energy decay property: E(u)≤E(u) for all n≥0?

Whilst the first issue already presents itself a fundamental problem for semiimplicit methods, the second one is even more serious. As it turns out the explicit or implicit treatment of the nonlinear term can lead to a fundamental change of the energy stability of the associated iterative system. The analysis of energy stability gives a clear picture why implicit methods(or partially implicit methods)are usually favored/adopted in the literature (see also recent [21]). To elucidate the discussion we shall compare the usual semi-implicit methods with the implicit methods in the next two subsections. For simplicity we assume the ideal scenario that all u∈(?1,1).

The usual semi-implicit discretization case

A typical semi-implicit discretization takes the form

Multiplying both sides by (??)(u?u) (note that u=u, and see (1.22)for the definition of (??)and u) and integrating by parts, we obtain

The usual implicit discretization case

A typical implicit discretization takes the form

where ξis a function with values sandwiched between uand u. Note that on the LHS of (1.14), we have the usual estimate

Now, from the above comparative discussion in the preceding two subsections,it is clear that there are nontrivial technical obstacles for the semi-implicit methods applied on the Cahn-Hilliard equation with logarithmic potentials. Nevertheless,the purpose of this work is to introduce a new framework to settle these open issues.

Consider the following semi-implicit discretization of (1.1):

3. Energy stability. E(u)≤E(u) for all n≥0.

To prove Theorem 1.1, we introduce a new strategy which concurrently establishes the strict phase separation and uniform Sobolev regularity of the iterates uthrough an inductive procedure. Besides using the discrete energy inequality to control H-norm of u, we employ several bootstrapping long time estimates on the discrete chemical potential

to gain uniform-in-time higher Sobolev bounds. This part of the argument is technical and we have to appeal to a delicate dichotomy argument to eliminate some sporadic drift of higher norms of K(see Subsection 2.3 for more details). The strict phase separation property of ucan be deduced through a uniform estimate on the quantity

which in turn is obtained by analyzing a nonlinear elliptic problem connecting gto K. A subtle point in the whole analysis is to obtain uniform in time estimates which are largely independent of the induction hypothesis. In order not to overburden the reader with notations and keep the analysis relatively simple,we do not optimize the regularity assumption on initial data,and we do not spell out the precise dependence of the time step constraint on various parameters. All these issues and further generalizations will be addressed in forthcoming works.

Remark 1.3. A variant of the scheme (1.17) is:

where

Theorem 1.1 also holds for this case. Compared with(1.17),a slight difference is the solvability of uin the numerical scheme. In the former case the time step has to be taken suitably small so that ucan be uniquely solved from u. In the latter case (i.e., (1.18)) the solvability is not an issue and one can uniquely solve ufor any τ>0.

Remark 1.4. From a more practical point view, one should consider the spectral Galerkin truncated system:

where Πis the projection into first N Fourier modes. With minor modifications our analysis can be extended to this case. Note that in this case for the phase separation property to hold, we need to impose it on u=Πusince Πis not a continuous operator in L. Alternatively by using the high regularity of u,one can show that

As an immediate application of Theorem 1.1 (and to make this paper selfcontained), we obtain the following wellposedness result for the continuous PDE solution to (1.1). As a matter of fact this approach can be refined to yield a new wellposedness and regularity theory for the continuous case which we will address elsewhere. For simplicity we do not lower the regularity assumption on the initial data.

for some δ∈(0,1).

Our final result is the error analysis for the semi-implicit scheme. A similar result also holds for the variant (1.18).

where τis the same as in Theorem 1.1. Define t=mτ, m≥1. Then

Here, C,C>0 depends on (u,δ,ν,θ,θ).

The rest of this paper is organized as follows. In Section 2 we give the proof of Theorem 1.1. Section 3 is devoted to the proof of Corollary 1.1. In Section 4 we complete the error analysis and give the proof of Theorem 1.2. In Section 5 we give some concluding remarks.

Notation 1.1. For any real number a∈R, we denote by a+ the quantity a+? for sufficiently small ?>0. The numerical value of ? is unimportant, and the needed smallness of ? is usually clear from the context. The notation a?is similarly defined.This notation is particularly handy for interpolation inequalities. For example we shall use the notation

2 Proof of Theorem 1.1

For simplicity we assume ν=1 in (1.1). Let us consider the following semi-implicit scheme:

where

Then

The inductive assumption is:

The choice of the constants Aand Awill become clear in the course of the proof.The base step n=0 clearly holds true. In the rest of the proof we shall focus on the induction step n?n+1 for general n.

Thus

2.1 Discrete energy estimate of un+1

Multiplying both sides of (2.1)by(??)(u?u)and integrating(Taylor expand ?F(u) around ?F(u)), we obtain

where ξis between uand u. Since

we obtain

Now note that

2.2 Preliminary estimate of Kn+1

Denote

Note that

Lemma 2.1. It holds that

Proof. We write

where

Since

we clearly have

where we have used the Poincar′e inequality

Thus, we complete the proof.

