Weak Solutions for a Sixth Order Cahn-Hilliard Type Equation with Degenerate Mobility
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.
We consider in , where is a bounded domain, , is mobility, , , and , are constants . 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 . Here is the scalar order parameter which is proportional to the local difference between oil and water concentrations. Pawow and Zajączkowski  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  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  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  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  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
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  and Lions ) 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 .
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  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
and the fact that is uniformly bounded, a generalized version of the Lebesgue convergence theorem yields
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.
J. L. Lions, Quelques méthodes de résolution des problèmes aux limites nonlinéaires, Dunod, Paris, France, 1969.