Radial symmetric solutions of the Cahn-Hilliard equation with

In this paper we study the radial symmetric solutions of the two-dimensional Cahn-Hilliard equation with degenerate mobility. We adopt the method of parabolic regularization. After establishing some necessary uniform estimates on the approxi- mate solutions, we prove the existence and the nonnegativity of weak solutions. Keywords. Cahn-Hilliard equation, radial solution, degenerate mobility, nonnega- tivity. AMS Classication: 5G25, 35Q99, 35K9, 82B26


Introduction
This paper is devoted to the radial symmetric solutions of the Cahn-Hilliard equation with degenerate mobility and m(s), A(s) are appropriately smooth and satisfy the following structure conditions for some positive constants p, q, µ, C s.We note that a reasonable choice of A(s) is the cubic polynomial, namely EJQTDE, 2001 No. 2, p.1 which corresponds to the so called double-well potential The Cahn-Hilliard equation was introduced to study several diffusion processes, such as phase separation in binary alloys, see [1,2].During the past years, such an equation has been paid extensive attention.In particular, there are vast literatures on the investigation of the Cahn-Hilliard equation with constant mobility, for an overview we refer to [3,4].However, there are only a few works devoted to the equation with degenerate mobility, see [5,6,7,8,9,10,11,12], among which Elliott and Garcke [6] was the first who established the basic existence results of weak solutions for space dimensions large than one.
In this paper, we study the radial symmetric solutions of the Cahn-Hilliard equation.We will study the problem in two-dimensional case, which has particular physical derivation of modeling the oil film spreading over a solid surface, see [13].After introducing the radial variable r = |x|, we see that the radial symmetric solution satisfies where It should be noticed that the equation (1.1) is degenerate at the points where r = 0 or u = 0, and hence the arguments for one-dimensional problem can not be applied directly.Because of the degeneracy, the problem does not admit classical solutions in general.So, we introduce the weak solutions in the following sense Definition A function u is said to be a weak solution of the problem (1.1)-(1.3),if the following conditions are fulfilled: (1) ru(r, t) is continuous in Q T , where Q T = (0, 1) × (0, T ); (2) rm(u)u rrr ∈ L 2 (P ), where We first investigate the existence of weak solutions.Because of the degeneracy, we will first consider the regularized problem.Based on the uniform estimates for the approximate solutions, we obtain the existence.Owing to the background, we are much interested in the nonnegativity of the weak solutions.For this purpose, we construct a suitable test function and discuss such a property under some conditions on the data.This paper is arranged as follows.We first study the regularized problem in Section 2, and then establish the existence in Section 3. Subsequently, we discuss the nonnegativity of weak solutions in the last Section.

Regularized problem
To discuss the existence, we adopt the method of parabolic regularization, namely, the desired solution will be obtained as the limit of some subsequence of solutions of the following regularized problem where Theorem 2.1 For each fixed ε > 0 and suitably smooth u 0 , under the assumptions (H 1 ), (H 2 ), the problem (2.1)-(2.3)admits a unique classical solution u in the space C 4+α,1+α/4 (Q T ) for some α ∈ (0, 1).
To prove the theorem, we need some a priori estimates on the solutions.We first have Lemma 2.1 For any α ∈ (0, 1  2 ] and β < α, there is a constant M independent of ε such that for all r, s ∈ (0, 1), where s ε = s + ε.
Proof.The first two estimates have already been seen from the arguments in Lemma 2.1.To prove (2.9), we need an integral estimate first.Multiplying the equation (2.1) by V and integrating with respect to r over (0, 1), we get ∂r dr i.e. dr, which, together with the first two estimates in this lemma, implies that Now, we begin to show (2.9).Without loss of generality, we assume that t 1 < t 2 and set ∆t = t 2 − t 1 .Integrating both sides of the equation (2.1) over (t 1 , t 2 ) × (y, y + (∆t) α ) and then integrating the resulting relation with respect to y over (x, x + (∆t) α ), we get By the mean value theorem, there exists x * = y * + θ * (∆t) α , y * ∈ (x, x + (∆t) α ), θ * ∈ (0, 1) such that the left hand side of the above equality can be expressed by For the right hand side, we have . By (2.8), (2.10) and the assumptions on m(u), A(u), we see that which implies, by setting α = 1/4 and using the properties of the functions in W 1,2 * ,ε (I), that The proof is complete.Proof of Theorem 2.1.Using Lemma 2.1 and 2.2, we see that r ε u is uniformly bounded in C 1/4,1/16 (Q T )-norm with the bound independent of ε.Similar to [14], we can further establish the estimates on the Hölder norm of Du.Then, using the classical Schauder theory, we may complete the proof of the remaining part in a standard way.

Existence
After the discussion of the regularized problem, we can now turn to the investigation of the existence of weak solutions of the problem (1.1)-(1.3).The main existence result is the following Theorem 3.1 Under the assumptions (H 1 ), (H 2 ), the problem (1.1)-(1.3)admits at least one weak solution.
Proof.Let u ε be the approximate solution of the problem (2.1)-( 2.3) constructed in the previous section.Using the estimates in Lemma 2.1 and 2.2, for any β < 1  2 , and (r 1 , t 2 ), (r 2 , t 1 ) ∈ Q T , we have with constant C independent of ε.So, we may extract a subsequence from {r ε u ε }, denoted also by {r ε u ε }, such that and the limiting function ru ∈ C 1/4,1/16 (Q T ).By (2.8), we also have r α u ∈ L ∞ (Q T ) with α > 0 and for any t ∈ (0, T ), u(•, t) ∈ W 1,2 * ,0 (I) with the norm u(•, t) * ,0 bounded by a constant independent of t.Now, let δ > 0 be fixed and set P δ = {(r, t); rm(u(r, t)) > δ}.We choose ε(δ) > 0, such that Then from (2.10) To prove the integral equality in the definition of solutions, it suffices to pass the limit as ε → 0 in In fact, for any fixed δ > 0, From the estimates (2.10), we have ∂V ∂r ∂ϕ ∂r drdt and hence By the arbitrariness of δ, we see that the limit (3.3) holds.
EJQTDE, 2001 No. 2, p.9 Finally, from the uniform convergence of r ε u ε to ru, we immediately obtain The proof is complete.

Nonnegativity
Just as mentioned by several authors, it is much interesting to discuss the physical solutions.For the two-dimensional problem (1.1)-(1.3),a very typical example is the modeling of oil films spreading over an solid surface, where the unknown function u denotes the height from the surface of the oil film to the solid surface.Motivated by this idea, we devote this section to the discussion of the nonnegativity of solutions.Theorem 4.1 The weak solution u obtained in Section 3 satisfy u(x, t) ≥ 0, if u 0 (x) ≥ 0.
Proof.Suppose the contrary, that is, the set is nonempty.