Well-posedness for a molecular beam epitaxy model

We study a general molecular beam epitaxy (MBE) equation modeling the epitaxial growth of thin films. We show that, in the deterministic case, the associated Cauchy problem admits a unique smooth solution for all time, given initial data in the space $X_0 = L^{2}(R^{d}) \cap \dot{W}^{1,4}(R^{d})$ with $d = 1, 2$. This improves a recent result by Ag\'elas, who established global existence in $H^{3}(R^{d})$. Moreover, we investigate the local existence and uniqueness of solutions in the space $X_0$ for the stochastic MBE equation, with an additive noise that is white in time and regular in the space variable.


Introduction
Molecular beam epitaxy (MBE) is a method of manufacturing very thin crystalline surfaces with well-defined orientations on substrates.As the name suggests, this is accomplished by sending very precisely controlled beams of molecules into a vacuum system containing the substrate.Originally discovered by Arthur and Lepore [9], MBE has become a leading method of manufacturing semiconductors used in various electronic components.
In an attempt to unify these models, Agélas [3] introduced the equation where for given parameters α > 0 and α j 0, j ∈ {1, ..., 4}.By making the change of variable t → t/α, we can from now on set α = 1.For this generalized model ( 1)-(2), Agélas proved global well-posedness in H s for s 3 under the assumption that α 1 > α 2 2 in the deterministic case η = 0.In the present work, we will extend Agélas' result, reducing the required regularity to obtain global existence.More specifically, it will be shown that the Cauchy problem for equation (1) in the deterministic case η = 0 is globally well-posed in the space L 2 ∩ Ẇ1,4 .Moreover, it is shown that the solution is smooth.These results will be stated more precisely in section 1.1, with the proofs contained in section 2.
An interesting fact about the literature on MBE is that the vast majority of results for any model of MBE are stated for the deterministic case η = 0, even though the model is inherently a stochastic one.Some notable exceptions to this exist, namely the works of Blömker et.al. [11,12,13].To add to this small collection, we also present a local well-posedness result for the stochastic Agélas model.This will be stated precisely in section 1.2, with the proof contained in section 3.
Finally, we conclude with some comments regarding the phenomenon of coarsening.Ideally, surfaces generated by MBE should be perfectly flat.However, the reality is that this is never the case.In fact, it has been observed experimentally that exotic structures can appear on these surfaces; see [2] for a scanning tunneling microscope image of these structures.Thus, a major goal of MBE modeling is to quantify the "roughness" of these surfaces.As is typical in engineering, deviations from the baseline are computed using the root-mean-square method.Since solutions represent deviations from the average height, it is standard practice to define the coarseness C(t) of a solution at time t to be its L 2 norm.In section 5, we make some comments on coarsening in these models, and point to some possible flaws in these models.

Remark 1. The local result can be shown to hold in any dimension, but for d 4, it is necessary to modify the initial data space to
The local well-posedness result in Theorem 1(i) is proved by applying a fixed point argument to the mild formulation of (3), in the space x , where we use the notation L q T Y := L q ([0, T ); Y) for a Banach space Y.Here S(t) = e −t∆ 2 is the linear propagator with Fourier symbol e −t|ξ| 4 , i.e.,

S(t)u
where is the kernel.As is well-known, it satisfies the time-decay estimate for all α ∈ N d , p 1 and some constant C = C(α, p) > 0. For the proof, see e.g., [10,Lemma 2.1].Similarly, one can show that for all s 0. By combining ( 6) and ( 7) with Young's inequality, we can obtain the following estimates for 1 r p ∞: and The proof of the local existence result relies on estimate (8).To obtain the smoothness result (4) in the spatial variable x, we use ( 8)-( 9) as well as the standard product estimate for the Sobolev spaces (see e.g., Corollary 3.16 of [14]), Smoothness in the time variable t then follows from the PDE (3) by a standard bootstrap argument.The global existence result in Theorem 1(ii) will follow from (4) and the global existence result in H 3 due to Agélas [3].
1.2.Stochastic MBE equation (η = 0).In addition to the deterministic case, we also consider the Cauchy problem for the stochastic MBE equation where η(x, t) is a real-valued Gaussian process with correlation function for some function c.The case c(x, y) = δ(x − y) corresponds to a space-time white noise.However, this is difficult to treat mathematically on unbounded domains, and therefore smoother correlation function will be considered here.
In order to give a mathematical definition of η and write (11) as an Itô integral, we introduce a probability space (Ω, F, P) endowed with a filtration {F t } t 0 , and a sequence {β k } k∈N of independent real valued Brownian motions on R + associated to the filtration {F t } t 0 .Given an orthonormal basis {e k } k∈N of L 2 (R d ) and a symmetric linear operator φ on L 2 (R d ), the process Note that φ = I corresponds to a space-time white noise with c(x, y) = δ(x − y).If φ is defined through a kernel L, which means that for any square integrable function f, In general, we assume that φ is a Hilbert-Schmidt operator from L 2 (R d ) to H s (R d ) for appropriate values of s, endowed with the norm Then the equivalent Itô equation to (11) becomes We shall construct a solution u satisfying the following mild formulation of (12): The last term on the right-hand side of ( 13) represents the effect of the stochastic forcing and is called the stochastic convolution, which we denote by Z, i.e., From the mild formulation ( 13)-( 14) we see that any solution u can be written as u = v + Z.We then study a fixed point problem for the residual term v = u − Z: Theorem 2. Let 1 d 3, u 0 ∈ X 0 and and φ ∈ HS(L 2 ; L 2 ).Then there exists a stopping time Remark 2. It is also possible to derive analogous a priori estimates for the general case α j 0 (j = 1, 2, 3, 4).However, we do not pursue this here.

