# Global Attractors of Sixth Order PDEs Describing the Faceting of Growing Surfaces

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## Abstract

A spatially two-dimensional sixth order PDE describing the evolution of a growing crystalline surface *h*(*x*, *y*, *t*) that undergoes faceting is considered with periodic boundary conditions, as well as its reduced one-dimensional version. These equations are expressed in terms of the slopes \(u_1=h_{x}\) and \(u_2=h_y\) to establish the existence of global, connected attractors for both equations. Since unique solutions are guaranteed for initial conditions in \(\dot{H}^2_{per}\), we consider the solution operator \(S(t): \dot{H}^2_{per} \rightarrow \dot{H}^2_{per}\), to gain our results. We prove the necessary continuity, dissipation and compactness properties.

## Keywords

Global attractor Cahn-Hilliard type equation Anisotropic surface energy## 1 Introduction

*W*given below, see (2), (3). We restrict our attention to special geometries and work with \(\Omega =(0,L)^d\), \(d=1,2\) and furthermore we assume periodic boundary conditions for

*h*.

We show existence of global attractors for the above system and we shall also explain, why this result seems the best we can hope for. The PDE was introduced by Savina *et al.*, see [20]. It has been derived by invoking Mullins’ surface diffusion formula [15], a normally impinging flux of adatoms to the surface and a strongly anisotropic surface energy formula. The reduced evolution equation is obtained by carrying out a long-wave approximation. The choice of periodic boundary conditions is realistic as the patterns of the nanostructures statistically repeat throughout the domain, which is much larger than the length-scales of interest. Numerical simulations imposing these kinds of boundary conditions show good agreement with the experimentally observed behavior of crystalline materials undergoing faceting and coarsening [7, 20]. We also notice that the analysis on periodic domains is easy to transfer for the numerical analysis of simulation schemes based on trigonometric interpolation. Such collocation methods are applied frequently to such problems.

*W*. If \(d=1\), then we take

Formula (3) gives a quadruple well that is responsible for the faceting of the growing surface in shape of pyramids with four preferred orientations and hence preferred slopes. A constant *A* may be chosen such that *W* is always nonnegative.

In two related works, [10, Theorem 1.1] and [11, Theorem 2.1], we proved the existence of global in time weak solutions to (1) with periodic boundary data. There were no size restrictions on the data.

In [10, 11] we showed only exponential bounds on the growth of solutions which is not particularly suitable for studying long time behavior. We will find the remedy here and we will show existence of a global attractor of (1) for \(d=1,2\). The destabilizing term does not give us much hope to establish convergence to an equilibrium state. However, if we had a Liapunov functional, then we could hope to use methods based on Łojasiewicz inequality to show convergence of solutions to a steady state, see [19].

Our plan is to study first the one-dimensional problem, so that we can develop ideas that are used later also in the more complex case. It turns out that the trick applied in [11, 20] works very nicely. Namely, after differentiating (1) with respect to *x* we obtain a *slope equation* for the new unknown quantity \(u=h_x\), see (4). One advantage is that we obtain a new conserved quantity, \(\int _0^L u\,\text {d}x =0\). This will imply that the semigroup generated by \(\Delta ^3\) has an exponential decay. Another advantage is, the resulting equation is similar to the convective Cahn–Hilliard equation, which has already been analyzed to some extent. Equation (1) may be interpreted as a convective Cahn–Hilliard (CCH) type equation of higher order, hence we call it the HCCH equation. Note that it is the gradient system perturbed by a destabilizing Kardar-Parisi-Zhang type term \(|\nabla h|^2\).

Here, we use ideas from the theory of infinite dimensional dynamical systems [4, 18] combined with the available results on convective Cahn–Hilliard equation, e.g. [1, 3, 12]. Eden and Kalantarov noticed, see [1], that the structure of the lower order convective Cahn–Hilliard equation permits to deduce bounds implying the existence of an absorbing set. The same method can be applied here. We deduce from it the existence of an absorbing set in the \(H^1\) topology and we extend this result to \(H^2\). Showing its compactness in \(H^2\) requires further improvement of the regularity of weak solutions. Once we have achieved this goal, we may conclude the existence of a global attractor, see [14, Theorem 1].

