ON UNIFORM IN TIME H-REGULARITY OF THE SOLUTION FOR THE 2D CAHN-HILLIARD EQUATION

In this article, we provide the uniform H2-regularity results with respect to t of the solution and its time derivatives for the 2D Cahn-Hilliard equation. Based on sharp a priori estimates for the solution of problem under the assumption on the initial value, we show that the H2-regularity of the solution and its first and second order time derivatives only depend on ε−1.

1. Introduction.Many dynamical problems in material science require deep understanding of the microstructure that develops in a mixture of two of more materials or phases.One such equation is the well-known Cahn-Hilliard equation.It was originally introduced by Cahn and Hilliard [5] to study the phase separation in binary alloys as the phenomenological models.The Cahn-Hilliard equations have been adopted and developed to model many other physical situations, e.g.interface dynamics in multi-phase fluids.
Given a smooth bounded domain Ω ⊂ R 2 , we consider the following Cahn-Hilliard equation: u(x, 0) = u 0 (x), (1.4) where u represents the concentration of one of the two metallic components of the alloy, w represents the chemical potential defined by (1.2), the parameter ε 5388 XINLONG FENG AND YINNIAN HE represents the interfacial width which is small compared to the characteristic length of the laboratory scale.The boundary of Ω is supposed to be sufficiently smooth, and is denoted by ∂Ω, with outward unit normal n.Clearly (1.1)-(1.4)represent a natural splitting of the original nonlinear fourth-order Cahn-Hilliard equation into two coupled problems which are second order in space.This model can be viewed as the gradient flow associated with the following energy functions respectively: in H −1 space, where F (u) is a given energy potential and f (u) = F (u) is usually a nonlinear term in this equation.The nonlinear term taken here is the double well potential in the Cahn-Hilliard equation as in most literatures.It is known that the solution of the phase field models possesses the properties that the total mass is conserved and the total free energy decreases with time.
It is clear that (1.1) and (1.3) imply Problem (1.1)-(1.4)has been widely analyzed by many authors.Elliott and Zheng [7] considered the global existence or blow up in a finite time to the solution of (1.1)-(1.4)as well as its related finite element Galerkin approximation.Alikakos et al. [2] proved that level surfaces of solutions to the Cahn-Hilliard equation tend to solutions of the Hele-Shaw problem under the assumption that the classical solutions of the latter exist.In these works the analysis is carried out under the assumptions that the solutions are bounded uniformly in time, so that one may essentially assume that the nonlinearity in f (u) satisfies a global or local Lipschitz condition with respect to Sobolev norms.Because of the lack of a maximum principle this means that one has to prove (or assume) that the solution is sufficiently smooth depending on the number of space dimensions.See [2,6,9,10,11,12,13,15] and the references cited therein.
Particularly, the authors in [11,12,15,16] give the assumption on the potential function F (u) satisfied the condition where L > 0 is a constant and f (u) = F (u).Although this condition is satisfied by many physically relevant potentials by restricting the growth of F (u) to be a lower order polynomial for large |u|, here it will be ideal to remove this restriction in the analysis.In [7], by use of the Gronwall inequality, Elliott and Zheng obtained the upper bounds for u and ∆u with exponential function form exp( T ε 2 ) and 1 ε 2 exp( T ε 2 ), respectively, for 0 < t ≤ T .It is observed that these bounds are too large for 0 < ε ≤ 1.Our goal is to present some H 2 -a priori estimates on (u, w), (u t , w t ) and (u tt , w tt ) with polynomial power of ε −1 .On the other hand, it is noted that "polynomial power of ε −1 " is done in [10], Feng and Prohl studied the Cahn-Hilliard equation on the bounded time interval [0, T ] and derived some energy estimates in various function spaces in terms of negative powers of ε for the solution of for given u 0 ∈ H 2+l (Ω), l = 0, 1.In order to trace dependence of the solution on the small parameter ε, they assumed three conditions on the nonlinear term f (u) and three conditions on the initial function u 0 .In this work, we only assume the initial function u 0 (x) ∈ H 5 (Ω) and satisfy a minor assumption, i.e.Assumption (A1) in Section 2.
In this paper, we first provide the H 2 -regularity depending on ε of the solution (u, w) of (1.1)- (1.4).Then, we provide the H 2 -regularity for the time derivatives (u t , w t ) and (u tt , w tt ) together with the L 2 -regularity ε −1 of u ttt .These estimates for the first and second order time derivatives are useful to the error estimate of the m-th order time discrete scheme with m = 1, 2 of problem (1.1)- (1.4).The analysis is based on sharp a priori estimates for the solution of problem (1.1)-(1.4)under the assumption on the initial function u 0 , particularly reflecting its behavior as t → 0 and as to t → ∞.In [3,4], Heywood and Rannacher presented a priori estimates for the nonstationary Navier-Stokes problem by incorporating the weight functions to obtain more precise information as t → 0 and as to t → ∞.Here we used the similar technique and introduced the weight functions σ(t) = min{1, t} and the exponential factor e t c 0 to the Cahn-Hiliard problem (1.1)-(1.4)and obtained the H 2 -regularity results of the solution and its first and second order time derivatives, the regularity results are uniform in time t and dependent of ε −1 .
This paper is organized as follows.In §2, some basic mathematical setting and the H 2 -regularity of the solution (u, w) are provided.In §3, the H 2 -regularity of the first order time derivative of solution (u, w) is obtained.In §4, the H 2 -regularity of the second order time derivative of solution (u, w) is provided.•, •) Ω to denote the inner product in L 2 (Ω).Throughout the paper we will frequently use the following inequalities( cf.[1,9]): here c 0 denotes a positive constant depending Ω, and c denotes a general positive constant depending on Ω, which can take different values at its different occurrences.
In this paper we always make the following assumptions on u 0 (x) and Ω. where ) and C 0 and C i with i = 1, 2, 3, 4 are positive constants which do not depend on ε.
For the purpose of numerical analysis, we need refinements of certain estimates for problem (1.1)-(1.4)previously given in the context of the existence and regularity theory.The required estimates will be proved below.We will omit most details which are already familiar from the existence and regularity theory, particularly as developed in [2,7,8,10].On the other hand, for the purpose of existence and regularity theory, it suffices to obtain estimates for the solution and its derivatives which grow exponentially as t → ∞, and these are usually what are given.However, here we wish to obtain error estimates useful for numerical approximations which are uniform in time when the solution being approximated is stable.This, of course, requires estimates of the solution's regularity which are uniform in time.
3. H 2 -regularity of (u t , w t ).In this section, we shall provide the H 2 -regularity of the first order time derivative of solution (u, w).
Proof.Taking q = w t in (2.Combining the above relations with (3.5), we deduce The above result together with Theorem 2.1 leads to (3.1).Next, taking q = −∆w t in (2.14) and v = −∆u tt in (2.15) yields 1 2 Multiplying this by σ(t)e t c 0 , one finds Integrating (3.8) from 0 to t, we deduce Moreover, it follows from (2.1)-(2.4)that Next we give estimate of the terms Π i on the right hand side of (3.9):The above inequalities together with (3.1), (3.9) and Theorem 2.1 yield (3.2).This completes the proof.