Proof of
Given initial data with norm u 0 X 0 R, we show that Γ is a contraction in the ball for sufficiently small T = T (R) > 0. For this, it suffices to prove for u, v ∈ B R,T .Therefore, the map Γ has a unique fixed point u ∈ R,T solving the integral equation (5).Continuous dependence on the initial data can be shown in a similar way.
As no additional difficulty arises when estimating the lower order terms, for the sake of clarity we set α 3 = 0 and α 4 = 0.Moreover, as the value of the parameters α 1 and α 2 are not of significant importance, we set Then by (8), Similarly, for α ∈ N d , such that |α| = 1, we have From the above estimates, we conclude that In a similar way, we can derive the difference estimate which holds when It follows that Γ u is a contraction on X T , from which existence and uniqueness of solutions follows.

Regularity of the local solution.
Our next goal is to prove the regularity result in part (i) of Theorem 1.As a first step, we prove the following lemma: Lemma 2. Let u ∈ X T be a solution of (5) associated with the initial condition Proof.For the sake of clarity, like in the proof of Theorem 1, we set α 1 = α 2 = 1 and α 3 = α 4 = 0. Using the semi-group property of the linear propagator S, we have that for any 0 t < t < T , The proof consists of four steps.
Step 1. Taking first t = 0, applying the operator |D| α to this equation ( 19) for α ∈ R, then taking the L 2 norm and using Hausdorff's inequality, we get Using the decay estimates (9) of S, we get So that, if α < 2, we get So for now, we have u ∈ L ∞ (0,T ] H 2− (R d ), since u ∈ X T (by Theorem 1).
Step 2. This step is required only in the case d = 2.In the case d = 1, one can go directly to the Step 3 of the proof.Let us apply the operator |D| α+ 1 2 , with α < 2, to (19).
Using again the decay estimates (9), we have ds.Now, in order to estimate the last term of the right hand side, we use (10), with s = 1 2 , s 1 = s 2 = 7 8 .Then s s j and s 1 + s 2 − 1 = 3 4 > s.We get where in the last line we choose α sufficiently large so that 2 > α > 1 Step 3. Let s 0 > d 2 and apply |D| α+s 0 to (19).We then have where 0 < t t < T .Then, using the decay estimates (9), we get where we used the algebra property of H s 0 , and where we take s 0 sufficiently small so that s 0 + 1 = d 2 + 1 + ε < 2 when d = 1, and Step 4. Let n d, let us suppose that u ∈ L ∞ (0,T ] H 2+ n 2 (R d ), which holds when From (19), we can apply the operator |D| γ 1 +γ 2 where γ 1 = 3 2 and γ 2 = 2 + n 2 − 1 (so that γ 1 < 2 and γ 2 + 1 = 2 + n 2 ) to obtain Then, using the decay estimates (9), we get where we used the algebra property of H γ 2 .This completes the proof.
Remark 3. The proof of Lemma 2 can be adapted to the case d = 3.
Step 1 would be the same.
Now, let us use Lemma (10) for the terms .

