Research Article | Open Access

# Weak Solutions for a Sixth Order Cahn-Hilliard Type Equation with Degenerate Mobility

**Academic Editor:**Ming Mei

#### Abstract

We study an initial-boundary problem for a sixth order Cahn-Hilliard type equation, which arises in oil-water-surfactant mixtures. An existence result for the problem with a concentration dependent diffusional mobility in three space dimensions is presented.

#### 1. Introduction

We consider in , where is a bounded domain, , is mobility, , , and , are constants [1]. From the physical consideration, we prefer to consider a typical case of the volumetric free energy ; that is, , in the following form [1, 2]: Equation (1) is supplemented by the boundary value conditions and the initial value condition Equation (1) is the sixth order parabolic equation which describes dynamics of phase transitions in ternary oil-water-surfactant systems [2]. Here is the scalar order parameter which is proportional to the local difference between oil and water concentrations. Pawow and Zajączkowski [2] proved that the initial-boundary value problem (1)–(4) with admits a unique global smooth solution which depends continuously on the initial datum. Wang and Liu [3] proved that the solutions of problem (1)–(4) with might not be classical globally. In other words, in some cases, the classical solutions exist globally, while in some other cases, such solutions blow up at a finite time. They also discussed the existence of global attractor. Liu and Wang [4] considered the optimal control problem for the problem (1)–(4) with . They proved the existence of optimal solution. The optimality system is also established. Since the mobility depends on the concentration in general, the equation with nonlinear main part reflects even more exactly the physical reality comparing to the one with linear main part. Schimperna and Pawow [5] studied (1) with viscous term , and logarithmic potential They investigated the behavior of the solutions to the sixth order system as the parameter tends to . The uniqueness and regularization properties of the solutions have been discussed. Liu [6] studied the problem (1)–(4) and he proved the existence of classical solutions for two dimensions and , , .

In this paper, we study the problem (1)–(4) with degenerate concentration dependent mobility. The main difficulties for treating the problem are caused by the degeneracy of the principal part, nonlinearity of the fourth order term, and the lack of maximum principle. Our method is based on Galerkin approximation and Simon's compactness results.

This paper is organized as follows. In Section 2, using Galerkin approximation, we prove the existence of the weak solution for positive mobility. In Section 3, we prove the existence of the weak solution for degenerate case.

#### 2. Existence for Positive Mobilities

In this section, we study the Cahn-Hilliard equation with a mobility which is bounded away from zero. We prove existence of weak solutions.

Consider the following sixth order Cahn-Hilliard equation: with the boundary conditions and the initial value

In this section, we assume is a bounded domain with Lipschitz boundary and such that

(i) and there is such that .

Under these assumptions we can state the following theorem.

Theorem 1. *Suppose . Then there exists a pair of functions such that*(1)*,*(2)*,*(3)*,*(4)*,**which satisfies equation (1) in the following weak sense:
**
for all , and
**
for all , and almost all .*

*Proof. *To prove the theorem we apply a Galerkin approximation. Let be the eigenfunctions of the Laplace operator with Neumann boundary conditions; that is,
The eigenfunction is orthogonal in the , , and scalar product. We normalize the such that . Furthermore we assume without loss of generality that . Now we consider the following Galerkin ansatz for (6):
This gives an initial value problem for a system of ordinary differential equations for :
By use of the Peano existence theorem, the initial value problem has a local solution. Now, we set
In order to derive a priori estimates we differentiate the and get
This implies
The last inequality follows the fact that .

On the other hand, we have
By the Young inequality
Combining the above inequalities and using , yield
This estimate implies that the are bounded and therefore a global solution to the initial value problem (6) exists. By Gagliardo-Nirenberg inequality (noticing that we consider only the three-dimensional case)
If we denote by the projection of onto span , we get
for all . This implies
Using compactness results, (Simon [7] and Lions [8]) we obtain for a subsequence (which we still denote by )
It remains to show the convergence of ; choosing in (14) gives
By (19) and Poincaré inequality, we obtain
and this implies weakly in .