We notice that, if we take the gradient of (1) with respect to the spatial variables in the two dimensional case, then the resulting system, see (8), has the structure which permits to carry the calculations we did for the one-dimensional problem. Thus, we establish the existence of the global attractor for the corresponding system, which is the result of the gradient of (1), and we call it the slope system, \(u=\nabla h\). Finally, we deduce from this existence result, the existence of a global attractor of the original Eq. (1), see Theorems 4, 5, 6.

We proceed as follows. In the next section we recall the notion of weak solutions and the necessary facts from [10, 11]. In addition we state the main results. In Sect. 3 we prove the existence of absorbing balls in \(H^1\) for the one-dimensional problem (4). This is done with the help of ideas taken from [1]. We also show in this section the necessary auxiliary facts. In Sect. 4, we study the system, which is obtained by taking the gradient of (1) and we call it the slope system. Its advantage is that we can use exactly the same method, as in the one-dimensional case to show the existence of an absorbing ball in \(H^1\). Next section is devoted to the proof of higher order regularity and compactness in \(H^2\) of the absorbing balls, we use the parameter variation formula for this purpose. This is done in both case \(d=1\) and \(d=2\).

Finally, we discuss the results and future plans in Sect. 6.

## 2 Preliminaries and Main Statements

### 2.1 Properties of Solutions and Main Statements

*x*. The resulting problem is

*weak solution*to (4) provided that it fulfills, (see [11]),

We showed the existence of such solutions:

### **Proposition 1**

- (a)
If initial condition \(u_0\) is in \(\dot{H}^1_{per}\), then for any \(T>0\) there is a weak solution to (4) on the time interval [0,

*T*). - (b)If in addition \(u_0 \in \dot{H}^2_{per}\), then a weak solution constructed in part (a) is unique and$$\begin{aligned} u \in L^2(0,T; \dot{H}^4_{per})\cap L^\infty \big (0,T; W^{1,\infty }(\Omega ) \big ), \end{aligned}$$(6)

For the two-dimensional problem (1) and (3) we established a similar result. We have shown the existence of a weak solution to (1) with periodic boundary conditions, understood as a function \(h \in C([0,T), H^3_{per})\) with \(h(\cdot ,0) =h_0\) and \(h_t \in L_\infty ((0,T),(H^{3}_{per})^*)\), such that *h* satisfies (1) in the distributional sense.

### **Proposition 2**

*W*will be carried out in Sect. 5.

For the purpose of proving our main results we should recast Eqs. (4) and (8) in the terms of the semigroup theory.

### **Proposition 3**

Let us denote by \(S(t)u_0\) the unique solution *u*(*t*) to (4) if \(d=1\), (respectively, (8) if \(d=2\)), with \(u_0\in (H^2_{per})^d\). Then, for each \(t>0\) operators \(S(t):(\dot{H}^2_{per})^d \rightarrow (\dot{H}^2_{per})^d\) are continuous. If we set \(S(0)=Id\), then the family \(\{ S(t) \}_{t\ge 0}\) forms a strongly continuous semigroup.

Continuity of *S*(*t*), \(t>0\) follows from results in [10, 11]. The uniqueness theorems imply that the family \(\{S(t)\}_{t\ge 0}\) has the semigroup property. It remains to establish strong continuity of the family \(\{S(t)\}_{t\ge 0}\) in the one-dimensional case. This will be done in Sect. 3.2. On the other hand, strong continuity of *S* in the two-dimensional case of (1) has been already established in [10].

The use of the language of the semigroup theory does not imply that we need to re-prove our existence results exploiting the analytical semigroup theory, see [5] or [2]. If we tried this, then we would repeat estimates specific for these problems presented in [10, 11]. However, we will need additional regularity estimates, which we will establish with the help of the constant variation formula see Sects. 2.3, 5.1.

Here are our main results.