Proof of Theorem 2
With the deterministic case complete, we now turn to the stochastic case.Before we begin, let us fix the notation.For an F-measurable function X : Ω → R and g : R → R, the expectation E[g(X)] is defined as ω -space is then defined via the norm Note that we have the embedding L q ω ⊂ L p ω whenever q > p, i.e., Lemma 3. Suppose that T > 0 and φ ∈ HS(L 2 ; H s ) for some s ∈ R. Then (ii) Z ∈ L q T Ẇ1,r (R d ) for q 1 and r 2, almost surely for ω ∈ Ω.Moreover, for any p 1, there exists C = C q,p,r,T > 0 such that Proof.Part (i) is contained in Theorem 6.10 of [1].So we only prove (ii).From ( 14), we see that ∇Z(x, t) is a linear combination of independent Wiener integrals.It is a mean zero Gaussian random process with variance where we used the Itô isometry to obtain the last equality.Now, recall that for mean zero Gaussian random X variable with variance σ, we have So for a positive integer p, we have by the embedding L 2p ω ⊂ L p ω , ( 21) and (20), Assume p max(q, r).By ( 22) and the decay estimate ( 9) 3.1.Local existence for (13).For simplicity, we take α 1 = α 3 = 1, α 2 = α 4 = 0.However, the argument works for any α j 0 (j = 1, 2, 3, 4).So N(u) becomes To prove local existence of solution (15) we apply a fixed point argument in , where For any v ∈ B R,T , we define the map Γ = Γ u 0 ,Z , by where First note that by Lemma 3, almost surely for ω ∈ Ω.In particular, there exists for A = A(q, T ) > 0 such that A almost surely for ω ∈ Ω.Thus, local existence of a unique solution in B R,T follows if we prove for v, w ∈ B R,T and sufficiently small T = T (R, A).
Proof of (23).Fix ρ > 8. Following the same argument as in the deterministic case, we obtain In the last inequality, we used Hölder inequality in time to obtain the last term.
Similarly, we may also obtain From the above estimates, we conclude that for some θ > 0, . This proves (23).

Proof of (24). Begin by writing
Using this identity, we see that x .
A similar computation yields . This completes the proof of (24).

Proof of Lemma 1
Suppose that v is a smooth solution of (15) on [0, T ].Then v satisfies where (by assumption α Then by (25) and Plancherel, By Young's inequality, the integrad in the first integral on the right is bounded by Therefore, the first integral on the right is bounded by Similarly, the second integral is bounded by where This proves Lemma 1.

On Coarsening in MBE Models
Continuing the discussion of coarseness in the introduction, let us state the formal definition of the coarsness of a solution.Definition 1.Let u(x, t) be a solution to any partial differential equation modeling MBE.We define the coarseness C(t) of u at time t by One of the most well-known asymptotic estimates for the growth rate of the coarseness is due to Rost and Krug, who showed that solutions to their model satisfy the estimate in the deterministic case.Interestingly, this estimate was obtained using only basic calculus and some mild assumptions on solutions based on physical principles.With this fact in mind, we observe that Theorem 3.1 of [3] implies that the solutions obtained in Theorem 1 must obey the a priori estimate where K is a non-negative constant which depends on the coefficients α 1 , . . ., α 4 .On the surface, this would appear to be a much weaker result.However, it should be noted that in a typical situation encountered in manufacturing, we start with a flat surface, so that u(0) = 0.This suggests that the solutions constructed in Theorem 1 should exhibit no coarsening.Since the model of Rost and Krug is a special case of equation ( 1), then the result should hold for that model, as well.Moreover, the uniqueness of solutions in the space X T implies that any solution corresponding to zero initial data which exhibits coarsening cannot be in X T .The only way that this can happen is if either u(t 0 ) L 2 x = +∞ or ∇u(t 0 ) L 4 x = +∞ for some t 0 > 0. The former case is particularly troubling, as such solutions would either have to go to infinity in the continuous case, or be discontinuous.Since both cases are physically unrealistic, this suggests that deterministic models should not lead to solutions which exhibit coarsening, which goes against the conventional wisdom and experimental results in the study of MBE.
To reconcile this discrepancy, we remark that in physically realistic situations, signals are often noisy.Thus, an accurate model of epitaxial dynamics should include noise.Based on the results of Theorem 2 and Lemma 1 (in particular, equation ( 26)), global solutions perturbed by a noise should satisfy the coarsening estimate C 2 (t) C 2 (0)e t + C T (e t − 1).For initially flat surfaces, this reduces to C 2 (t) C T (e t − 1).
As is well known, e t − 1 = O(t) as t → 0. We may thus conclude that C(t) Ct for short times, so that we recover Rost and Krug's result.This suggests that any model of MBE must contain a noise term to exhibit coarsening.Subsequently, it also suggests that by improving manufacturing methods, it may be possible to reduce the coarsening of surfaces.