The strong convergence of in and the fact that in give . By weakly in and weakly in we can pass limit in (9). Using (22), we know . Then we can pass the limit in (10).

#### 3. Existence for the Degenerate Case

Our approach is to approximate the degenerate problem by nondegenerate equations. By Theorem 1, we know the existence of weak solution to the problem where .

We denote the solution by ; from now on we assume either or is convex.

Lemma 2. *The solution belongs to the space and .*

*Proof. *Since
for all , and almost all , from elliptic regularity theory, we get . By , (22) and (23), we have
Then we get
which implies . Therefore, we get
for all .

In the next step we prove the energy estimates.

Lemma 3. *There exists an such that for all , and the following estimates hold with a constant C independent of *

*Proof. *The is a valid test function in (33). Therefore, we obtain
Define functions
where we set when . It is easily proved that
for at least a subsequence . Furthermore we can show strongly in . For any we have
Since
it follows that
Using we have for almost all
Pass to the limit in the equation, where we apply the convergence properties of . Hence, we get
and (2) follows easily from (1), and this finishes the proof of Lemma 3.

Theorem 4. *Let either or convex and suppose that . Then there exists a pair such that*(1)*,*(2)*,*(3)*,*(4)*,**which satisfies in ; that is,
**
for all and
**
in the following weak sense:
**
for all which fulfill on .*

*Proof. *From Lemma 2, we get
Now we apply the compactness result [7] with , , and to conclude the existence of a subsequence of such that
Furthermore using standard compactness properties we obtain the convergence
It remains to show that fulfills the limit equation. The weak convergence of and gives in the limit
for all . Now we have identified . Therefore, we want to pass to the limit in
where with on and the left hand side converges to . Since may not have a limit in , we integrate the first term on the right hand side of (50) by parts to get
It is easily proved that uniformly. Hence, we have
Since weakly in and is uniformly bounded, we conclude
Now we pass to the limit in . We consider the case . As for , we have uniformly, which gives
By using
and the fact that is uniformly bounded, a generalized version of the Lebesgue convergence theorem yields
Hence
where we use the fact that .

To complete the proof of Theorem 4 we have to show
In fact, we have
On the other hand, we know
Hence, when , the right hand side of the inequality tends to zero. Similarly, we can pass to the limit, and this finishes the proof of Theorem 4.

#### Conflict of Interests

The authors declare that they have no conflict of interests in this paper.

#### References

- G. Gompper and J. Goos, “Fluctuating interfaces in microemulsion and sponge phases,”
*Physical Review E*, vol. 50, no. 2, pp. 1325–1335, 1994. View at: Publisher Site | Google Scholar - I. Pawłow and W. M. Zajączkowski, “A sixth order Cahn-Hilliard type equation arising in oil-water-surfactant mixtures,”
*Communications on Pure and Applied Analysis*, vol. 10, no. 6, pp. 1823–1847, 2011. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Z. Wang and C. Liu, “Some properties of solutions for the sixth-order Cahn-Hilliard-type equation,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 414590, 24 pages, 2012. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Liu and Z. Wang, “Optimal control for a sixth order nonlinear parabolic equation,”
*Mathematical Methods in the Applied Sciences*, 2013. View at: Publisher Site | Google Scholar - G. Schimperna and I. Pawłow, “On a class of Cahn-Hilliard models with nonlinear diffusion,”
*SIAM Journal on Mathematical Analysis*, vol. 45, no. 1, pp. 31–63, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - C. Liu, “Regularity of solutions for a sixth order nonlinear parabolic equation in two space dimensions,”
*Annales Polonici Mathematici*, vol. 107, no. 3, pp. 271–291, 2013. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. Simon, “Compact sets in the space ${L}^{p}(0,T;B)$,”
*Annali di Matematica Pura ed Applicata. Serie Quarta*, vol. 146, pp. 65–96, 1987. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. L. Lions,
*Quelques méthodes de résolution des problèmes aux limites nonlinéaires*, Dunod, Paris, France, 1969.

#### Copyright

Copyright © 2014 Aibo Liu and Changchun Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.