### **Theorem 4**

(1D Attractor in \(\dot{H}^2_{per}\)) Let us consider \(\Omega =(0,L)\) with \(L>0\) arbitrary. The semigroup \(S(t):\dot{H}^2_{per} \rightarrow \dot{H}^2_{per}, u_0 \mapsto S(t)u_0 = u(t) \) generated by the HCCH Eq. (4) with periodic boundary conditions has a compact global attractor.

### **Theorem 5**

( 2D Attractor in \((\dot{H}^2_{per})^2 \)) Let us consider \(\Omega =(0,L)^2\) with \(L>0\) arbitrary. The semigroup \(S(t):(\dot{H}^2_{per})^2 \rightarrow (\dot{H}^2_{per})^2, u_0 \mapsto S(t)u_0 = u(t) \) generated by Eq. (8) with periodic boundary conditions has a compact global attractor.

Once we show these results we may address the question of the behaviour of the solutions to the original problem (1). We notice that one can easily recover a continuous function *f* from its derivative and its mean. Thus, the above results imply:

### **Theorem 6**

The semigroup generated by Eq. (1) has a global attractor in \(H^3_{per}\) for \(d=1\) and \(d=2\).

*u*stays roughly below 1, independently of the value for \(\delta \). Our analytical result, however, gives us information of different nature. We can take bigger initial conditions and still the absorption is in the same ball. This property is indicated in Fig. 2 where another typical evolution of Eq. (4) is shown together with the decrease of the norm of the discrete solution and the three phase spaces \((u,u_x), (u,u_{xx})\) and \((u_x, u_{xx})\). However, these runs calculated with a pseudospectral method discussed elsewhere [11], are particular examples with special initial conditions, fixed domain length and deposition parameter. The theory establishes a general result. We would like to remark, that more simulation results, also for the two-dimensional setting, can be found in [20].

### 2.2 Tools of Dynamical Systems

We will use the methods of the infinite dimensional dynamical systems, see the books by Hale, [4], Temam, [22] or Robinson, [18]. However, we will use the theorem guaranteeing existence of a compact global attractor as stated in [14]. The general theory stipulates that \(S(t):H \rightarrow H\) is a semigroup, where *H* is a Hilbert space. Following [14], we recall the necessary notions.

*absorbing set*if for any bounded \(B \subset H\) there is time \(t_{K,B}\ge 0\) such that

### **Theorem 7**

(see [14, Theorem 1]) Let us suppose that \(S(\cdot )\) has a compact attracting set *K*. Then there is a compact global attractor for \(S(\cdot )\) and \( \mathcal{A}= \omega (K)\).

Alternatively, we could establish first that \(S(t): (H^1_{per})^d \rightarrow (H^2_{per})^d\) is compact for \(t>0\). Then, we could draw the same conclusion slightly differently.

**Theorem** 7’ (see [4, Chapter 2], [17, Theorem 2.29]) *Let us suppose that* \(S(\cdot ): (H^1_{per})^d \rightarrow (H^2_{per})^d\) *is compact for* \(t>0\) *and there is an absorbing set in* \((H^1_{per})^d\). *Then there is a compact global attractor for* \(S(\cdot )\).

Our line of argument, however, is based on Theorem 7. This result will imply our Theorems 4 and 5 once we show its assumptions are fulfilled. For this purpose we need Proposition 3. We also have to show the existence of a compact attracting set *K*. This will be achieved in two steps. First, we will establish existence of an absorbing set in \(H^1\). Next, by application of a different method, the existence of an absorbing set in \(H^2\) and its compactness will be proved.

Note that \(\dot{H}^2_{per}\) is the correct choice of space for the slope systems (4) and (8), because we could not work with solution operators acting on lower order spaces due to the lack of uniqueness.

### 2.3 The Integral Representation of Solutions

The energy estimates become more tedious in two dimensions. Therefore, we choose a different approach to prove the higher oder absorption.

*h*in [10]. Here the exponential operator is defined by

## 3 The One-Dimensional Problem

In the following subsections we prove, by using Gronwall estimates, that there exists an absorbing ball in \(H^1\). Throughout the calculations, we denote by *C* a constant that may change from estimate to estimate, but does not depend on the initial condition. This quantity may rely on the domain length and the deposition related parameter, *L* and \(\delta \), respectively. Numbers whose actual value is needed for balances with other estimates are denoted by \(C_j\), where *j* is an integer index, and these numbers are fixed.