By Lemma 2.1, (2.8) and (2.7), it follows that for sufficiently small τ, we have

2.3 Long time estimate of Kn+1

Now we consider the evolution equation for K. We have

Multiplying both sides by ??Kand integrating, we obtain

where

Note that α≥0. We then have

By Sobolev embedding, we have

Also observe that

Now taking ?>0 sufficiently small, we obtain

Lemma 2.2. Recall

Assume

where C>0 is a constant. Then we have

where C>0 depends on C.

Clearly,

The desired estimate (2.16) then easily follows.

Now note

Collecting all the estimates, we have

Now using (2.18) and summing backwards in n until one meets a good n or n=0,we then obtain

2.4 Control of gn+1?gn+12

2.5 Control of gn+1 and Kn+1

Now

2.6 Control ofuH3, gH3, f?(u)H3, ∞,?(∞and2(∞?

We shall explain the argument for j=n+1. It is clear from the argument below that the estimates will be uniform in j.

2.7 Control of un+1H5

Here we shall exploit the discrete smoothing effect. Denote

Iterating the above gives

In the estimate below, we shall use the uniform estimate:

2.7.1 Discrete smoothing estimates

We first prove two auxiliary lemmas needed for the higher order estimates later. In a slightly more general setup, we assume for some s≥0,

Define

Lemma 2.4. We have

Consequently

where α>0 depends only on α.

Proof. Observe that

Lemma 2.5. Assume 0<τ ≤1 and 4≤Jτ<5. Then

Proof. Observe that J ≥4 and

2.7.2 Higher order estimates

Now we discuss two cases.

Case 1: nτ ≤20. In this case we take J=n. By Lemma 2.4, we have

3 Proof of Corollary 1.1

In yet other words, vis the piece-wise linear interpolation of (u). Observe that for each t∈(nτ,(n+1)τ), we have

Proof. Without loss of generality, we assume T =1. From the argument below together with a further diagonal argument, one can easily cover the general case T>0.

This together with the regularity of u implies that u is the desired solution. Note that the strict phase separation and uniform Sobolev regularity of u on the time interval [0,∞) follows by taking the limit. Thanks to strict phase separation, it is routine to check that our constructed solution is unique. We note that the general case s>5 can be obtained by a simple bootstrapping argument. We omit further details here and leave them to interested readers.

4 Proof of Theorem 1.2

In this section we carry out the error estimate in L.

4.1 Auxiliary L2 error estimate for near solutions

where C>0 is a constant depending only on (δ,θ,θ).

The desired result follows from the standard Gronwall inequality.

Next we state and prove two lemmas needed for the proof of Theorem 1.2.

Lemma 4.1 (Discretization of the PDE solution). Let t=nτ, n≥0. Let u be the exact PDE solution to (1.1). Denote

We have

where

Similarly for a slightly different discretization, we have

where

Proof. Integrating the PDE for u on the time interval [t,t], we obtain

Note that for a one-variable function h=h(t), we have the formula

By using the above formula, we have

Thus

where

The derivation of (4.10) is similar. We omit details.

Lemma 4.2. Let u be the PDE solution constructed in Corollary 1.1. Then we have

Next to obtain L-in time integrability of ??u, we recall

Clearly,

Thanks to strict phase separation, we have ?f(u)≥0. Multiplying both sides of(4.13) by ??μ and integrating by parts, we obtain

Thus, we complete the proof.

Proof of Theorem 1.2. We need to consider

We first rewrite the PDE solution u in the discretized form. By Lemma 4.1,we have

where

Thus we obtain (1.20).

5 Concluding remarks

In this paper we studied the Cahn-Hilliard equation with singular logarithmic potentials on the two-dimensional periodic torus. We analyzed a first order in time,semi-implicit numerical discretization scheme which treats the linear fourth-order dissipation term implicitly and the nonlinear term explicitly. Prior state of the art literature are concerned with implicit or partially implicit methods for which phase separation and energy stability are established under nearly optimal conditions. For semi-implicit type methods, these issues were long standing open problems. In this work we developed a new theoretical framework and proved strict phase separation and energy stability for all time under mild constraints on the time step and initial data. We also carried out a rigorous error analysis which is done for the first time for semi-implicit methods on Cahn-Hilliard equations with singular potentials. It is expected our theoretical framework can be refined and generalized to cover many other similar problems. Research is now underway to investigate several directions including the stability and error analysis of higher-order methods, general thin-film type problems with singular potentials,various time-splitting methods,and adaptive time-stepping methods.

Acknowledgements

The first author’s work was supported in part by Hong Kong RGC grant GRF Nos. 16307317 and 16309518. The second author’s work is partially supported by the NSFC grants Nos.11731006 and K20911001,NSFC/RGC No.11961160718,and the Science Challenge Project (No. TZ2018001).