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Classical solutions for the Cahn-Hilliard equation with decayed mobility
© Huang et al.; licensee Springer 2014
Received: 8 October 2014
Accepted: 8 December 2014
Published: 19 December 2014
We consider the Cahn-Hilliard equation with concentration dependent mobility in two dimensions. Global existence and uniqueness of classical solutions are established for a mobility with some delayed structure and general potential including for both and .
Combining (1.7), (1.8), and (1.9), and by a simple calculation, we obtain the desired Cahn-Hilliard equation (1.1).
The Cahn-Hilliard equation with constant mobility, i.e., has been intensively studied. In one spatial dimension, a well-known work is by to Elliott and Zheng , who showed that the sign of γ is crucial to the global existence of solutions. Exactly speaking, if then solutions exist always globally in time; while if , then solutions must blow up in a finite time for large initial data. From the physical point of view, the mobility should depend on the concentration. In general, for , there is no restriction with positive lower bound, but there is a possibility with degeneracy, see for example , –, where, for , the existence of weak solutions for the degenerate case and classical solutions for the uniformly parabolic case are established, respectively. Our interest lies in the case that the mobility but decays as . Some discussion in this topic for one spatial dimension was given in our previous work , while the present paper is focused on the discussion for two dimensions. The case with decayed mobility not only gets rid of some properties for the case , but it also exhibits some new features compared to the case Indeed, the existence result can be established for both and without essential restrictions on the initial data, see  for some information in one spatial dimension. However, compared to the one-dimensional case , the present work will encounter more difficulties in the arguments of the regularity of solutions. For this reason, we employ the framework based on Campanato spaces to obtain the Hölder continuity of higher order derivatives of solutions.
The main result of this paper is the following theorem.
Then the problem (1.1)-(1.4) admits a unique classical solution for a small smooth initial value.
This paper is organized as follows. Section 2 is devoted to the a priori Hölder norm estimates; the gradient Hölder norm estimates are given subsequently in Section 3. Finally, in the last section we prove the existence and uniqueness of classical solutions to the problem (1.1)-(1.4).
2 Hölder norm estimates
In this section, by means of energy estimates, we establish the a priori Hölder norm estimates on solutions to the problem (1.1)-(1.4).
where C is a positive constant.
where q is a positive constant depending on p, and here we used the smallness of δ to conclude that .
holds for any given . The proof of this proposition is complete. □
3 Gradient Hölder norm estimates
In this section, we establish the gradient Hölder norm estimates on solutions to the problem (1.1)-(1.4).
where C is a positive constant.
Since the Hölder norm estimate of u has been already established in the previous section, we may assume that is a known Hölder continuous function. For a qualitative calculation, without loss of generality, we may also assume that and are sufficiently smooth, otherwise we replace them by their approximation functions.
and is the ball centered at with radius ρ.
By the classical linear theory, the above decomposition is uniquely determined by u. The following lemmas will be used to establish the a priori estimates of the solutions in the Campanato space.
The above inequality implies the desired result of this lemma. The proof of this lemma is complete. □
holds for any given.
Then by using (3.2) itself we can obtain the desired estimate at once. The proof of this lemma is complete. □
By the definition of and χ, we immediately obtain the desired first inequality of this lemma, and thus we complete the proof. □
The proof of this lemma is complete. □
The following technical lemma is required to estimate the Hölder norm of ∇u. One can find its proof in Giaquinta .
where C is a positive constant depending only on α, β, and A.
where is a constant. For in Lemma 3.5, we can choose such that whenever . Then, by Lemma 3.5, one can complete the proof of this lemma immediately. □
Now we can give the proof of the main result in this section.
Proof of Proposition 3.1
for any given .
denote the normal and tangential derivatives, respectively. The remaining part of the proof is similar to that in the proof of the previous lemmas, and we omit the details here. The proof of this theorem is complete. □
4 Existence and uniqueness
In this section, we give the proof of the existence and uniqueness of classical solutions to the problem (1.1)-(1.4).
Proof of Theorem 1.1
By classical linear theory, the above problem admits a unique solution in the space . So the operator T is well defined and compact. Moreover, if for some , then u satisfies (4.1), (1.3), (1.6), and . Thus, from above discussion, the norm of u in the space can be determined by some constant C depending only on the known quantities. By the Leray-Schauder fixed point theorem, the operator T has a fixed point u, which is the desired classical solution of the problem (1.1)-(1.4).
It follows from the arbitrariness of the function f that a.e. in . Then, by the continuity of and , we have in . The proof of this theorem is complete. □
The research of Huang and Yin was supported in part by NNSFC (No. 11071099) and SRFDP (No. 20114407110008). The research of Mei was supported in part by the Natural Sciences and Engineering Research Council of Canada.
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