In the second part of this section we will show that the semigroup \(S(t): \dot{H}^2_{per} \rightarrow \dot{H}^2_{per}\) is indeed strongly continuous. This will be done by a series of a priori estimates of Galerkin approximations and passing to the limit.

### 3.1 Absorbing Ball in \(H^1\)

Consider the HCCH Eq. (4) with periodic boundary conditions on a domain \(\Omega = (0, L)\) and initial condition \(u(x, 0) = u_0(x)\). We extend the analysis from [11] by showing that the solutions are in fact absorbed into a ball whose radius does not depend on the initial value’s norm. To prove this result we will need to combine several estimates that we want to formulate as separate statements. Subsequently, we write the \(L^2\)-norm as \(\Vert \cdot \Vert = \Vert \cdot \Vert _{L^2(0,L)}\) and the \(L^2\) scalar product by \((\cdot , \cdot )\). Other norms are equipped with a corresponding subscript.

### **Lemma 8**

### *Proof*

*L*dependent constant, we get

The result shown above can be used for proving the existence of absorbing sets in \(\dot{H}^1\), therefore one needs to take care of the right hand side in (11). This is done in the following lemma.

### **Lemma 9**

### *Proof*

*u*as a test function in the same transformed Eq. (12),

*L*. \(\square \)

Now we are able to prove the existence of the first absorbing set.

### **Theorem 10**

(Absorbing balls in \(\dot{H}^1_{per}\)) The semigroup \(S(t):\dot{H}^2_{per} \rightarrow \dot{H}^2_{per}\), \(u_0 \mapsto S(t)u_0 = u(t) \) generated by Eq. (4) with periodic boundary conditions (i.e. the existence and uniqueness of weak solutions is guaranteed) has an \(H^1\) absorbing ball \( \mathcal{B}= \{u \in \dot{H}^1_{per} : \Vert u\Vert _{\dot{H}^1_{per}} \le \rho \}\), i.e. for a set \(B\subset \dot{H}^2_{per}\) bounded in the \(\dot{H}^1_{per}\) topology there is \(t_B\ge 0\) such that \(S(t)u_0 =u(t) \in \mathcal{B}\) for \(u_0\in B\) and \(t\ge t_B\).

### *Proof*

### *Remark*

We now know that \(\Vert u\Vert ^2 \le C, \Vert u_x\Vert ^2 \le C\) and \( \Vert u\Vert ^4_{L^4} \le C\), for a constant *C* independent of the initial condition that is undershot after a transient time. Since this case is one-dimensional this result leads to a uniform \(L^\infty \) bound on *u*. Furthermore, it was neither necessary to impose any restrictions to the deposition related parameter \(\delta \) nor to the domain length *L* to achieve the result.

By the same method we can establish the existence of an absorbing set in the \(H^2\) topology, but the argument is more involved. Possibly, we may show its compactness. However, this is of no use in the two dimensional case. This is why we will use a more general tool capable of handling both dimensional cases simultaneously. However, the starting point is the specific estimate like (18).

### 3.2 Strong Continuity of \(S(\cdot )\)

We need to show that in the one-dimensional case Eq. (4) generates a strongly continuous semigroup. Since the original argument in [11] is based on the Galerkin method applied to Eq. (4), we will use it here.

### **Proposition 11**

Let us suppose that \(u_0 \in \dot{H}^2_{per}\), then \(S(t)u_0 \equiv u(t)\) converges to \(u_0\) in the \(\dot{H}^2_{per}\) topology, as \(t\rightarrow 0^+\), where *S*(*t*) is the semigroup operator defined by (4).

### *Proof*

*p*is interpreted as 0.

### **Lemma 12**

*B*is a bounded subset of \(\dot{H}^1_{per}\). Then, weak solutions to Eq. (4) with \(u_0\in B\) for \(t\ge t_B\) fulfill

### *Proof*

*a*and

*b*implies

### **Lemma 13**

Let us suppose that *u* is a weak solution to (4) with initial condition \(u_0\) in \(\dot{H}^2_{per}\). Then, \(u\in L^2(0,T;\dot{H}^5 _{per})\).

### *Proof*

*t*we get,

*N*due to the existence result established for the HCCH equation, we can pass to the limit and conclude that indeed our claim holds. \(\square \)

## 4 The Slope System in the Two Dimensional Setting

For the purpose of analysis of the two-dimensional spatial domain, we rewrite Eq. (1) as a system of slope equations. The surface with height *h* over the reference plane depends on the domain size, it grows due to coarsening that leads to an increase of the average size of the evolving structures. The slopes have a more dissipative character as the anisotropy of the surface energy forces the slopes to stay at a certain level that is independent of the domain size.

*W*with respect to its arguments \(u_1\) and \(u_2\), we denote it here by \(D_u\),

### **Lemma 14**

### *Proof*

*u*as a test function in (30) and adding the components yields

*C*depends on the domain parameter

*L*as we applied Young’s inequality to \(u^2 \cdot 1\). Overall we have derived

## 5 The Global Attractor

### 5.1 Additional Regularity

*u*has zero mean. Namely, after setting

### **Lemma 15**

*y*. We used \(y = (t-s) (|\xi |^6-\lambda _0)\). Thus,

We may now establish new results based on (36).

### **Lemma 16**

### *Proof*

### 5.2 Compactness of Absorbing Balls

Using Lemma 15 we do not only show the existence of absorbing sets in \(H^2\) but also their compactness. Therefore we make the following key observation.

### **Proposition 17**

### *Proof*

\(\square \)

We may complete the *proofs of Theorems* 4 *and* 5 in one stroke. Proposition 17 yields compactness of an absorbing ball in \(H^2\) topology. On the other hand we have already established the strong continuity of the semigroup *S*(*t*). Thus, an application of Theorem 7 finishes the proof. \(\square \)

Now, we prove the final assertion. We transfer the above results to the problem expressed in terms of the shape *h* in (1).

### *Proof of Theorem 6*

## 6 Conclusions and Outlook

We have established the existence of global attractors in \(\dot{H}^2_{per}\) for the slope Eqs. (4) and (8). This enable us to show the existence of global attractors in \( H^3_{per}\) for (1) in the 1+1D and 1+2D settings. On the way, we showed that solutions to (4) and (8) enjoy further regularity. For the one-dimensional case we succeed in deriving proper uniform estimates by repeated application of Gronwall inequality. As we needed uniform constants for the estimates, the work may seem somewhat tedious at certain points, e.g. during the application of Gagliardo-Nirenberg’s inequality. Because of its repeated application this approach is not feasible the two-dimensional setting. Instead we reconsidered the constant variation formula from our previous work [10] to improve the regularity result. It turns out, once this approach is understood, the semigroup ansatz seems more elegant for this problem.

We are content with the results obtained for the presented equations. They coincide with the observations made with the help of a pseudospectral numerical method in the previous work [11], though we were not yet able to show or negate the existence of stationary or traveling wave solutions, which have been discussed in this publication. As we are not able to find a Lyapunov function, we were not in the position to use approaches based on the Łojasiewicz-Simon inequality (e.g. [13, 21]).

We do not know much about the \(\omega \)-limit set, but as Fig. 1 has already indicated, we expect to have time-periodic or stationary solutions for smaller values of \(\delta \) and a strange attractor for increased values of the deposition rate dependent parameter. Note that once the structures form, the solutions in this figure stay in an \(\dot{H}^2_{per}\) ball as predicted. The numerical simulations suggest that at least for small initial data the \(L^\infty \) norm of *u* stays roughly below 1, independently of the value for \(\delta \).

## Notes

### Acknowledgments

MK would like to acknowledge the financial support by the DFG Research Center Matheon. Furthermore MK thanks the University of Warsaw for the hospitality during two visits in 2012. The work of PR was supported in part by NCN through 2011/01/B/ST1/01197 Grant. The authors thank the referee for his/her extensive comments which helped to improve the text.

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